Physics ― Electromagnetism

Transformers and Power Transmission (ch 33)

*Vˢ[out] = (Nˢ/Nᵖ) Vᵖ[in]* If Nˢ > Nᵖ or Vˢ > Vᵖ, then it is a STEP-UP transformer. If Nˢ < Nᵖ or Vˢ < Vᵖ, then it is a STEP-DOWN transformer.

Resonance of an AC Circuit (ch 33)

A series RLC circuit is an electrically oscillating system. Such a circuit is said to be IN RESONANCE when the driving frequency is such that the rms current has its maximum value. In general, the rms current can be written as: Iʳᵐˢ = Vʳᵐˢ / Z == Vʳᵐˢ / √[R² + (Xᴸ - Xᶜ)²] The angular frequency ω₀ at which Xᴸ - Xᶜ = 0 is called the *RESONANCE FREQUENCY* of the circuit: *ω₀ = 1 / √LC*

Electromagnetic Waves, Displacement Current, and the General Form of Ampère's Law (ch 34)

Ampère's original law does not account for time-varying situations. James Clerk Maxwell updated Ampère's Law to account for it. This new statement includes a factor called the DISPLACEMENT CURRENT, which is: *Iᵈ = ε₀ (dΦᴱ/dt) == dq/dt* THIS IS NOT CONDUCTION CURRENT, which is the 'normal' current carried by charge carriers in a conductor, given by μ₀I. Combining the conduction and displacement currents gives the corrected and full form of Ampère's Law: *∮⟨B⟩ ∙ d⟨s⟩ = μ₀(I + Iᵈ) == μ₀I + μ₀ε₀(dΦᴱ/dt)*

Gauss's Law (ch 24)

Consider a positive point charge 'q' located at the center of a sphere of radius 'r.' We know from Coulomb's Law that the magnitude of the electric field everywhere on the surface of that sphere is E = kₑq/r². The field lines are directed radially outward and hence perpendicular to the sphere's surface at every point on that surface. That also means ⟨E⟩ is parallel to the vector ∆⟨A⟩ᵢ representing a local element of area ∆Aᵢ surrounding the surface point. Therefore, *⟨E⟩ ∙ ∆⟨A⟩ᵢ = E ∆Aᵢ* And since the Gaussian surface is a closed surface, electric flux is *Φᴱ = ∮⟨E⟩ ∙ d⟨A⟩ = E ∮dA* 'E' has moved outside the integral because, by symmetry of the spherical surface, 'E' is constant all over the surface. The value of E = kₑq/r². Since the surface is spherical, ∮dA = A = 4πr². Recalling that kₑ = 1/4πε₀, this allows for the electric flux to be expressed as: *Φᴱ = q / ε₀* This is *THE NET FLUX THROUGH ANY CLOSED SURFACE SURROUNDING A POINT CHARGE, which is INDEPENDENT OF THAT SURFACE'S SHAPE.* Gauss's Law is officially written as: *Φᴱ = ∮⟨E⟩ ∙ d⟨A⟩ = qᵢₙ / ε₀* Where 'qᵢₙ' is the sum of the charges inside the surface. Keep in mind, while 'qᵢₙ' refers to the charges within the surface, ⟨E⟩ is the TOTAL electric field.

Law of Conservation of Electric Charge (KA)

If any set of charged particles is broken up (by decay, collision, etc) into any number of other particles, the total charge of the initial particles will be THE SAME as the total charge of the final particles. Example: If a photon with 0 Coulombs collides with a surface and breaks into an electron and a positron, the electron and positron can have any charge so long as they are opposite of each other and add to 0 Coulombs.

"A Model for Electrical Conduction" or Drift Velocity and Resistivity (ch 27)

In a classical model of electrical conduction in metals, the electrons are treated as molecules of gas. In the absence of an electric field, the average velocity of the electrons is zero. When an electric field is applied, the electrons move (on average) with a *drift velocity* ⟨vᵈ⟩ that is opposite the electric field. The drift velocity is given as: *⟨vᵈ⟩ = T q⟨E⟩/mₑ* Where: : q = Electron's charge : mₑ = Mass of the electron : T = Average time interval between electron-atom collisions According to this model, the resistivity of the metal is given as: *ρ = mₑ/nq²T* Where: : n = Number of free electrons per unit volume

