Physics II exam 2

*How ignition system of cars work!*

12 V (battery) → 30 000 V to ignite gas-air mixture → spark cars only work with DC -some chargers (like a computer charger) will change from AC -> DC

*Same power (conservation of energy) but different*

current (I) *Ip/Is = Ns/Np* -changing magnitude of current -need same power on both sides (P = VI)

*Cos angle decreases:*

flux decreases → induced current

*if there are 3 times more loops/turns around, that means*

voltage is 3 times more

Example 1.) Part of a single rectangular loop of wire is situated inside a region of uniform magnetic field of 0.55 T. The total resistance of the loop is 0.23 ohms. Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 3.4 m/s.

B = 0.55 T R = 0.23 ohms V = 3.4 m/s l = 0.35 m F = ? F = (B^2l^2v)/R F = (0.55)^2 (0.35)^2(3.4) / 0.23 *F = 0.55 N*

*ΦB =*

B A cosθ angle between B and normal to the surface -Maximum when B || to normal to A zero when B ⊥ to normal to A

Switch closed:

current

Switch open:

current slowly decreases

*If voltage stepped up:*

current stepped down

*If voltage stepped down:*

current stepped up

*What about 45º angle?*

(cos45)^2 = 1/2

*Focal Point (Convex Mirrors)*

-In convex mirror rays diverge -Rays diverge as coming from single point -Ray projections cross each other at focal point (behind mirror) -aka diverging mirror because reflected rays diverge from the center

Uc =

1/2 1/C Q^2

*totally plane polarized light:*

100% intensity at some angles and 0% at others

*Visible spectra: λ =*

400 to 700 nm (1 nm = 10^-9 m) very small

*small period (T) =*

large frequency = smaller λ

*Emitted or reflected light from each point of object*

reaches our eyes

*Law of reflection:*

the angle of reflection equals the angle of incidence (both in respect to normal to the surface)

*Frequency:*

# of oscillations per second (1/s = Hz)

Power delivered by external force to move rod

(B^2l^2v^2)/R

*Energy in E&M Waves: Electric field energy density =*

(U / volume) uE = 1/2 ε0 E^2 -small u = energy density

*di = fdo/(do-f)* *only when do < f*

(for concave mirror) always for convex mirror

*Geometrically optics*

(no waves for now) -ray model works to explain shadows

low frequency limit circuit

(side 5) inductor's current relaxed like it's not even there

*Mirror Equation (Concave Mirror)*

(slide 2 pictures, class 23) -Ray diagrams aren't easy to draw nor precise -Use geometry (similar triangles) to obtain formulas -θI = θR -triangles have the same angle

Example: light dimmers

(turn nob → move rod in coil → increase inductance → increase reactive inductance → decrease current → dim light) no power lost

use vectors

* Vmax = √(Vmax of C)^2 + (Vmax of R)^2

*Faraday's Law: ε =*

*-N (ΔΦB/ΔT)* N is the number of loops (where induction happens) ΔΦB/ΔT is rate at which flux changes *minus sign*: emf opposes change in flux (to come ...)

*if ho/-hi = do/di AND ho/-hi = do-f/f* and hi <0 then

*1/do + 1/di = 1/f* -h is height -do is distance from the object to the mirror -di is distance from the image to the mirror

*since uE = uB*

*E = cB*

Force applied on surface: F =

*F = Δp/Δt = I A / c* *Fav = Iav A / c*

*When angle between polarizer transmission axis and wave polarization plane θ is not 90º (not perpendicular to the first sheet)*

*I = I0 cos^2 θ* since I = cu = c ε0 E^2 -use this if the 2 sheets are at an angle with each other

Intensity related to power =

*I = P/A*

I =

*I = uc*

average intensity related to power =

*Iav = Pav/A*

Iav =

*Iav = uav c*

*From Faraday's law:*

*L = N (ΔΦB/ΔI)*

Average power consumed by AC RC circuit

*Pav = Irms^2 R* same current in the loop: Irms = Vrms/Z -> Pav = Irms (Vrms/Z) x R using cosφ = R/Z

*Voltage oscillates between - Vmax and Vmax (peak voltage)*

*V = Vmax sin ω t* where the angular frequency is *ω = 2πf*

Use vectors

*Vmax = Imax Z* *Z = √R^2 + (XL−XC)^2* where impedance Z is "resistance" of circuit (units Ω - Ohms) -ohm's law always for AC (V = IR) -Z can be R, XL, or Xc -impedance = harder to establish current -reactance = resistance for resistor and capacitor

for impedance, Vmax =

*Vmax = Imax Z* where the impedance is *Z =√R^2 + Xc^2* (units Ω - Ohms) *cosφ = R/Z*

*Voltage across inductor*

*Vmax,L = Imax XL*

*Inductive reactance (unit Ω - Ohm)*

*XL = ωL*

XL =

*XL = ωL*

Xc =

*Xc = 1/(ωC)*

Circuit Summary

*Z = √R^2 + (ωL - 1/ωC)^2* *ω = 2pif* *Pav = Irms Vrms cosφ*

*c =*

*c = 3x10^8 m/s* Electric permittivity ε, magnetic permeability μ → c depends on medium (slowed down)

f ' ' =

*f ' ' = f ' (1 ± u/c)* *f ' ' = f (1 ± u/c)^2*

*Convex mirror: For small mirror compared to radius of curvature and small angles, f =*

*f = -R/2*

*Convex mirror: For small mirror compared to radius of curvature and small angles, f =*

*f = R/2* R is the radius (from center to mirror)

*Same problem solving: m =*

*m = hi/ho = -di/do*

*Index of Refraction (larger than 1)*

*n = c/v* -Speed of light depends on medium -Upper limit c = 3 x 108 m/s in vacuum -In water v = 3/4 c -n depends on wavelength -n = speed of light in vacuum/vol

*Snell's Law*

*n1 sinθ1 = n2 sinθ2* where θ1 is incident angle and θ2 is refraction angle Angles measured with respect to normal to the surface

momentum (p) =

*p = U/c* big U = energy

Radiation Pressure: Pressure on surface

*pressure = F/A* *pressure = I/c* *pressure av = Iav/c* have a force = can define a pressure

Phase angle between total voltage and current

*tanφ = (XL-Xc) / R* *cosφ = R/Z* -triangle in notes: φ is the angle, R is adjacent, XL-Xc is opposite, and Z is hypotenuse

*electromagnetic field that has both electric field and magnetic field: Total energy density*

*u = 1/2 ε0 E^2 + 1/2 1/μ0 B^2*

*Average energy density for E and B*

*uav = 1/2 ε0 Erms^2 + 1/2 1/μ0 Brms^2* *uav = ε0 Erms^2 = 1/μ0 Brms^2* *average of energy density NOT zero*

Baseball scouts often use a radar gun to measure the speed of a pitch. One particular model of radar gun emits a microwave signal at a frequency of 10.525 GHz. What will be the increase in frequency if these waves are reflected from a 60.0 mi/h fastball headed straight toward the gun? (Note: 1 mi/h = 0.447 m/s )

*Δf = 1880 Hz*

Amount of momentum absorbed by surface (Δp) =

*Δp = ΔU/c = I A Δt/c* *Δpav = Iav A Δt/c*

Frequency Limits

- Extremely high frequency: extremely high XL (open circuit), extremely low XC (ideal wire) - Extremely low frequency: extremely low XL (ideal wire), extremely high XC (open circuit) like battery !

converging lens ray diagram steps

-1st parallel ray refracts through focus -2nd ray passes through focus and refracts parallel -3rd passes through center of the lens *-Image forms where rays cross* *-Real image in this case (light passes through image)* *-Image would be virtual for object between F and lens* -refraction happens on both walls -real image even though it is one the other side (real means light goes to the actual image)

*Lenses*

-Circular lenses with 2 sides: plane, concave, convex combination -Used for glasses, cameras, binocular, telescopes, microscopes, ... -Made of glass or clear plastic (higher n) -converge or diverge (double convex) -light slows down = bends

Maxwell Equations facts

-Define electromagnetism theory -As fundamental as Newton's laws -Relativistic (c comes naturally as a constant) -Math involves calculus -relativistic because speed of light is a constant

*video (slide 5, class 20): Tour of the Electromagnetic Spectrum (NASA)*

-EM waves travel through space at the constant speed of light -wavelength measured in m or nm -waves per second = Hz = frequency -long waves = low frequency -adding energy = increases frequency = shorter wave -gamma rays = shortest with highest frequency -some wavelengths are reflected and some are absorbed in visible spectrum -wavelength reflected in a range of nm = see that color -different waves across EM spectrum allow us to view non-visible waves -radio waves = longest = least energy -hotter temperature = bluer end of visible light -all electromagnetic radiation is light -

Loudspeakers

-Electric energy → sound energy -Current alternates at audio frequency f -Force due to magnetic field on current -Coils of wire attached to speaker cone -Cone moves back and forth at same f -Compressions/rarefactions of air form sound waves

*Electromagnetic Waves*

-Electromagnetic waves predicted by Maxwell (1831 - 1879) -Developed unified "symmetric" theory of electricity and magnetism (and light) -Before, light understood as wave that propagated in "ether" (wrong!) -Heinrich Hertz produced and measured the 1st electro- magnetic wave (not visible) in 1887 -normal waves need a medium to propagate (light is a wave -made up wave for light = ether (not seen) -light doesn't need a medium because the speed of light is experimentally constant: independent of the movement of the source or detector or the direction in which it travels. Light contrasts with sound, which travels through the air (or some other material medium) -light waves move with the speed of light

*Focal Point (Concave Mirrors)*

-For object infinitely far away, light rays are parallel -In concave mirror they converge -For sharp images rays have to converge to single point -Works for small mirrors with large radius of curvature (not too curved)

Resonance

-Forcing a system to oscillate at a frequency near its natural frequency -Large displacement -imposing a frequency = amplitude grows

*Plane Mirror/Flat Mirror*

-Forms virtual image: seems to be coming from behind the mirror (cannot be projected in screens) -Image upright with same height as object -Image with right / left reversed -Image seems to be at same distance from mirror as object -Rays leave object form top to bottom -Many rays leave object: some reflected rays reach eye -Your brain thinks rays traveled in straight line -Rays seem to be coming from behind the mirror (image respective points) -top and bottom lines = each the top and bottom of eyes -virtual = where we project the image -nothing behind the mirror, but our vision perceives it that way -mirror = reversed image

Measuring "c"

