Physics Ch 23

M0= "Mue not"

4π x10⁻⁷ T xm/A

Electrical Motor

An electric motor is exactly the opposite of a generator - it uses the torque on a current loop to create mechanical energy if we apply right-hand rule number two we point our hand in the direction of our electric current let's suppose it's going this way so we pointed this way and then we rotate our hand in the direction of our magnetic field our magnetic field begins on the north side of the magnet and ends on the south side of our magnet so travels in the following direction and this magnetic force will Cree a torque that will essentially oppose our original torque we see that the higher our electric current is the greater this counter torque is so we see that in order for our AC electric generator to actually generate an alternating electric current the amount of energy that we input our input torque has to be greater than the counter torque that is produced by this external magnetic field English (auto-generated)

Induced EMF Equation known as Faraday's law of electromagnetic induction

Typically deal with only one loop of wire let's suppose we take and number of loop of wires that have the same exact area and we stack them on top of one another so if we have n loops of wire If a circuit contains N tightly wound loops and the flux changes by ΔΦ during a time interval Δt, the average emf induced is given by If zero there will be no induced emf

Angular velocity

W= Angular velocity T= time period

RL Circuit

the same way that we can place capacitors and resistors into an electric circuit we can also place our inductors and resistors into an electric circuit because an inductor consists of a wire and because wires carry resistance that implies that inductors as with any wire carries at least some amount of resistance so circuits that contain resistors and inductors are known as RL circuits because we have no battery source that basically means we don't have any voltage difference within our electric circuit and so electric current will not flow however at the moment we insert a battery into our electric circuit current will begin to flow so if we take a battery and place it into our electric circuit and the voltage on that battery is given by V naught our electric current will begin to flow from the high potential electrode to the low potential electrode of our battery **can we describe the rate of change of our electric current mathematically once we actually place our battery into our electric LR circuit?* Kerkoff second rule basically states that the sum of the voltage differences across any loop within our circuit is always equal to zero "Tau" is simply our time constant it's given by the ratio of L our inductance divided by our our resistance now recall that towel simply gives us the quantity of time that it takes our electric current to reach 63% of its maximum value T= L/R curve represents the rate of change of our electric current once we place our battery inside our electric circuit our electric current will begin to increase according to the following curve so this curve represents the growth of our electric current inside our L our circuit a time of infinity at an infinitely large time our I is a maximum value equaling V naught divided by R and that's given by this line here so this line represents the maximum possible value that can be reached inside our electric circuit: t=∞ V0= voltage on battery R=Resistance

Motional EMF

there are three ways by which a change in magnetic flux can take place one of these ways is by increasing or decreasing the area of the loop of conducting wire you shape conducting wire as shown and we take a metal rod and we place it on top of our U shaped conducting wire so that we form the following closed conducting loop metal rod is allowed to move across our x axis so either to the left or to the right - this entire system is placed into an external magnetic field that is assumed to be uniform given by B which points out of the board as shown by the following blue dots now we want to examine the if the rod moves at a speed of V over an infinitely small time interval given by DT then that implies by the distance equation the rod will move a total distance given by DT multiplied by V remember velocity times the time gives us our length of distance that our object travels

Magnetic Flux

used to define Faraday's law of electromagnetic induction given by the Greek symbol Phi with the M or B symbol which stands for magnetic so the magnetic flux is equal to the product of the two vectors the magnetic field and our area vector the product of these two vectors is equal to the product of the magnitude of these two vectors multiplied by the cosine of the angle theta which is the angle between these two vectors equation only works as long as our magnetic field is assumed to be uniform

Three ways of inducing an EMF

we can induce an EMF if we change our magnetic flux B while keeping a and the angle constant keep the B constant and the angle constant and we either increase or decrease our A we can induce an EMF is by changing the angle theta between these two vectors a and B this induced EMF will in turn produce an electric current through that conducting wire and finally that electric current will produce its own magnetic field let's call this magnetic field the induced magnetic field so notice we have two different magnetic fields: magnetic field that induces that EMF now that EMF induces an electric current and that electric current will create the second magnetic field which we are calling the induced magnetic field the induced magnetic field will point out of the board so in the opposite direction of the external initial magnetic field B that creates that induced electric current