Capacitors in an AC Circuit (ch 33)

In a simple C-circuit, Kirchhoff's loop rule is written ΔV - q/C = 0, or ΔV = q/C Replacing ΔV with Vᵐᵃˣ sin(ωt) gives: *Vᵐᵃˣ sin(ωt) = q/C* Therefore, the charge in the capacitor is: *q = C Vᵐᵃˣ sin(ωt)* Differentiating this with respect to time gives the current at any moment: *I = dq/dt == ω C Vᵐᵃˣ sin(ωt)* Therefore, maximum current in a capacitor is: *Iᵐᵃˣ = ω C Vᵐᵃˣ == Vᵐᵃˣ / (1/ωC)* == Vᵐᵃˣ / Xᶜ Because (1/ωC) depends on the applied frequency ω, the capacitor 'reacts' differently for differing frequencies in terms of offering opposition to current. This means that (1/ωC) has the same units as resistance! For this reason, (1/ωC) is defined as *CAPACITIVE REACTANCE*: *Xᶜ = 1/ωC* Voltage across the capacitor is then: *ΔV = Vᵐᵃˣ sin(ωt) == IᵐᵃˣXᶜ sin(ωt)* Current and voltage in an AC capacitor are *OUT OF PHASE* with each other, and *CURRENT LEADS VOLTAGE* by π/2 radians (90 degrees).

Inductors in an AC Circuit (ch 33)

In a simple L-circuit, Kirchhoff's loop rule is written ΔV - L(dI/dt) = 0, or ΔV = L(dI/dt) Replacing ΔV with Vᵐᵃˣ sin(ωt) gives: *Vᵐᵃˣ sin(ωt) = L(dI/dt)* Solving this to get current: (Vᵐᵃˣ/L) sin(ωt) dt = dI ∫(Vᵐᵃˣ/L) sin(ωt) dt = ∫dI *I = -(Vᵐᵃˣ/Lω)t cos(ωt)* == (Vᵐᵃˣ/Lω)t sin(ωt - π/2) Therefore, maximum current in an inductor is: *Iᵐᵃˣ = Vᵐᵃˣ/ωL* == Vᵐᵃˣ/Xᴸ Because ωL depends on the applied frequency ω, the inductor 'reacts' differently for differing frequencies in terms of offering opposition to current. This means that ωL has the same units as resistance! For this reason, ωL is defined as *INDUCTIVE REACTANCE*: *Xᴸ = ωL* Voltage across the inductor is then: *ΔV = -L(dI/dt) == -Vᵐᵃˣ sin(ωt) == IᵐᵃˣXᴸ sin(ωt)* Current and voltage in an AC inductor are *OUT OF PHASE* with each other, and *CURRENT LAGS VOLTAGE* by π/2 radians (90 degrees).

[Analysis Model] Particle in a Magnetic Field (ch 29)

In addition to containing an electric field, the region of space surrounding any MOVING electric charge also contains a magnetic field. ⟨B⟩ is the traditional variable used to represent a magnetic field (recall that ⟨E⟩ is used for an electric field). The magnetic field is measured in units of *1 Tesla = 1 N / Cm/s == 1 N / Am* Just like electric force (Fᴱ) was measured by using a test charged particle, the magnetic force (Fᴮ) can be measured in the same way. *⟨Fᴮ⟩ = q⟨v⟩ × ⟨B⟩* and *|| Fᴮ || = qvB sinΘ* Where: : Θ = Smallest angle between ⟨v⟩ and ⟨B⟩ *Similarities between Fᴱ and Fᴮ are:* * Fᴮ is proportional to the speed (v) of the particle. * Fᴮ on a negative charge is directed opposite to the force on a positive charge moving in the same direction. * Fᴮ is proportional to the magnitude of the magnetic field vector ⟨B⟩. *Differences between Fᴱ and Fᴮ are:* * Fᴮ is proportional to the speed (v) of the particle. * If the velocity vector makes an angle Θ with the magnetic field, the magnitude of Fᴮ is proportional to sinΘ. * When a charged particle moves PARALLEL to the magnetic field vector, the Fᴮ on the charge is 0. * When a charged particle moves in a direction NOT PARALLEL to the magnetic field vector, the Fᴮ acts in a direction perpendicular to both the velocity and magnetic field vectors (ie. the Fᴮ is perpendicular to the plane formed by ⟨v⟩ and ⟨B⟩). *How to use the Right-Hand Rule:* 1) Point your hand in the direction of ⟨v⟩ with your palm facing ⟨B⟩ 2) Curl fingers toward ⟨B⟩ 3) Thumb will point in the SAME direction of ⟨Fᴮ⟩ for a +q charge, and it will point in the OPPOSITE direction for a -q charge.