-Galileo (1564 - 1642) tried to measure the speed of light (flash of lamp between hilltops) and found it too high to be perceived -Ole Roemer (1644-1710) tried to measure the speed of light (dif- ference in the period of Jupiter's moon when Earth moving away from Jupiter) and found it to be ~ 2 x 108 m/s -The speed of light was finally measured precisely by Michelson around 1900

Image from ray tracing with concave mirror

-Image extremity I' lies where lines meet -Rays from other points of object form I -I is a real image for this particular case (light passes through image) -I would be virtual for object between F and mirror -concave mirror = can have upside-down image

Image from ray tracing with convex mirror

-Image top lies where extended lines of reflection meet -Rays from other points of the object form I -I is a virtual image (no light through image) -focus behind the mirror -image is where the extensions meet (behind the mirror)

LC Circuits

-Inductor and capacitor (no generator) -Begins with charged capacitor -Current flows through inductor as capacitor discharges -Current keeps flowing after capacitor discharges (induced current resists change) -Capacitor charges the other way ... Oscillations ! Continue forever without dissipation -ignoring resistance because it's so small -HAVE to begin with AC with a charged capacitor = let it uncharge = current through inductor = magnetic field = current in other way decreasing overall current = charge capacitor in the other way (oscillations one way then the other way)

Doppler Effect for Light

-No medium for propagation (space) -Speed of light wave crests/troughs with respect to observer is constant! But frequency changes -Independent of who moves -light ONLY changes in frequency, not speed

*Converging Lenses*

-Non parallel surfaces of object deflect light -Rays bend towards normal / away from normal (Snell's law) -Object thicker at center than at edges converges light -light bends towards the normal going in and away from the normal coming out -lights in from far away will converge to the focal point

*Refraction: What if the light changes medium (different n)?*

-Part reflected / part goes through -Rays (not perpendicular ones) change direction -light travels through air = slows down = bends (angles with respect to the normal)

*Dispersion*

-Prisms separate white light into all colors -Refraction bends different wavelengths (frequencies) at different angles -Shorter wavelengths (violet) suffer more bending -Different materials have different dependence of refraction on wavelength -Examples: drops of water, diamonds (lots of total internal reflexion due to high n) -Rainbow: red on top - rays reach your eye first while drops fall -bottom = refraction exchange -double rainbow = second one will be upside down (opposite color pattern)

*Speed of Light "c"*

-Speed of a car on the highway: 30 m/s -Speed of sound in air: about 300 m/s -Speed of light: 300 000 000 m/s -Sound waves oscillate particles of air, water, ... -Electromagnetic waves don't oscillate medium, only fields and can propagate in vacuum -From Maxwell's equations (in vacuum) -light travels faster than sound -"c" is light itself

Optical Fiber

-Total internal reflection -Transparent plastic or glass with high n surrounded by low n coating -Light propagates in zigzag path -Flexible: curved path -Endoscopes, bronchoscopes, cardioscopes -Carry lots of information fast without lost / interference -Tens of thousands of phone conversations in 1 cable

Liquid Crystal Displays

-Used in calculators, digital watches, cell phones, computer screens, tv screens, ... -Pixels (picture elements) are tiny rectangles that can be light or dark -Color pixels have three cells (red, green, blue) -Each pixel is a glass sandwich with liquid crystal (gooey substance with long molecules that like to stay parallel to each other) -different intensities to make different colors -each pixel = sandwich of molecules

3 different voltages across R, C, L

-V across R in phase with current -V across L 90o ahead current -V across C 90o behind of current -current is the same everywhere (only changed with producing/eliminating charge)

impedance is the "resistance" of an AC RC circuit

-current is the same everywhere in the circuit with one loop -have value R and Xc, but if R is greater than Xc then the voltage is almost in phase, but if Xc is greater than R then voltage is almost perfectly out of phase (usually find something in between)

speed of light measured by

-directed light at a face of fast rotating eight sided mirror -reflected light traveled to distant mirror and back -rotation speed of mirror → c

Radar used to calculate path of objects

-e&m pulses emitted in all directions with frequency f -wave reflects and comes back with different frequency f '' -speed of approaching storm, plane, ... calculated -wave emitted reflects off the moving object = comes back to you -ex: radar send a wave out and the object detected sends a wave back with the same frequency (video on slide 11) -ex: radar gun detects changes in frequency

Application: plane mirrors

-everyday use -ex: corner reflectors on moon: reflect light back at same angle even if incident angle is not 90o ...

*when light moves it can..*

-keep moving if nothing happens to it -can be reflected -can go through objects

Example 2.) Consider a spiral galaxy that is moving directly away from Earth with a speed V = 5 ×10^5 m/s at its center. The galaxy is also rotating about its center, so that points in its spiral arms are moving with a speed v = 6.3 ×10^5 m/s relative to the center. If light with a fre- quency f = 8.08 ×10^14 Hz is emitted in both arms of the galaxyf a.) what frequency is detected by astronomers observ- ing the arm that is moving toward the Earth? b.) and in the one moving away from Earth?

-rotates = top part is coming toward you and the bottom is moving away -v > V, so top speed: v-V bottom speed: v+V f ' = f (1 ± u/c) u = v ± V a.) f ' = (8.08 ×10^14) (1 + (6.3 ×10^5 - 5 ×10^5 / 3x10^8) *f ' = 8.084x10^14 Hz* blue-shifted b.) f ' = (8.08 ×10^14) (1 - (6.3 ×10^5 + 5 ×10^5 / 3x10^8) *f ' = 8.050x10^14 Hz* red-shifted -red-shifted means receding from us = decrease in frequency = moving away (blue-shifted is the opposite)

A 60-W light bulb radiates electromagnetic waves uniformly in all directions. At a distance of 1.0 m from the bulb, the light intensity is I0, the average energy density of the waves is u0, and the rms electric and magnetic field values are E0 and B0, respectively. 1.) At 2.0 m from the bulb, what is the light intensity? 2.) At 2.0 m from the bulb, what is the average energy density of the waves? 3.) At 2.0 m from the bulb, what is the rms magnetic field value?

1.) (I0/2) (cos45)^2 = *1/4 I0* 2.) 1/4 u0 3.) u = 1/2 1/μ0 B^2 -> *1/2 B0*

An object O is placed at the location shown in front of a convex spherical mirror. Use ray tracing to determine the location and size of the image in the mirror. As you work, keep in mind the following properties of principal rays:

1.) A ray parallel to the axis, after reflection, passes through the focal point F of a concave mirror or appears to come from the (virtual) focal point of a convex mirror. 2.) A ray through (or proceeding toward) the focal point F is reflected parallel to the axis. 3.) A ray along the radius through or away from the center of curvature C intersects the surface normally and is reflected back along its original path. 4.) A ray to the vertex V is reflected, forming equal angles with the optic axis.

A concave lens refracts parallel rays in such a way that they are bent away from the axis of the lens. For this reason, a concave lens is referred to as a diverging lens 1.) Consider the following diagrams, where F represents the focal point of a concave lens. In these diagrams, the image formed by the lens is obtained using the ray tracing technique. Which diagrams are accurate? 2.) If the focal length of the concave lens is -7.50 cm , at what distance do from the lens should an object be placed so that its image is formed 3.70 cm from the lens? 3.) What is the magnification m produced by the concave lens described in Part B? 4.) Where should the object be moved to have a larger magnification?

1.) Figure 1 and 3 (A concave lens always forms an image that is on the same side of the lens as the object) 2.) 1/-7.5 = 1/do + 1/-3.7 -> *do = 7.3 cm* (di negative because image on same side of lens as object) 3.) m = -di/do m = -(-3.7)/(7.3) -> *m = 0.507* 4.) the objects should be moved closer to the lens

Ray Tracing (Convex Mirrors)

1.) Ray goes from O' horizon- tally and is reflected along a line that extends back through focus 2.) Ray goes from O' and reflects horizontally (extending through focus) (ray goes from the object has to extend through the focus but reflects horizontally) 3.) Ray goes from O' perpendicular to mirror (extending through C) and reflects on itself

In this problem we consider the resonance curve for a circuit with electrical components L, R, and C and resonant frequency ω0. 1.) For given values of R and C, if you double the value of L, how does the new resonance curve differ from the original one? 2.) For given values of R and C, if you double the value of L, how does the new rms current at resonance Irms differ from its original value? Assume that the voltage amplitude of the ac source is the same.

1.) The peak height won't change, and the peak frequency will be 1/2√ times as great. In an R-L-C circuit, the resonance peak depends only on R, while the resonant frequency is determined by both L and C. In particular, for a given value of C, the resonance frequency is inversely proportional to the square root of L. Similarly, for a given value of L, the resonance frequency is inversely proportional to the square root of C. 2.) Irms is unchanged In a series R-L-C circuit, for a given voltage, the rms current is always inversely proportional to the circuit impedance. Since at resonance the impedance depends only on R, the rms current in the circuit remains constant when either L or C is changed. Note that this is true only at resonance.

*A section of a sphere has a radius of curvature of 0.66 m.* *1.) If this section is painted with a reflective coating on both sides, what is the focal length of the convex side?* *2.) If this section is painted with a reflective coating on both sides, what is the focal length of the concave side?*

1.) f = -R/2 f = -(0.66)/2 f = -0.33 m 2.) f = R/2 f = (0.66)/2 f = 0.33 m

You take a picture of a rainbow with an infrared camera, and your friend takes a picture at the same time with visible light. 1.) Is the height of the rainbow in the infrared picture greater than, less than, or the same as the height of the rainbow in the visible-light picture? 2.) Choose the best explanation

1.) greater 2.) The height will be greater because the top of a rainbow is red, and so infrared light would be even higher.

A lens produces a real image of a real object 1.) Is the image inverted or upright? 2.) Is the lens diverging or converging? 3.) Is the image enlarged or reduced in size? 4.) If two convex lenses identical in size and shape are manufactured from glass with two different indices of refraction, would the focal length of the lens with the greater index of refraction (lens 1) be larger or smaller than that of the other lens (lens 2)? 5.) If lens 1 from Part 4 were placed in exactly the same location as lens 2, would the image produced by lens 1 be larger or smaller than the image produced by lens 2?