Motional EMF Equation

because magnetic flux is given by taking the dot product of our magnetic field B and our influency small change in area da we get the following result now because da is equal to L times V times DT we replace da with the following result and notice we have DT appears on the top and the bottom so we can cancel those out and we see that our induced EMF that is created inside our loop of conducting wire as a result of our moving conductor is equal to the product of the magnetic field be the length of our movable metal section given by L multiplied by V the speed of our movable conducting section this equation gives us the induced EMF on a moving conductor and it works as long as our B our V and L are mutually perpendicular with respect to one another now notice the B represents the external magnetic field in which our system our conductor is found in the L is the length of our conductor in this case it's this distance and the V represents the speed of our moving conductor

Induced emf - Faraday's Experiment

can a magnetic field induce an electric current inside a conducting wire? wire number one that is connected to a battery to a voltage difference Wire number two which does not have a battery but it has a galvanometer Iron ring essentially is connected to wire number one and two wire number two and we use this ring of iron to essentially magnify the magnetic field that is produced by the electric current moving in why number one

Inductance L= (induced EMF) E=

coil of wire so this coil consists of some number of loops now what will happen inside our coil of wire when we allow an alternating electric current to flow through this wire as shown by the following orange arrows now when our electric current travels through the loops of wire it will create a magnetic field within the loop as shown by the following blue magnetic field lines in the same way that magnetic field is created within a solenoid now because our electric current is changing because it's alternating that means the magnetic field produced will also be an alternating or a change in magnetic field and that implies that a changing magnetic flux will exist inside the loops of coil now changing magnetic flux will induce an EMF within a coil of wire so that will induce an emf within this coil of wire and that induced e-m-f will in turn induce an electric current within that loop of wire now inductance is equal to the product of the number of loops of wire found in this particular coil let's say that's given by n multiplied by the magnetic flux within these loops of wire given by Phi M divided by I the electric current that is traveling through these wires given by I now this is known as our inductance and inductance has the unit's given by Henry's which is given by uppercase H Faraday's law of electromagnetic induction by Faraday's law we know that our induced EMF within our coil is equal to the negative of the product of the number of loops given by and multiplied by the derivative of our magnetic flux with respect to time purpose of an inductor when we take an inductor which is essentially a coil of wire and place it inside an electric circuit with some source of voltage

galvanometer

device that is able to detect an electric current

Electric Generators aka AC Generator

electric generators are responsible for producing alternating electric current North Pole and a South Pole and these magnets will essentially produce an external magnetic field given by B external that will begin on the North Pole and will point towards the South Pole anytime we have a moving conducting loop of wire inside a magnetic field that will induce an EMF and that EMF is known as motional EMF so as a result of that induced EMF and alternating electric current will begin to flow within the following coil of wire and that induced EMF can be read between the following two points a b Will use Faraday's law of electromagnetic induction to show that our induced EMF varies sinusoidally and that creates a sinusoidal electric current that is it creates an alternating electric current suppose that the coil is rotating with an angular velocity given by Omega

power loss inside resistors P=I²R P=IV

is equal to I square times R where R is the resistance and I is our electric current if we increase the voltage will decrease the electric current and that will decrease the power loss alternating electric current is generated inside power plants and that is generated via a certain amount of voltage now the voltage is first increased via a step-up transformer so that the electric current inside the Pera lines travels via a very high voltage now before that voltage gets the city we use a step-down transformer to decrease the voltage back to a usable quantity

Eddy Currents

result of electromagnetic induction Showing a metal disc that is rotating about an axis of rotation and it's rotating in a counterclockwise direction as shown by the following two arrows as our disk rotates through this region that has a magnetic field there is an EMF that is induced within that region and that EMF will in turn induce an electric current that will flow through that region those electric currents are known as a t currents and by Lenz's law we can determine the direction of those electric currents right hand rule number 1 the induced current will point in a counter clockwise direction orient our fingers we curl our fingers so that they come out of the board as shown by the following motion we extend the thumb and the thumb will create the following electric current that will point in the counterclockwise direction and this induced electric current is known as an eddy current

Motional EMF Continued

since the length of the loop and the length of our movable metal rod is equal to L implies the change in our area of this initial loop of wire is given by the orange region so we take the height given by L multiplied by the width given by V times DT and that gives us our infinitely small change in area that takes place as a result of our moving conducting metal because this initial loop of conducting wire experiences our change in area as a result of our moving metal rod that means there is a changing magnetic flux that is taking place and because we have a change in magnetic flux that implies by Faraday's law of induction there will be an induced EMF inside our conducting wire and an induced electric current will flow through this conducting wire electrons will begin to move