1.) inverted 2.) converging 3.) cannot be determined 4.) smaller 5.) smaller same as convex mirror -object beyond F -inverted -reduced or enlarged -real

A humorous scene in Akira Kurosawa's classic film The Seven Samurai shows the young samurai Kikuchiyo wading into a small stream and plucking a fish from it for his dinner. 1.) As Kikuchiyo looks through the water to the fish, does he see it in the general vicinity of point 1 or point 2 in the figure? 2.) If the fish looks up at Kikuchiyo, does it see Kikuchiyo's head in the general vicinity of point 3 or point 4?

1.) point 1 2.) point 4

Electromagnetic wave 1 has a maximum electric field of 60 V/m , and electromagnetic wave 2 has a maximum magnetic field of 1.4 μT 1.) Which wave has the greater intensity? 2.) Calculate the average intensity of the first wave. 3.) Calculate the average intensity of the second wave.

1.) second wave 60 vs (1.4x10^-6)(2.998x10^8) 2.) IE = 60^2/2μ0c -> *IE = 4.8 W/m^2* 3.) IB = (419.72)^2 / 2μ0c -> *IB = 230 W/m^2*

An electric charge on the x axis oscillates sinusoidally about the origin. A distant observer is located at a point on the +z axis. 1.) In what direction will the electric field oscillate at the observer's location? 2.) In what direction will the magnetic field oscillate at the observer's location? 3.) In what direction will the electromagnetic wave propagate at the observer's location?

1.) x 2.) y 3.) + z

Ultraviolet light is typically divided into three categories. UV-A, with wavelengths between 400 nm and 320 nm, has been linked with malignant melanomas. UV-B radiation, which is the primary cause of sunburn and other skin cancers, has wavelengths between 320 nm and 280 nm. Finally, the region known as UV-C extends to wavelengths of 100 nm. 1.) Find the range of frequencies for UV-B radiation. 2.) In which of these three categories does radiation with a frequency of 7.9 × 10^14 Hz belong?

1.) λf = c 2.8x10^-7 (f) = 3x10^8 -> f = 1.07x10^15 3.2x10^-7 (f) = 3x10^8 -> f = 9.38x10^14 2.) λf = c λ (7.9 × 10^14 ) = 3x10^8 λ = 380 nm *UV-A*

UL =

1/2 L I^2

KE =

1/2 M v^2

*Pav =*

1/2 Vmax Imax = Vrms Irms 1/2 I^2 R = I^2 R 1/2 (Vmax^2/R) = (Vrms^2/R)

PE =

1/2 k X^2

k→

1/C X→Q M→L v→I

*f =*

1/T *f = v/λ = c/λ* Inverse relation with frequency

A convex lens is held over a piece of paper outdoors on a sunny day. When the paper is held 30 cm below the lens, the sunlight is focused on the paper and the paper ignites. What is the focal length of the lens?

1/do = 0 (distance of the object is essentially infinite) 1/f = 1/di + 0 1/f = 1/30 *f = 30*

*Xc =*

1/ωC equivalent to Ohm's law

A laser beam is reflected by a plane mirror. It is observed that the angle between the incident and reflected beams is 26º If the mirror is now rotated so that the angle of incidence increases by 5.0º, what is the new angle between the incident and reflected beams?

26/2 = 13 13 + 5 = 18 18 x 2 = 36º

*Visible spectra: f =*

4.3x10^14 to 7.5x10^14 Hz c is very large

*In US f =*

60 Hz (60 oscillations per second)

A person riding in a boat observes that the sunlight reflected by the water is polarized parallel to the surface of the water. The person is wearing polarized sunglasses with the polarization axis vertical. If the wearer leans at an angle of 27.5º to the vertical, what fraction of the reflected light intensity will pass through the sunglasses?

90-27.5 = 62.5 I/I0 = (cos62.5)^2 *I/I0 = 0.213*

A magnetic field is oriented at an angle of 58º to the normal of a rectangular area 6.5 cm by 7.8 cm If the magnetic flux through this surface has a magnitude of 5.0×10^−5 T•m^2 , what is the strength of the magnetic field?

A = 0.00507 m^2 θ = 58º ΦB = 5.0×10^−5 T•m^2 ΦB = B A cosθ 5.0×10^−5 = B (0.00507) cos58 B = 19 mT

*Besides changing B, we can change*

A or relative θ Same when increasing area or cos angle (See conceptual checkpoint 23 -1 (textbook)) -ΦB changes with induced current -change in area or angle = ΦB changes = induces current

Average Power

AC circuit also uses power Power transformed in resistor (always positive)

Production of Electromagnetic (E&M) Waves

AC generator connected with antenna (long wire) -light = EM waves -generator was attached to a closed loop before, but now there is no loop in this case = current will move down and positive charges are moved to one side -> generator inverts current -> positive charges are moved the other way -speed of light is not infinite -current is a wave -electric field propagates out

*Upright image:*

ALWAYS negative distance (same side for lens)

*Inverted image:*

ALWAYS positive distance (other side for lens)

Chapter 24:

Alternating Voltage/Current and Capacitors in AC Circuits

Intensity (I)

Amount of energy passing through area A per unit of time

Example 2.) The rms voltage across a 0.018 μF capacitor is 2 V at a frequency of 60 Hz. a) What is the rms current through the capacitor? b) What is the maximum current through the capacitor?

C = 0.018 x 10^-6 F Vrms = 2V f = 60 Hz ω = 2πf a.) Irms = ω C Vrms Irms = 2π(60)(0.018 x 10^-6)(2) *Irms = 13.57 x 10^6 A = 13.57 μA* b.) Imax = √2 frms Imax = √2 (13.57) *Imax = 19.19 μA*

E and B in phase

Can propagate in vacuum (no medium necessary) as E increases, B decreases can't hear an explosion in space

*Induction Problem Solving*

Change in magnetic flux close to closed conducting loop: i. Determine if flux decreases/increases ii.Determine magnetic field of induced current (oppose flux change) iii. Use magnetic right-hand rule to find inducted current

*Power P = V I*

Changes sign *Average power is zero* because capacitor not drawing energy (ideal)

class 17

Chapter 24: RC Circuits and Inductors in AC Circuits VVVVVV

Total voltage?

Complicated: voltages out of phase ! (maximum / minimums not synchronized) -cannot add R and Xc because the voltages are out of phase

*Capacitors in AC Circuits*

Current in circuit with AC generator and capacitor (no resistor): not trivial -stronger generator = stronger current -charge capacitor = no more current through -high frequency = doesn't really charge capacitor = current never goes to zero

*wavelength*

Distance between 2 consecutive wave crests / troughs

Sinusoidal voltage:

E and B sinusoidal they are in the shape of waves basically

Current up:

E down, B in (on right side) -E moves with flow of charge (+ to -) and B is perpendicular to E

Detection of E&M Waves

E incoming wave → force on charges → AC current -antenna produces and detects waves = incoming wave (B present, corresponding) -when it hits the antenna connected to the circuit = charges were moving with field then changed direction

Current changes direction:

E, B changes direction

Current reduced:

E, B reduced

chapter 25

Electromagnetic Spectrum VVVVVV

chapter 25

Energy and Momentum in Electromagnetic Waves and Polarization

*Root mean square of E and B*

Erms = √(E^2)av = Emax/√2 Brms = √(B^2)av = Bmax/√2

External force to move rod with speed v

F = (B^2l^2v)/R

*Do magnetic fields generate currents?*

Faraday experiment -inducing current by creating a magnetic field inside coil (weak to strong field)

*Symbol AC generator*

Generates AC current Generates AC voltage

*When the other component is eliminated:*

I = 0

Example 1.) Suppose a current is given by the equation I = 1.8 sin (210t) where I is in Amperes and t in seconds. a) What is the frequency f? b) If this is the current through a 42 Ω resistor, write the equation that describes the voltage as a function of time.

I = 1.8sin210 Imax = 1.8 A ω = 210 a.) ω = 2πf 210 = 2π f *f = 33.42 Hz* b.) V = Vmax sin ωt / RI = RImax sin ωt V = (42)(1.8) sin210 t *V = 75.6 sin210t V*

*When one component is eliminated:*

I = 1/2 I0

Vertically polarized light with an intensity of 0.40 W/m^2 passes through a polarizer whose transmission axis is at an angle of 85º with the vertical. What is the intensity of the transmitted light?

I = I0 cos^2θ I = (0.40) (cos85)^2 *I = 0.00304 W/m^2*

Consider the circuit shown in the figure R = 5.5 ohms voltage = 9.0 V Assuming the inductor in this circuit has the value L = 8.0 mH , how much energy is stored in the inductor after the switch has been closed a long time?

I = V/R I = 1.6 UB = 1/2 LI^2 UB = 1/2 (8) (1.6)^2 *UB = 11 MJ*

*Using Ohm's law*

I = V/R = (Vmax/R) sin ω t *I = Imax sin ω t* where Imax is the peak current -I and V are values that change over time (oscillate)

Example 2.) Unpolarized light passes through five successive polarizing sheets, each of whose axis makes a 45º angle with the previous one. What is the intensity of the transmitted beam?

I0 = I0/2 cos45 = 1/2 so (I0/2) (cos45)^2 = I0/4 (I0/4) (cos45)^2 = I0/8 (I0/8) (cos45)^2 = I0/16 (I0/16) (cos45)^2 = *I0/32*

*Image Formation*

Image is what we see

class 15

Inductance

*Inductors in AC Circiuts*

Inductor in series with AC generator -Sort of opposite behavior than capacitor *picture is one side with ω labeled Vrms and the other side with an inductor* -XL, inductors have something called reactance of the inductor = over time it is harder for you to establish a current (like a resistor)

In ac circuits, I and V are not measured directly. Instead, ac ammeters and voltmeters are designed to measure the root-mean-square values of I and V:

Irms = I/√2 and Vrms = V/√2

Pav =

Irms Vrms cosφ

*Self-Inductance (back or induced emf)*

Isolated coil Change in I → change in Φ → I' inducted in same wire! As I increases → B' points left → I' points the other way (increase I = increase B) Self-inducted current always opposes changes

Chapter 23

Lenz's law, Work, Electrical Energy, Generators and Motors

Total Internal Reflection

Light bends when changing media (different n) n1 sinθ1 = n2 sinθ2 Higher n → lower n: light bends away from normal At critical θc: θ2 = 90º *θc = asin(n2/n1) -picture on slide 10: horizontal refraction = all light staying in medium -light always reflected and refracted, but eventually refraction -thick->thin medium = water->air -only light is the one from inside the glass

The light-year (ly) is a unit of distance commonly used in astronomy. It is defined as the distance traveled by light in a vacuum in one year. 1.) Express 1 ly in km. 2.) Express the speed of light, c, in units of ly per year. 3.) Express the speed of light in feet per nanosecond.

Light moves at a velocity of about 300,000 kilometers (km) each second 1.) 9.46 x 10^12 2.) v = distance / time = 1 lightyear / 1 year = *1 lightyear/year* 3.) speed of light = 299,792,458 meters per second * = 0.984 ft/ns*

*Magnetic Flux (ΦB)*

Magnetic flux ∝ area -related to the intensity of the magnetic field

chapter 26

Mirror Equation and Refraction

Example 1.) A coil with 1000 turns has a cross section of 1 cm^2 and a length of 0.1 m. a) Find the inductance b) Find the emf in the coil assuming the current increases from 0 to 1 A in 10^-3 s

N = 1000, A = 1 x (10^-2)^2 m^2, l = 0.1 m a.) L = μ0 N^2 A / l -> L = (4π x 10^-2) (1000)^2 (10^-4) / 0.1 -> *L = 1.26 x 10^-3 H* b.) ΔI = Δt - 0 = 1 A -> Δt = 10^-3 s εinduced = -L (ΔI/Δt) -> 1.26 x 10^-3 (1/10^-3) -> *ε = -1.26 V*

Doppler Effect for Sound

Observer hears sounds of approaching sources with higher pitch (frequency) than receding sources -car moves closer to last emitted wavefront -meeting sound and move = waves compress on the side the object is moving towards and stretch away from the staring point -ex: hear a higher pitch when the firetruck is coming towards you and a lower pitch as it drives away

Impedance

Opposition to current Resistance is constant -resistance never changes = impedance never changes = resistor (same impedance with increased frequency) -capacitor (Xc) = increase in frequency will decrease impedance -inductance (XL) will increase in impedance while frequency increases

Example 1.) A 57 kW radio station broadcasts its signal uniformly in all directions. a) What is the average intensity of its signal at a dis- tance of 260 m from the antenna? b) What is the average intensity of its signal at a dis- tance of 2600 m from the antenna?

P = 57x10^3 W A = 4πr^2 a.) Iav = Pav/A -> (57x10^3) / 4π(260)^2 -> *Iav = 6.71x10^-2 W/m^2* b.) Iav/100 -> 6.71x10^-2/100 -> *Iav = 6.71x10^-4*

Spring:

PE → KE → PE

Average power for inductors

Pav = Irms Vrms cos90º = 0 like in circuit with capacitor

average power equation in ACRC circuit

Pav = Irms Vrms cosφ

*Polarization of E&M Waves*

Polarization of E&M wave refers to electric field slide 8, class 21: -picture on the left has a wave polarized up and down in the direction z -picture on the right is polarized in a 60º angle in repeat to y

Chapter 25

Production and Propagation of Electromagnetic Waves

Smallest impedance:

R → 0, XL = XC

Example 1.) A generator connected to an RLC circuit has an rms voltage of 140 V and an rms current of 34 mA. If the resistance in the circuit is 3.1 kΩ and the capacitive reactance is 6.7 kΩ, what is the inductive reactance of the circuit?

RCL circuit Vrms = 140 V Irms = 34x10^3Ω Xc = 6.7x10^3 Ω XL = ? XL = (6.7x10^3) +/- √(140/34x10^3)^2 - (3.1x10^3)^2 *XL = 9.41 kΩ* OR *XL = 3.99 kΩ*

Applications

Radio and TV -Tuning stations (frequencies) Low resistance -Turn nob → change capacitance → change natural frequency → pick up signal of corresponding frequency -Other frequencies produce small currents Metal detectors -Person passing with metal increases L of inductor → decrease ω0 → large change in current -frequencies picked up = changing system (change with knob = change L = change ω) -change in frequency = change in resonance

*There are e&m waves that we cannot see (like Hertz's wave)*

Radio, microwave, infrared, ultraviolet, X-rays and -rays

*Magnification*

Ratio between heights and distances of image and object (from slide 2) *m = hi/ho = -di/do*

chapter 26

Ray Tracing for Lenses, Thin Lens Equation and Dispersion

*Ray Model of Light*

Rays are lines perpendicular to wave fronts (crests) -source is emitting waves in all directions

chapter 26

Reflection of Light, Plane mirrors and Spherical Mirrors

RC Circuits

Resistor and capacitor in series with AC generator *picture in slide has ω instead of battery*

RLC Circuits

Resistor, inductor, capacitor in series with AC generator

*Polarization*

Ropes can be plane polarized: oscillate in vertical, horizontal, ... planes We can block undesirable waves oscillations (e.g. horizontal) using a slit (e.g. vertical) Works for transverse waves only -horizontally or vertically polarized = important because it depends o on ability to block wave

*ω =*

S x 2π stands for angular frequency

*Thin Lens Equation*

Same mirror equation from similar triangles *1/f = 1/do + 1/di*

*Reflection*

Silvered mirror (flat glass with reflective metallic coating): 95% reflected -flat surface = reflects at same angle -angled surface = normal and angles change

As the solid metal disk in the figure(Figure 1) swings to the right, from the region with no field into the region with a finite magnetic field, is the induced current in the disk clockwise, counterclockwise, or zero? -Choose the best explanation from among the following:

The induced current is counterclockwise to generate a field within the disk that points out of the page.

Electric Generators

Transform mechanic energy → electric energy/current -Many loops of wire around armature that rotates -Something turns the axle (water, steam, gasoline...) -Conductor moves through B -> change in area -> induced emf -Rotation easier -Conductor rotates through B -> change in angle -> induced emf -In picture at a: flux ↓: I at a into screen to increase back B -After a passes by left: flux ↑: I at a into screen to de- crease back B -After a passes by top: flux: ↓: I at a out of screen to in- crease back B -After a passes by right: flux ↑: I at a out of screen to de- crease back B -Current flips direction each 1/2 cycle: AC current -Frequency of current is 60 Hz in US (flips 60 times in 1 sec) -rotates clockwise = current inward -alternating current (AC) = same in positive and negative -if flux is moving up, then current is moving down (opposed)

Root Mean Square Values

Type of average "rms"

LC circuit:

UC →UL →UC

*How can we obtain polarized light (very small wavelength) ?*

Use crystals or polarizing sheets (very small long molecules parallel to each other) -Almost all light with matching polarization passes through transmission axis -Almost all perpendicular polarization is absorbed -sheets are called polarizers -extremely thin slits are created on the sheets in a certain direction -sheet blocks 1/2 of the light passing through -need right orientation for the light to pass through the sheet

The analogy to Ohm's law is then

V = IZ

*P =*

V I = Vmax Imax sin^2 ω t I^2 R = I^2 R sin^2 ω t V^2/R = (Vmax^2/R) sin^2 ω t

for both a capacitor and resistor, V and I compared

V lags I by 45º φ = negative

for only a capacitor, V and I compared

V lags I by 90º φ is negative

A capacitor is designed to store energy by allowing charge to build up on its plates. Although a capacitor has no resistance in an ac circuit, there is a potential difference vC across the plates of the capacitor. The maximum values of i and vC do not occur at the same time. The voltage reaches a maximum after the current. The maximum potential difference across the inductor is

V of C = IωC. By defining the quantity 1/ωC as the capacitive reactance XC, V of C can be rewritten as V of C = IXC. As with the case for the inductor, this equation is similar to Ohm's law.

*R =*

V/I = Vmax/Imax = Vrms/Irms

Mastering Physics hw 7

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class 16

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radiation pressure would be

Very small for flashlight, but very large for something orbiting the earth in space -ex: Solar sail (uses pressure from the sun) -light can apply a pressure, but it is so little you cannot tell the difference

Maximum voltages for

Vmax = Imax R (for a resistor) Vmax = Imax Xc (for reactance) -in an alternating current you are given Vmax, Imax, or, Vrms, Irms -I = V/R, and V and I can be replaced by ^^^ for a resistor -I = V/Xc for a capacitor

Pmax =

Vmax Imax

Recall that an inductor is designed to oppose any change in current in the circuit. Although an inductor has no resistance, there is a potential difference vL across the ends of the inductor. Unlike the case for the resistor, i and vL do not reach maximum values at the same time. The voltage reaches a maximum before the current. The maximum potential difference across the inductor is

Vof L = IωL. By defining the quantity ωL as the inductive reactance XL, V of L can be rewritten as V of L = IXL. This equation is similar to Ohm's law.

Power in Circuits with Inductors

Voltage "leads" the current by 90º (opposite to capacitor) -things are out of phase switch direction of current = strong change in current = induced current in other direction

To understand basic calculations involving L-R-C ac circuits. Because the currents and voltages vary, ac circuits are more complex than dc circuits. Consider a circuit consisting of a resistor of resistance R, an inductor of inductance L, and a capacitor of capacitance C connected in series to an ac power source. (Figure 1) As with all circuit components connected in series, the same current flows through each of these elements. This is an ac circuit, so the current is changing with time. The current i at a time t can be found using the relationship: i=Icos(ωt), where I is the maximum current and ω=2πf (f is the frequency of the current source). The relationship between the current and voltage in an ac circuit works according to Ohm's law. Consider just the resistor in the circuit. Because the current changes in time, the voltage across the resistor vR also changes. However, both i and vR will be at a maximum at the same time. The maximum voltage across the resistor is given by

Voltage of resistor (V) = IR

which can be used in the equation V=IZ to yield

Vrms = Irms Z

Ray Tracing (Concave Mirrors)

What about an object O that is not at infinity? Draw at least 2 lines 1.) Ray goes from O' horizontally and is reflected through focus 2.) Ray goes from O' through focus and reflects horizontally 3.) Ray goes from O' perpendicular to mirror (through C) and reflects on itself ) -converge and form an image at the focus -image comes from 3 rays (picture in notes)

Reflection

When light wave is reflected by moving obstacle -Moving object acts like moving observer (f → f') -Moving object acts like moving source (f' → f'')

Energy Stored in B

Work must be done to establish current Same way electric energy is stored in capacitor (in electric field) *UB = 1/2 LI^2* -magnetic energy is stored in inductor (in magnetic field) -current can't decrease slowly by itself = needs energy stored by B

*Capacitance reactance*

Xc (like resistance for capacitor) Units Ω (Ohm) Does not let capacitor fully charge -reactance (Xc) = resistance in capacitor = harder to establish current -larger frequency = smaller Xc

An L-R-C circuit, operating at 60 Hz, has an inductor with an inductance of 1.53×10^−3H, a capacitance of 1.67×10^−2F, and a resistance of 0.329 Ω. What is the capacitive reactance of the circuit?