Diagram on Magnetic Flux

square loop of wire as shown and the side length of the square loop of wire is given by L now the area of this loop of wire is given by taking the product of L and L so L squared gives us the magnitude of our area vector area vector whose Direction is always perpendicular to the face (aka field) of our loop so the angle between this loop and our area vector is always 90 degrees and the angle theta that we described in this equation is the angle between our magnetic field vector and our area vector if the angle is zero that basically assume that our magnetic flux is equal to the dot product of these two vectors (B*A) which is simply the area of our loop of wire multiplied by the cosine of the angle when the angle is zero degrees that implies our magnetic flux is at a maximum and that is equal to B times A on the other hand if we take our loop of wire and we orient our area vector so that it points at a 90 degree angle with respect to our magnetic field lines that will imply that cosine of 90 is zero so our magnetic flux will be zero and we see that the minimum quantity of magnetic flux is equal to zero

Electric Generators aka AC Generator: Continued E=

suppose that the coil is rotating with an angular velocity given by Omega angular velocity is equal to the rate of change of angular displacement with respect to time Omega is equal to D theta divided by G time where theta is our angular displacement induced EMF varies sinusoidally and that's exactly why our electric current also varies sinusoidally with respect to time so it alternates back and forth induced EMF due to n loops of wire is given by the following equation also known as our amplitude voltage it's the peak EMF it's the highest value that our voltage attains and it's given by e with the knot symbol on the bottom

Faraday Showed:

the more rapidly the magnetic field changes the greater than deuced EMF and he was able to show that the magnitude of the induced EMF also depends on the area of the loop of wire through which that changing magnetic field travels through which the magnetic field lines travel now

Calculating Current using Lenz's Law |E|= I=

Lenz's law states that an electric current created by an induced EMF points in a direction so that the magnetic field it produces opposes the change in magnetic flux tells us the direction of our induced electric current inside our conducting wire If our magnetic flux decreases then our electric current will point in such a direction so that our magnetic field produced by that induced electric current points in the opposite direction as the the original magnetic field If we increase our magnetic flux then the direction of the electric current will point in such a direction so that our magnetic field produced by that new electric current points in the opposite direction of the original magnetic field EXAMPLE: magnetic flux decreases EMF also decreases Lenz's law tells us that an electric current will be induced within our electric wire and the electric current will point in such a direction so that the magnetic field it produces opposes the change in flux By decreasing the area of the loop we are decreasing the magnetic flux and hence because there is a change in magnetic flux there is an EMF that is induced now this induced EMF will in turn produced and an electric current that will point in the direction that opposes the change in flux since the flux decreases the induced electric current, it will create a magnetic field B that will increase it so therefore the induced magnetic field will point into the board and I is clockwise flux will decrease and this electric current will try to increase that flux back to its original amount

Lenz's Law: Increasing the angle (from zero) between B and A and seeing how the magnetic flux changes

Lenz's law tells us that the induced magnetic field b will point in the same direction as our external magnetic field and that's because we have a decreasing magnetic flux right hand and now we wrap our hand around our wire so that our fingers curl in the same direction as the induced magnetic field which points out of the board in the same direction as the external magnetic field so we twisted this way we extend the thumb and in this case the thumb points in the following counterclockwise direction because our magnetic flux is not changing that implies that there is no induced EMF and so there is no induced electric current inside our loop of wire remember only a change in flux will induce an electric current inside our closed loop of wire

Electromagnetic Induction,

an electric current moving inside a conducting wire will produce a magnetic field only a changing magnetic field a non-uniform magnetic field can induce an electric current a uniform or a constant magnetic field that is produced when we have a constant electric current does not induce an electric current so electromagnetic induction is essentially the process by which we induce an EMF an electric potential difference using a changing magnetic field

Transformers Vs/Vp= IpVp=

Within power plants the electric generators usually create an EMF that has a very high value and before households and businesses can use that voltage that voltage must be decreased within the power lines and the devices that decrease the voltage are known as transformers the reason we laminate the iron core is to reduce the occurrence of eddy currents our transformer so our electric generator is found somewhere in that region it creates that alternating electric current which then travels through the following while you're eventually reaching the primary coil as that changing electric current passes through our loops of wire it essentially creates a changing magnetic field in the same way that a magnetic field is created inside a solenoid so that change in magnetic field then travels to the following region of our iron and when it travels through the loops of the secondary wire changing magnetic field induces a change in magnetic flux and by Faraday's law whenever we have a change in magnetic flux that will essentially induce an EMF inside the wire so an EMF a voltage will be induced in the secondary coil and that voltage will have the same exact frequency as the voltage in the primary coil equation number one describes the primary coil while equation number two describes the secondary coil Vp= Primary voltage Vs= Secondary Voltage These two quantities are exactly the same ratio of the secondary voltage to the primary voltage is equal to the ratio of the secondary number of loops to the primary number of loops and this is known as the transformer equation only works as long as we have an alternating electric current if our voltage on the secondary is higher than the voltage on primary that means we have a greater number of coils on the secondary coil than on the primary coil now these voltages can be either RMS voltages or the peak voltages RULES ON IMAGE: two different types of transformers we have step-up (voltage is essentially increased) transformers and step-down (voltage will decrease) inside an ideal transformer the amount of power that is inputted is equal to the amount of power that is outputted because power is equal to the product of the electric current and the voltage we see that our product of the electric current in the primary coil multiplied by the voltage in the primary coil is equal to the electric current in a secondary coil multiplied by the voltage in the secondary coil this is known as our power equation for transformers