Xc = 1/(ωC) Xc = 1/376.9911184 x 1.67×10^−2 *Xc = 0.159 Ω* Because XC is inversely proportional to the frequency, high-frequency ac currents pass through a capacitor more easily than low-frequency ac currents. Hence circuits containing capacitors are often used as filters that allow high frequencies but not low frequencies to pass. These filters are called high-pass filters.

Can we get DC current from a generator?

Yes, using split-ring commutators

If this circuit were connected to a standard 120 V ac outlet, what would the rms current in the circuit be? Z = 0.532

Z = 0.532 Vrms = 120 Vrms = Irms Z 120 = Irms (0.532) *Irms = 226 A* The current is high because the total impedance is relatively low. Actually, plugging such a circuit into a 120-V outlet would most likely burn out the circuit elements.

The reactance of a capacitor is 50 Ω at a frequency of 67 Hz . What is its capacitance?

Z = 1 / C ω 50 = 1/C (2π67) *C = 47.5 μF*

The capacitive reactance, the inductive reactance, and the resistance of the circuit can be combined to give the impedance Z of the circuit. The impedance is a measure of the total reactance and resistance of the circuit and is similar to the equivalent resistance that can be found from various resistors in a dc circuit. Because vL and vC do not reach maximum values at the same time as i, the impedance is not found by adding R, XL, and XC. Instead, the impedance is found using

Z = √R^2+(XL −Xc )^2

An L-R-C circuit, operating at 60 Hz, has an inductor with an inductance of 1.53×10^−3H, a capacitance of 1.67×10^−2F, and a resistance of 0.329 Ω. What is the total impedance of the circuit?

Z = √R^2+(XL −Xc )^2 Z = √(0.329)^2 + (0.577 - 0.159)^2 *Z = 0.532 Ω*

If X →0:

Z→R, cosφ = 1, φ→0º, P = I Vrms -P is only a positive number for a capacitor

If X →0:

Z→R,cosφ = 1, φ→0º

If R→0:

Z→X , cosφ = 0, φ→-90º, P = 0 -P only = 0 for capacitor

If R→0:

Z→X ,cosφ = 0, φ→-90º

*Rays travel on*

a straight line -Proof: shadow

Example 2.) An object is placed 39 cm in front of a converging spheri- cal mirror of radius 24 cm. a) Use a ray diagram to locate the image formed by this mirror b) Discuss the characteristics of the image

a.) picture drawn in notes (10/19) = where they meet is where the image is b.) upside down, real, and reduced upside down because points meet below the line of center

*Light can be*

absorbed, transmitted and/or reflected by objects

*Diverging Lenses*

aka double concave -Thinner in the center than at the edges -Parallel rays diverge -Focal point where incident rays seem to emerge from -bends away from normal going out

*LC circuit →*

alternating current → energy transferred to separated circuit → waves with ~ speed of light -LC = inductor and capacitor -Beginning of wireless communication

+ u/c if

approaching -coming towards a point

*Plane waves*

are wave fronts far from source (look flat) -start perpendicular and get flatter moving farther from the source (parallel instead of perpendicular)

Power in RC Circuits

average power consumed

Electromagnetic waves move

away from antenna -comes out in all directions -B perpendicular to antennas

*Unpolarized light can be stopped*

by crossed polarizers -1 sheet vertical and the other horizontal = no light through

*ωt = π/2 → π:*

capacitor delivers energy to generator

*ωt = 0 → π/2:*

capacitor draws energy from generator

Tuning changes L or C circuit →

changes f0 -change circuit by changing capacitor (C and L give you frequency)

*Induced current depends on*

circuit ε = R I

*Electromagnetic Spectrum Created in*

circuits, accelerating / decelerating electrons, nuclear transitions

chapter 24: RLC Circuits and Resonance in Electric Circuits

class 18

*Summary slide*

class 22, slide 14 -convex = arbitrary image location, upright image orientation, reduced image size, and virtual image type -concave = object location between F and C, inverted image orientation, enlarged image size, and real image type

*the mirror equation applies to*

concave and convex mirrors

Optical illusions: Objects seem to be shorter in water

consequence of refraction -ex: limbs will look funny in water -ex: trying to catch a fish straight on

*For ray diagrams of converging lens*

consider lenses to be infinitely thin -Consider rays to bend sharply at center line instead of refraction in each surface

*Standing still magnet:*

constant B → nothing

If you view a clock in a mirror, do the hands rotate clockwise or counterclockwise?

counterclockwise

when inductance increases

current decreases

original circuit and frequency

current splits in picture (slide 5) -dealing with limits -taking limit of high frequency = frequency switches so often = never really charges the capacitor

*Curved convex mirrors*

decrease size of image -ex: rear view mirrors -makes the image appear smaller

Current in RL circuit:

decreases with frequency (XL increases with frequency)

Capacitive reactance Xc = 1/ωC

decreases with frequency (less charge)

*Virtual image:*

di < 0 (behind mirror) distance negative

*For Virtual image mirror equation*

di < 0 (behind mirror)

*Real image:*

di > 0 (in front of mirror) distance positive

*For real image mirror equation*

di > 0 (in front of mirror)

*Negative focal distance for*

diverging lenses

example 1.) A candle is 20 cm in front of a diverging mirror that has a focal length of -15 cm. Find the location and characteristics of the image using the mirror equations.

do = 20cm, f = 15cm, di = ? 1/do + 1/di = 1/f 1/20 + 1/di = 1/15 di = -86cm *(di<0) virtual* m = -di/do m = -8.6/20 m = 0.43 *(m<f) reduced* m = hi/ho m = -8.6/20 m = 0.43 *(hi>ho) upright*

Example 2.) An object is 24 cm in front of a diverging lens that has a focal length of -15 cm. Find the image location and characteristics of the image.

do = 24cm, f = -15cm, 1/f = 1/do + 1/di 1/-15 = 1/di + 1/24 di = -9 *virtual image (di < 0)* m = -di/do -> m = -(-9)/(24) -> m = 0.38 *reduced (|m| < 1)* m = hi/ho -> m = 0.38 upright (hi > 0)*

An object with a height of 27 cm is placed 3.0 m in front of a concave mirror with a focal length of 0.70 m. 1.) Find the location of the image produced by the mirror using the mirror and magnification equations. 2.) Find the magnification of the image produced by the mirror using the mirror and magnification equations.

do = 3, ho = 0.27, f = 0.7 1.) m = hi/ho = -di/do AND 1/do + 1/di = 1/f 1/3 + 1/di = 1/0.7 *di = 0.91 m* 2.) hi/ho = -di/do hi/0.27 = -(0.91/3) hi = -0.082 -0.082/0.27 = m *m = -0.303 m*

*Highly distorted image at*

do = f

Accelerated charges also create

electromagnetic waves (coming !)

Observed frequency =

f ' = f (1 ± u/c) f is the frequency of source u is relative speed between source and observer

An RLC circuit has a resonance frequency of 1.7 kHz . If the inductance is is 0.11 mH , what is the capacitance?

f = 1.7x10^3 Hz L = 0.11x10^-3 H C = ? ω0 = 1/√LC (2π 1.7x10^3) = 1/√(0.11x10^-3) C *C = 80 μF* LC = 1/ω^2

Example 1.) A converging lens has a focal length of 12 cm. For an object a) 15 cm b) 8 cm from the lens, where is the image formed, and what are its characteristics?

f = 12cm, 1/f = 1/do + 1/di a.) 1/15 + 1/di = 1/12 -> di = 60cm *real image (di > 0)* m = -di/do -> m = -60/12 -> m = -4 *enlarged (|m| > 1)* m = hi/ho -> m = -4 *upside down (hi < 0)* b.) di = -24 *virtual image (di < 0)* m = 3 *enlarged (|m| > 1)* m = 3 *upright (hi > 0)*

An L-R-C circuit, operating at 60 Hz, has an inductor with an inductance of 1.53×10^−3H, a capacitance of 1.67×10^−2F, and a resistance of 0.329 Ω. What is the inductive reactance of this circuit?

f = 60 L = 1.53×10^−3 H C = 1.67×10^−2 F R = 0.329 Ω XL = ? ω = 2πf -> ω = 376.9911184 XL = ωL -> *XL = 0.577 Ω* because XL is proportional to the frequency, low-frequency ac currents pass through an inductor more easily than high-frequency ac currents. Hence circuits containing inductors are often used as filters that allow low frequencies but not high frequencies to pass. These filters are called low-pass filters.

Example 2.) Find the frequency at which a 35 μF capacitor has the same reactance as a 35 mH inductor. What is the resonance frequency of an LC circuit made with this inductor and capacitor?

f = ? C = 35x10^-6 F L = 35x10^3 H Xc = XL (1/ωC = ωL -> 1/CL = ω^2) f = ω/2π *f = 1.44x10^2 Hz* is the resonance +

Large current when

f is f0 (natural frequency of current) -resonance = very large current = antenna picks it up

*Flat mirror:*

f → infinity: di → do -mirror almost flat when f is infinity

A frequent application of L-R-C ac circuits is the tuning mechanism in a radio. The L-R-C ac circuit will have a resonant frequency that depends on both the inductance and capacitance of the circuit according to the formula

f0 = 1/(2π √LC) This is the frequency at which the impedance is the smallest, which causes the largest current to appear in the circuit for a given Vrms. The radio picks up this resonant frequency and suppresses signals at other frequencies. A variable capacitor in this circuit causes the resonant frequency of the circuit to change. When you tune the radio you are adjusting the value of the capacitance in the circuit and hence the resonant frequency.

To see whether the L-R-C ac circuit from Part A would be suitable for a tuner in a radio, find the resonant frequency of this circuit. An L-R-C circuit, operating at 60 Hz, has an inductor with an inductance of 1.53×10^−3H, a capacitance of 1.67×10^−2F, and a resistance of 0.329 Ω

f0 = 1/(2π √LC) f0 = 1/2π √(1.53×10^−3)(1.67×10^−2) *f0 = 31.5 Hz* This frequency does not correspond to either the standard AM or FM band

*For convex mirror equation*

f<0 f = focal length = 1/2 radius -f is negative in this case

Changes in E, B felt later

farther away (constant v = c)

*Polarizers can be used to*

filter light or analyze polarization of light

*Area decreases:*

flux decreases → induced current

*Distance between focal point and convex mirror is*

focal length - f -focus is behind the mirror = add a negative sign to the distance

*Distance between focal point and convex mirror is*

focal length f

*Rays cross each other at*

focal point

*in converging lenses, parallel rays converge to*

focus or below/above it -Works if diameter of lens is small compared to radii of curvature -Only rays from far away arrive parallel (converge to focus) -Can be used to find the focus! -Same focus on both sides of lens (double convex) -Focus depends of both media (air and glass)

Steam power plant:

fossil fuel (coal, oil) burns → water boils → high pressure steam turns turbine → generator axle turns!

*Inverted image:*

h < 0 height negative

*Upright image:*

h > 0 height positive

Mirages:

hot air has lower n

sign of the angle determines

how much ahead or behind voltage is compared to current -ex: 0º = in phase

*Problem Solving: Ray Diagrams*

i. Draw a ray diagram with 2 (better 3) rays (to check results in the end) ii. Use the mirror and magnification equation (1/do + 1/di = 1/f AND m = hi/ho = -di/do iii. Check signs (h and d) iv. Compare with ray diagram

Maxwell Equations

i. Gauss law: relates electric field to electric charge ii. sameformagneticfield,butwithclosedfieldlines iii. Faraday's law: change in magnetic field produces electric field iv. Ampere law: electric current or change in electric field (not covered) produce magnetic field

high frequency limit circuit

ideal limit (slide 5)

for only a resistor, V and I are

in phase φ = 0

*reflected angle changes with*

incidence angle

*Moving magnet in solenoid:*

increases B → induces current -closed path of electrons

Current in RC circuit:

increases with frequency (XC decreases with frequency)

Inductive reactance XL=ω L

increases with frequency (more inducted voltage)

Current in R circuit:

independent of frequency

*Polarizers change both*

intensity and polarization of light

*Real images are*

inverted

*Normal light (bulb, Sun, etc.)*

is unpolarized (E oscillates in many planes)

Rank these electromagnetic waves on the basis of their frequency from largest to smallest

largest X-ray green light yellow light infrared FM radio AM radio smallest

*Snell's Law: Rays bend away from normal (larger θ) if n is*

less like going from water into air

When the ac generator in the figure (Figure 1) operates at high frequency, is the rms current in the circuit greater than, less than, or the same as when the generator operates at low frequency? Choose the best explanation.

less than Less current flows at high frequency because in that limit the inductor is like an open circuit and current has only one path to flow through.

*electromagnetic waves =*

light -any frequency, visible or not

Polarized light rotated by

liquid crystal -unpolarized light -> pass through a polarizer -> liquid crystal rotates the light making it vertical instead of horizontal -> add diagonal polarizer = light goes through

Rank these electromagnetic waves on the basis of their wavelength from longest to shortest

longest AM radio FM radio infrared yellow light green light X-ray shortest

*Curved concave mirrors*

magnify image -ex: cosmetic mirrors -makes the image appear closer

*I and V in phase*

maximum and minimum values at same times -voltage and current oscillate between values (choose max and min values) -I = V/R -> one oscillates = the other oscillates (corresponds in phase, meaning the max, min, and zeros correspond when comparing voltage and current graphs) -Imax = Vmax/R -R is a number

*ωt = π:*

maximum voltage across capacitor (maximum charge) → zero current

Maximum current change:

maximum voltage across inductor -current increasing = Vmax

*refract*

means to go through into a different medium -light bends -opposite of reflect

*More loops in secondary coil:*

more V in secondary coil *(step-up transformer)*

*partially polarized light:*

more intensity at certain angles and less at others

*Refraction: Rays bend away from normal if*

n is less (higher speed)

*Refraction: Rays bend towards normal if*

n is more (lower speed)

*Snell's Law: Rays bend towards normal (smaller θ) if*

n is more like going from air into water

example 2.) A beam of light in air strikes the glass top of a coffee table at an angle of incidence of 45o. The glass has an index of refraction of 1.5. a) What is the angle of refraction for the light transmitted into the glass? b) Prove that the beam emerging from the other side of the glass is parallel to the incident beam. c) What about if the glass was 4 cm thick (instead of 2 cm)?

n1 = 1, n2 = 1.5, θ1 = 45, θ2 = ? a.) n1sinθ1 = n2sinθ2 θ2 = 28º b.) θ2 = θ3 θ3 = 28º θ4 = ? n3sinθ3 = n4sinθ4 θ4 = 45º c.) distance does nothing to the angles

A submerged scuba diver looks up toward the calm surface of a freshwater lake and notes that the Sun appears to be 30º from the vertical. The diver's friend is standing on the shore of the lake. At what angle above the horizon does the friend see the sun?

n1 = 1.33 n2 = 1 θ1 = 30º θ2 = ? n1sinθ1 = n2sinθ2 θ2 = 41.7º 90 - 41 = *48º*

Adjust frequency of generator to

natural frequency of circuit ω0 = 1/√LC Maximum current -system with frequency and current = impose frequency = high current

out of phase means

no power

For θ1 > θc:

no refraction ! Only reflection

*di > 0 if image is*

on opposite side from object: real image -for lens = positive distance is on the opposite side

*di < 0 if image is*

on same side as object: virtual image

*Unless switch is used in the primary coil:*

open/close → change in flux → current

*Specular reflection (mirror):*

parts reflect at same angle and image only reach you if you are at certain angle

*Diffuse reflection (rough surface):*

parts reflects at different angles and image can reaches you at many angles

E and B

perpendicular to each other and direction of wave propagation (transverse waves) -transverse = move forward as well as up and down (like a wave on a moving rope)

- u/c if

receding -going away from a point

Example 1.) An AC generator with a frequency of 150 Hz and an rms voltage of 25 V is connected in series with a 11 kΩ resistor and a 0.27 μF capacitor. What is the rms current in this circuit?

resistor and capacitor = RC circuit with AC f = 150 Hz (ω = 2πf) Vrms = 25 V R = 11 x 10^3 Ω C = 0.27 x 10^-6 F Irms = Vrms /Z -> Irms = Vrms/√R^2 + Xc^2 -> Irms = Vrms/√R^2 + (1/ΩC)^2 Irms = 25 / √(11x10^3)^2 + [(1/11x10^3)(0.27x10^-6)]^2 *Irms = 2.14 mA*

Example 2.) An AC generator with a frequency of 1.04 kHz and an rms voltage of 20.2 V is connected in series with a 2.2 kΩ resistor and a 325 mH inductor. a) What is the phase difference (between V and I) for this circuit? b) What is the average power consumed by this cir- cuit?

resistor and inductor = RL circuit with AC f = 1.04 x 10^3 Hz (ω = 2πf) Vrms = 20.2 V R = 2.2 x 10^3 Ω L = 325 x 10^-3 H a.) φ = ? -> cosφ = R/Z -> cosφ = R/√R^2 + XL^2 -> cosφ = R/√R^2 + (ω/L) -> cosφ = R/√R^2 + (2πf/L) cosφ = 2.2 x 10^3 / √(2.2 x 10^3)^2 + (2π 1.04 x 10^3/325 x 10^-3)^2 cosφ = 0.72 *φ = 44º or 0.77 radians* b.) Pav = Irms Vrms cosφ -> Pav = (Vrms/Z) Vrms cosφ -> Pav = (Vrms/√R^2 + ((2πf)L)^2) Vrms cosφ Pav = (20.2/√(2.2x10^3)^2 + [(2π 1.04x10^3)(325 x 10^-3)]^2 (20.2) (0.72) *Pav = 0.096 W*

RL Circuits

resistors and inductors (RL) -Circuit containing resistor and inductor in series with AC generator -Voltage resistor in phase with current -Voltage inductor 90º ahead of current -voltage not in phase, so add them up using equations -current same everywhere, V is different

*For ray diagram of diverging lenses, draw*

same 3 rays -Projections meet on the incident side -Virtual image -Upright image -Reduced image -rays come n parallel from far away -> focus

*diverging lenses characteristics*

same as concave mirror -object location arbitrary -upright -reduced -virtual

*converging lenses characteristics*

same as convex mirror -object beyond F -inverted -reduced or enlarged -real

*unpolarized light:*

same intensity at all angles of polaroid sheet

Same rms voltage + same rms current:

same reactance 1/ω0C = ω0 L = *ω0 = 1/√LC* angular frequency ω0 = √k/M

*Ultraviolet:*

seen by some animals (cats, bees) causes cancer -should be absorbed by ozone layer

*Secondary voltage only when*

switch is opened/closed

The Voyager I spacecraft has traveled farther than any other man-made object, and in August 2012 it entered into interstellar space when it was a distance of 1.8×10^13m from Earth. How many hours elapse between the time a command is sent from Earth and the time the command is received by Voyager when it entered interstellar space?

t = d/c t = 1.8×10^13 / 3 x 10^8 *t = 60000 s = about 17 hours*

*C is*

the center of curvature (center of sphere of which mirror is a section)

*F is*

the focal point -Image from infinite distance object forms at F

*The more curved the mirror,*

the more blurred the image (spherical aberration)

*Inserting a 45º angle polarizing sheet changes*

the transmitted intensity I = 0 → I = 1/8 I0 -unpolarized light with a sheet in front = 1/2 the intensity -add one = opposite direction = no light through -add a third one diagonally = see more light -after passing through the first it has half the intensity vertically -> add a sheet horizontally = no intensity (zero) -> add a diagonal sheet = wave of light through

*Image height follows*

the usual convention

*visible light*

there is a very small amount of visible light, but a lot before and after it -wavelength increases going left -frequency increases going right

Current passing through liquid crystal aligns molecules in different position so

they don't rotate light

*period (T) =*

time between 2 consecutive wave crests / troughs

*Unpolarized beam can be thought of as*

two plane polarized beams of equal magnitude perpendicular to each other

*Energy in E&M Waves: Magnetic field energy density*

uB = 1/2 1/μ0 B^2

*But, E and B have the same contribution*

uE = uB 2 times one of them = total *u = ε0 E^2 = 1/μ0 B^2*

*Virtual images are*

upright

*X-rays:*

used for imaging bones, metal, ... very harmful

*Radio:*

used in radio and TV transmissions, amateur radio, cell phones, wireless internet, GPS signal usually created in circuits

*Infrared:*

used in remote controls, night-vision seen by some animals (vipers) -makes cells of our skin resonate (warms us) -anything warm = radiation

*Microwave:*

used in telephones, kitchen, satellite TV usually created in circuits

*y-rays:*

used to kill microorganisms, radiation cancer therapy, nuclear imaging -extremely harmful - very energetic

Part B: Repeat part A for the case in which you walk toward a mirror but at an angle of 27º to its normal.

v = (cos27) (2.5) (2) = 4.46 m/s

Part A: How rapidly does the distance between you and your mirror image decrease if you walk directly toward a mirror with a speed of 2.5 m/s?

v = 2.5 x 2 = 5 m/s

Example 1.) How long does it take light to reach us from the Sun, 1.5 x 10^11 m away?

v = d/t -> c = d/t -> t = d/c t = (1.5 x 10^11) / (3 x 10^8) *t = 500 s = about 8 minutes* -this shows how it takes a while for light to reach us

Rank these electromagnetic waves on the basis of their speed (in vacuum), from fastest to slowest

vAM radio = vFM radio = vinfrared = vyellow = vgreen = vX-ray all the light waves have the same speed, c; The speed of light in a vacuum

Which color of light has the higher frequency, red or violet?

violet

What type of image of the object will the convex mirror create?

virtual

*In AC capacitor circuit:*

voltage and current not in phase -min, max, and zero points don't align/coincide

Hydroelectric power plant:

water falls from waterfall → water turns turbine → generator axle turns!