Two things Determined by Faraday's experiment

a direct current produces a constant magnetic field such a magnetic field does not induce an electric current in wire number two at the moment that we connect or disconnect our battery from wire number one we see that an electric current is produced that changes that is not constant and this non constant electric current produces a non constant or non uniform magnetic field and we see that a non uniform magnetic field will in fact produce an electric current which will be picked up by our galvanometer

Energy Storage in LC Circuits and Electromagnetic Oscillations

a time of zero seconds all that energy is stored inside our plates of our capacitor but as time progresses as time increases the charge is discharged from the plate of the capacitor and energy begins to store inside the magnetic field of our inductor inside a capacitor energy is stored within the electric field produced as a result of our separation electric charge between the two plates of our capacitor inside our inductor energy is stored inside the magnetic field produced as a result of the flow of an alternating electric current through the loops of our inductor energy stored inside the capacitor is equal to Q squared divided by 2 C where C's the capacitance and Q is the electric charge Q is equal to the following equation Q naught cosine of our Omega T plus Phi where Phi is our phase angle T is the time Omega is the angular velocity and Q naught is the quantity of charge stored on the plates of the capacitor at time of 0 seconds

SI units of Magnetic flux

are T m² = Wb (Weber)

Using RHR#2 to determine the direction of force in an eddy current

we extend it in the same direction as the motion of our eddy current so at this point it points up then we curl our fingers in the direction of our external magnetic field which is into the board we extend our thumb and the thumb points in the direction of our magnetic force so that means by right-hand rule number two the force will point to the left this magnetic force will create a torque that will point in the opposite direction of the initial torque that creates that rotation so our initial torque points in the counterclockwise direction as shown but this force will create a torque that points in the clockwise direction and that will oppose this initial torque now this will in turn slow the rotation of our motion sphere is initially raised it to a location where there is no magnetic field so you raise it to this location and then we let go it will begin to oscillate back and forth it will begin to travel now notice initially when it enters our magnetic field as we saw in this region when our disc rotates from point one to point two there will be an increase in our magnetic flux and that basically means we're going to have an eddy current that will feel a force a magnetic force that points in the opposite direction of the velocity of the motion and that force will impede the motion of our simple pendulum enters this location (no magnetic field) when it goes from our magnetic field region to no magnetic field region there is a decrease in our magnetic flux and that basically means that instead of being counter clockwise our direction of the current of the eddy current will be in the clockwise direction pendulum is moving back and forth because of the presence of this magnetic field and because this is a conducting object the eddy currents will essentially act to create a force that will oppose the force that creates the motion and so that means eventually our Bob will slow down

Inductance of Solenoid

when our electric current is traveling through the loops of wire that will create a magnetic field inside our solenoid and because our current is alternating that means our magnetic field will also be changing and that will create our changing magnetic flux now by Faraday's law of electromagnetic induction that will basically imply that an induced EMF will be created within our loops of wire and that is known as self inductance or simply inductance magnetic field B is equal to the permeability of free space from you not x and the number of loops of wire multiplied by I our electric current divided by L the length of our thin long solenoid B= M0NI/L

Lenz's Law: Loop increasing in size; effect of magnetic field and magnetic flux

with the right hand rule #1: curl our fingers in the same direction as our induced magnetic field witch's into the board so we wrap our hand around the wire and curl the fingers into the board extend the thumb and the thumb points in the clock in the clockwise direction

Electrical Motor Equations

ε₀= NBAωsin E=NBAωsin (ωt) ω= 2πf ω " omega" y-axis is the voltage it's the induced EMF and the x-axis is the time we see that our EMF changes sinusoidally so we have pink EMF in the following two locations