Matrix of wires "choses"

which pixels to turn off adding = makes pixels not work

Average power consumed by capacitor (AC):

zero

*Average current is*

zero (same positive and negative)

*Average E and B is*

zero (sinusoidal)

*ωt = π/2:*

zero voltage across capacitor (no charge) → maximum current

*ωt = 3π/2:*

zero voltage across capacitor (no charge) → maximum opposite current ...

Maximum current:

zero voltage across inductor -average is zero

Power P =

ΔU/Δt

*reflexion equation*

θr = θi angle of incidence = angle of reflection

*v =*

λ / T = c

*c =*

λf

Calculate the frequency of blue light with a wavelength of 470 nm.

λf = c 4.7x10^-7 (f) = 3x10^8 *f = 6.38x10^14 Hz*

Calculate the frequency of red light with a wavelength of 680 nm.

λf = c 6.8x10^-7 (f) = 3x10^8 *f = 4.41x10^14 Hz*

A cell phone transmits at a frequency of 1.64×10^8 Hz. What is the wavelength of the electromagnetic wave used by this phone?

λf = c λ (1.64×10^8) = 3x10^8 *λ = 1.83 m*

if XL < XC:

φ < 0, total voltage legs current

if XL = XC:

φ = 0, total voltage in phase with current

if XL > XC:

φ > 0, total voltage leads current

Irms =

ωCVrms = Vrms/Xc equivalent to Ohm's law

*Irms =*

√(I^2)av = (1/√2) Imax

*Vrms =*

√(V^2)av = (1/√2) Vmax

Using vectors V =

√(Vmax of R)^2 + (Vmax of L)^2 = √(Imax R)^2 + (Imax XL)^2 = Imax √R^2 XL^2 *Vmax = Imax Z* *Z = √R^2 XL^2* *cosφ = R/Z* -Z now combining Xc and XL (instead of Xc and R like the previous class)

Remember that Z =

√R^2+(XL −Xc )^2 and cosφ = R/Z (no info sign)

Mastering physics homework 6

VVVV

mastering physics homework 5

VVVV

*For a *solenoid* (starting with no current)*

*L = μ0 N^2 A / l* *L = μ0 n^2 volume* defining the # of loops per length n = N/l and volume = l A l = length n = N(number of loops) x length -induce voltage in other direction (faster change in current =

A 75.0 Hz generator with an rms voltage of 120 V is connected in series to a 3.40 kΩ resistor and a 1.15 μF capacitor. 1.) Find the rms current in the circuit. 2.) Find the phase angle, ϕ, between the current and the voltage.

1.) Irms = 31 mA 2.) φ = -28.5º

*-2 different currents (secondary and primary) =*

2 different numbers of turns (affects N)

*For DC current there is no change in*

B and no induction!

Example 2.) An ideal 600 W transformer has 50 turns on its primary coil and 100 turns on its secondary coil. a) Is this transformer a step-up or a step-down ar- rangement? b) If the primary coil is connected to a 120 V source, what are the output voltage and current of this transformer?

P = 600 W, Np = 50, Ns = 100 a.) Vs/Vp = Ns/Np -> Vs = Vp (Ns/Np) -> Vs = 2Vp -> *step up* b.) Vp = 120 V -> Vs = 2(120) -> *Vs = 240 V* P/Vs = Is -> 600/240 = Is -> *Is = 2.5 A*

*Units of Magnetic Flux*

T•m^2 = Wb (weber)

Find the impedance of a 50.0 Hz circuit with a 70.0 Ω resistor connected in series with a 85.0 μF capacitor.

Xc = 1 ⁄ (2πƒC Xc = 1 / 2π(50)(85x10^-6) Xc = 37.45 Z = √R^2 + Xc^2 Z = √(70)^2 + (37.45)^2 *Z = 79.39 Ω*

*L is self-inductance*

[V.s/A]=[Ω.s]=[*H(henry)*] Generally L is small (mH)

*Lenz's law:*

a current produced by an induced emf moves in a direction so that the magnetic field opposes the original change in flux -Magnetic flux depends on B, A, θ *emf = -N(ΔΦ/Δt)*

*Moving magnet out of solenoid:*

decreases B → induces current the other way

*equation for self-inductance*

*ε = -L (ΔI/Δt)*

In a solenoid: UB =

*UB = (1 / 2μ0) B^2 A l* *UB = UB/volume = 1/2 (B^2/μ0)* (both formulas valid for any shape of conductor)

Mutual Inductance

- Change in current in wire 1 → change in Φ → inducted current in wire 2 - As I1 increases → B1 points left → B2 points right → I2 points the other way - M is the mutual inductance - Depends on material, size, shape, # of turns of coils, relative distance, iron inside - How transformers work -inductors = "L" -2 different types of inductance -change current = change magnetic field = induce another current

Motion emf - horizontal

-Assume uniform B perpendicular to area -Increase in conductor area (moving rod with constant v) *|ε| = Blv* for one loop -Induced current clockwise in picture on slide 6, class 14 -change in magnetic field or area by pulling or pushing the bar -current is moving clockwise = force is perpendicular going in

LR Circuit

-Battery connected in series to resistor and inductor -Change in current inducts opposite current in inductor -Net current smaller → inducted current smaller ... until current approaches Imax = ε/R -Time ∝ L / R -current only max current (current in other direction cancels out original current) = start out with lower current (inducing current to bring flux back = slowly reduced current)

*Transformers*

-Change AC voltage -Decrease voltage from power lines to be used at home (120 V) -Increase the voltage to be used in TV, ... -2 coils of -wire: primary and secondary -Change in B in the 1st coil (AC current) -Induces current in 2nd coil -Iron core (99% efficient) -Same frequency but different V -2 coils, change something in 1 = induce current -AC = alternating current = current switches direction really fast (easier to transform) = B switches = changes flux = induced current in other direction

*review: direction of current*

-Change in magnetic flux = induced current (emf) -Current flows so as to oppose what caused it = current is perpendicular to magnetic field -Induced I generates B with same pole when magnet approaches / opposite pole when magnet recedes -direction of current chosen in a way to oppose magnetic field (picture in notes) -clockwise field = current coming out of the page

Electric Field Generation

-Change in magnetic flux induces current / emf -There is an induced electric field (charges move) -Conservation of energy W E=W B associated with rod -using displacement in direction of force: *E = Bv* -Direction ? Along the current

Credit Cards

-Credit cards are digital magnetic tapes -Include information about your account -Swiping it allows the reading: change in flux → current -Strong magnetic field can demagnetize cards

*Alternating Current*

-Electrons in direct current (DC) move in only one direction -Provided by batteries/adapters -Electrons in alternating current (AC) move in one direction and then another (many times per second) -Provided by electric company, most generators (can be transformed !) -Current is sinusoidal -Voltage is sinusoidal -AC no naturally created -graph is more curved (time vs current) -graph goes from maximum value (1) to minimum value (-1) = sinusoidal

*Electromagnetic Induction*

-Magnetic field exerts force on moving charges (current) -Currents generate magnetic field -no current generated in magnetic field

*Transmission of Power*

-Power plants far from cities -Power loss minimized by low current / high voltage -Voltage changed by transformers P = RI^2 -lose current (I) with distance because increasing resistance (R) -high voltage = lowers current = allows for optimal power -cables not touching the ground = no electrocution *high V (low I) → less dissipation*

Tapes and Discs

-Record audio and video -Thin plastic tape covered with layer of magnetic oxide -Signal (current) sent to recording head (tiny electromag- net): magnetizes regions with opposite polarities -Head also reads change in magnetic field (moving tape): induces current: further amplified -Used in hard-drives -cassette tape = magnetizable

Microphones

-Sound waves move membrane → change in flux -Induced current in coil -Frequency sound waves = frequency emf -Signal is amplified -Sent to loud speakers, computer, ... -Same for seismograph (waves from earthquakes) -membrane = sheet with a coil glued to it -magnet generates a field -vibration from sound waves = changes the area of the sheet = current through the system

Electric Guitars

-Strong magnet magnetizes strings (diamagnetism) -Plucking strings changes magnetic flux -Induces current in coil -Frequency sound waves = frequency emf -Signal is amplified

Motors

-Transforms electric energy → mechanical energy -Cylinder with several large coils around (rotor) -Voltage source generates current in the coils -Magnetic field generates force → torque -Rotor moves around axle -After coils moves pass top/bottom torque changes direction -Unless current is reversed (AC) -What about DC currents (like car)? split-ring commutators -rotates by supplying force -force moves with current

Eddy Currents

-Uniform B perpendicular to area -Gravity: metal plate accelerates down -Decreases B / magnetic flux -Induced Eddy current increases flux -Magnetic force opposes gravity: controls v -Magnetic breaking (like friction) -Used in fishing reels, exercise bikes, roller coasters, etc.

Motional emf - vertical

-Uniform B perpendicular to area -Gravity: rod accelerates down -Decreases conductor area / magnetic flux -Induced I increases flux -Magnetic force opposes gravity -Conservation of energy - GPE -> KE -> U -> light/heat -force upward so current opposite direction

review: Faraday's law

-change in magnetic flux = generates a current -magnetic field moves N -> S -magnetic flux = how much magnetic field (B) goes through a certain area -don't know current direction = found through Lenz's law

*circuits on slide 2, class 13*

-left: I on left generates B -right: Does B generate I on right? (no touching!) No (no current) But when current on left is turned on / off (change in B) there is current on right ! -Change in magnetic field induces current (*changing the magnetic field creates a current*)

A 0.16 μF capacitor is connected to an ac generator with an rms voltage of 14 V . For what range of frequencies will the rms current in the circuit be less than 2.0 mA ?

0 ≤ f < 142Hz

The figure shows a graph of the output from an AC voltage source. (Figure 1) 1.) What is the maximum voltage Vmax of the source? 2.) What is the average voltage Vavg of the source? 3.) What is the root-mean-square voltage Vrms of the source? 4.) What is the period T of the source? 5.) What is the frequency f of the source? 6.) What is the angular frequency ω of the source?

1.) 3 V 2.) V = Vmax sin ω t V = 3 sin0 ω (0.02) *V = 0V* 3.) Vrms = (1/√2) Vmax Vrms = (1/√2) 3 *Vrms = 2.12 V* 4.) 0.08 s 5.) 1/0.08 = *12.5 Hz* 6.) ω = 2πf ω = 2π(12.5) *ω = 78.5 rad/s*

A coil has N turns enclosing an area of A. In a physics laboratory experiment, the coil is rotated during the time interval Δt from a position in which the plane of each turn is perpendicular to Earth's magnetic field to one in which the plane of each turn is parallel to the field. The magnitude of Earth's magnetic field at the lab location is B. 1.) What is the magnitude Φinitial of the magnetic flux through one turn of the coil before it is rotated? 2.) What is the magnitude Φfinal of the magnetic flux through one turn of the coil after it is rotated? 3.) What is the magnitude of the average emf induced in the entire coil?

1.) BA 2.) zero 3.) NBA/Δt

The inductance of a solenoid with 420 turns and a length of 24 cm is 7.1 mH 1.) What is the cross-sectional area of the solenoid? 2.) What is the induced emf in the solenoid if its current drops from 2.7 A to 0 in 57 ms?

1.) L = μ0 N^2 A / l 7.1 x 10^-3 = (4π x 10^−7) (420)^2 A / 0.24 *A = 7.7 x 10^−3 m^2* 2.) εinduced = -L (ΔI/Δt) εinduced = -(7.1 x 10^-3) (2.7/0.057) *εinduced = 0.34 V*

The rms current in an ac circuit with a resistance of 160 Ω is 0.15 A . 1.) What is the average power consumed by this circuit? 2.) What is the maximum power consumed by this circuit?

1.) Pav = I^2 R Pav = (0.15)^2 (160) *Pav = 3.6 W* 2.) Pmax = Imax^2 R Pmax = (0.15 x √2)^2 (160) *Pmax = 7.2 W*

You hold a wire coil so that the plane of the coil is perpendicular to a magnetic field B⃗ . 1.) If the magnitude of B⃗ increases while its direction remains unchanged, how will the magnetic flux through the coil change? 2.) B⃗ is kept constant but the coil is rotated so that the magnetic field, B⃗ , is now in the plane of the coil. How will the magnetic flux through the coil change as the rotation occurs?

1.) The flux increases because the magnitude of B⃗ increases (The magnetic flux through a coil is directly proportional to the magnitude of the magnetic field.) 2.) The flux decreases because the angle between B⃗ and the coil's axis changes. (As the orientation of the coils changes, the magnetic flux through the coil decreases. It reaches its minimum value (zero) when the coil is parallel to the field.)

In the Alcator fusion experiment at MIT, a magnetic field of 50.0 T is produced. 1.) What is the magnetic energy density in this field? 2.) Find the magnitude of the electric field that would have the same energy density found in part A

1.) UB = 1/2 (B^2/μ0) UB = 1/2 (50^2/4π x 10^−7) *UB = 9.95 x 10^8 J/m^3* 2.) U = (1/2)(E)^2(ε) 9.95 x 10^8 = 1/2 E^2 (8.85 x 10^-12) *E = 1.50 x 10 ^10 V/m*

Six transformers have the rms primary voltages (Vp), number of primary turns (Np), and number of secondary turns (Ns) listed below 1.) Which of the transformers are step-up transformers? Which of the transformers are step-down transformers? 2.) Rank the transformers on the basis of their rms secondary voltage. 3.) 100 A of rms current is incident on the primary side of each transformer. Rank the transformers on the basis of their rms secondary current.

1.) Vs/Vp = Ns/Np Vs/480 = 2000/4000 -> Vs = 240 = step down Vs/480 = 1000/4000 -> Vs = 120 = step down Vs/240 = 500/1000 -> Vs = 120 = step down Vs/240 = 2000/1000 -> Vs = 480 = step up Vs/120 = 2000/500 -> Vs = 480 = step up 2.) Vs/√2 240/√2 = 169.7 120/√2 = 84.85 480/√2 = 339.4 3.) Is = Ip * Np / Ns Is = 100 x (2/1) -> Is = 200 Is = 100 x (4/2) -> Is = 200 Is = 100 x (1/2) -> Is = 50 Is = 100 x (1000/500) -> Is = 200 Is = 100 x (500/2000) -> Is = 25

For each of the actions depicted below, a magnet and/or metal loop moves with velocity v⃗ (v⃗ is constant and has the same magnitude in all parts). Determine whether a current is induced in the metal loop. If so, indicate the direction of the current in the loop, either clockwise or counterclockwise when seen from the right of the loop. The axis of the magnet is lined up with the center of the loop. 1.) For the action depicted in the figure, (Figure 1) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop). 2.) For the action depicted in the figure, (Figure 2) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop). 3.) For the action depicted in the figure, (Figure 3) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop). 4.) For the action depicted in the figure, (Figure 4) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop). 5.) For the action depicted in the figure, (Figure 5) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop).

1.) counterclockwise 2.) counterclockwise 3.) zero 4.) zero 5.) clockwise

When a long copper wire of finite resistance is connected to an ac generator, as shown in the figure (a),(Figure 1) a certain amount of current flows through the wire. The wire is now wound into a coil of many loops and reconnected to the generator, as indicated in the figure (b). 1.) Is the current supplied to the coil greater than, less than, or the same as the current supplied to the uncoiled wire? 2.) Choose the best explanation

1.) less than 2.) Less current is supplied to the circuit because the coiled wire acts as an inductor, which increases the impedance of the circuit.

chapter 23

Induction, Magnetic Flux and Faraday's Law

depicts the following scenario. The lightbulb in the circuit shown in the figure has a resistance of 25 Ω and consumes 5.9 W of power; the rod is 1.15 m long and moves to the left with a constant speed of 3.5 m/s . The strength of the magnetic field is 3.0 T . 1.) Find the current that flows in the circuit. 2.) What speed must the rod have if the current in the circuit is to be 1.4 A ?

R = 25 Ω, P = 5.9 W, l = 1.15, v = 3.5 m/s, B = 3 T 1.) P = I^2 R *I = 0.49 A* 2.) E = IR E = 35 E = BlV 35 = (3) (1.15) v *v = 10 m/s*

*Assuming 100% efficiency (same (ΔΦB/Δt))*

Vp/Np = Vs/Ns -> Vs/Vp = Ns/Np

A 500 Ω resistor and a 240 mH inductor are connected in series with an ac generator with an rms voltage of 19.0 V and a frequency of 65.0 Hz What is the rms current in this circuit?

Vrms = Irms (√R^2 XL^2) 19 = Irms (√(500)^2 (0.24)^2) Irms = 0.038 A *Irms = 38 mA*

*Induced current on secondary coil*

Vs = Ns (ΔΦB/Δt)

Work done to move rod

W = ((B^2l^2v^2)/R) Δt

Example 2.) When a magnet is plunged into a coil at speed v, a voltage is induced in the coil and a current flows in the circuit. a) If the speed of the magnet is doubled, what happens to the induced voltage? And current? b) The same magnet is plunged into a coil that has twice the number of turns as before. If the speed of the mag- net is again v, what happens to the induced voltage? And current?

a.) U vs 2U E = IR b.) same because everything is the same

Example: 1.) A 15 cm diameter circular loop of wire is placed in a 0.5 T magnetic field. a) When the plane of the loop is perpendicular to the field lines, what is the magnetic flux through the loop? b) The plane of the loop is rotated until it makes a 35º angle with the field lines. What is the angle to be used in the equation ΦB=B A cosθ in this case? . c) What is the magnetic flux through the loop at this angle?

a.) ΦB = |B| |A| cos θ -> 0.5 x pi (15 x 10^2) cos0 -> ΦB = *8.89 x 10^-3 Wb* b.) 90 - 35 = *65* c.) ΦB = 1/2 x pi (15 x 10^2 / 2)^2 cos65 -> ΦB = *5.07 x 10^-3 Wb*

Example 2.) Assume the moving rod is 12 cm long and it is pulled at a speed of 15 cm/s. If the magnetic field is 0.8 T, calculate a.) the emf developed. b.) the electric field felt by electrons in the rod.

a.) ε = BlV ε = (0.8)(0.12)(0.15) *ε = 0.0144 V* b.) E = BV E = (0.8)(0.15) *E = 0.12 V/m*

*Coils with high inductance are called*

inductors

*Faraday's Law*

law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force —a phenomenon called electromagnetic induction -Induced electromotive force (not a force !)

*Less loops in secondary coil:*

less V in secondary coil *(step-down transformer)*

*more turns =*

more voltage = less current (current doesn't matter for appliances, but voltage does)

Nuclear power plant:

nuclear reactions occur → energy released → water boils → high pressure steam turns tur- bine → generator axle turns!

*But, flux change in primary current depends on*

voltage Vp = Np (ΔΦB/Δt)

A 0.57 T magnetic field is perpendicular to a circular loop of wire with 58 turns and a radius of 14 cm . If the magnetic field is reduced to zero in 0.12 s , what is the magnitude of the induced emf?

ε = -N (ΔΦB/ΔT) ε = -(58 x 0.14) (0.057/0.12) ε = 17.09 V = ΔΦ/Δt V = (58 x 0.57 x pi x 0.14^2) / 0.12 V = 17

A metal rod 0.85 m long moves with a speed of 2.3 m/s perpendicular to a magnetic field. If the induced emf between the ends of the rod is 0.65 V , what is the strength of the magnetic field?

ε = BlV (0.65) = B (0.85) (2.3) B = 0.33 T