Motional emf (ch 31)

Motional emf is the emf induced in a conductor that is moving through a constant magnetic field. Using Faraday's Law and noting that x changes with time at a rate dx/dt = v, we find that the induced motional emf is: *emf = -dΦᴮ/dt == -d[BLx]/dt == -BL(dx/dt) == -BLv* Where: : Lx = Area enclosed by the circuit created by the moving conductor The magnitude of the induced current is: *I = |emf| / R == BLv / R* Where: : R = Resistance in the circuit (that is stationary) The power delivered by the applied force that is in equal and opposite direction of the magnetic force on the moving conductor is given as: *P = Fᵃᵖᵖv == (ILB)v == (BLV)²/R = E²/R*

Mutual Inductance (ch 32)

Often, the magnetic flux through the area enclosed by a circuit varies with time because of time-varying currents in nearby circuits. This condition induces an emf through a process known as *mutual induction,* so named because it depends on the interaction of 2 circuits. In two nearby coils of wire, the current in coil 1 will have a magnetic flux that passes through coil 2, and vice versa. The mutual induction can be expressed as: *M₁₂ = N₂Φ₁₂/I₁* The emf induced in each coil is expressed as: *emf₂ = -M (dI₁/dt)* and *emf₁ = -M (dI₂/dt)*

Capacitance (ch 26)

The capacitance (C) of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors: *C = Q / ∆V* Capacitance is ALWAYS POSITIVE, and is defined as 1 Farad (F) = 1 Coulomb (C) / Volt (V) Capacitance of... ...an isolated charged sphere: C = 4πε₀a ...parallel plates: C = ε₀A/d ...a cylindrical capacitor: C = L / [ 2 kₑ ln(b/a) ] ...a spherical capacitor: C = ab / [ kₑ(b-a) ] Potential energy stored in a charged capacitor: *U or W = 1/C ∫[0,Q] q dq == Q²/2C == 1/2 C (∆V)²* == 1/2 QV

Electromotive Force (ch 28)

The electromotive force is unfortunately named, because it is not a force at all, but rather a potential difference in volts (aka. a voltage). *emf = IR + Ir* *I = emf / R+r* Power: *Iemf = I²R + I²r*

Energy in a Magnetic Field (ch 32)

The total energy stored in the inductor of an RL circuit is given as: *Uᴮ = LI²/2* The magnetic energy density of an inductor (ie. the energy stored per unit volume in the magnetic field of the inductor) is given as: *uᴮ = B²/2μ₀*

Alternating Current & RMS Values (ch 33)

Voltage also alternates in accordance with the following sinusoidal function: *v(t) = Vᵐᵃˣ sin(ωt)* Where: ω = 2πf == 2π/T 66 Hz = 377 rad/s To convert to RMS values: Iʳᵐˢ = Iᵐᵃˣ/√2 == 0.707Iᵐᵃˣ Vʳᵐˢ = Vᵐᵃˣ/√2 == 0.707Vᵐᵃˣ RMS values are used commonly, and when a voltage is given for an outlet it is in RMS. The max voltage of an outlet is actually about 170V.

Electric Current (ch 27)

Whenever there is a net flow of charge through some region, an electric current exists. *Iᴬⱽᴳ = ∆Q / ∆t == dQ / dt == nqvᵈA* Where: : q = Charge on each carrier : A = Cross-sectional area of the conducting material : vᵈ = Drift speed, or average speed of all the charge carriers passing through the area (A) : n = Number of mobile charge carriers per unit volume (ie. the charge carrier density) The current is measured as 1 Coulomb (C) / Second (s) = 1 Ampere (A) Current density (J) is defined as: *J = I / A == nqvᵈ*, with units A/m²

The Magnetic Field of a Solenoid (ch 30)

∮⟨B⟩ ∙ d⟨s⟩ = BL == μ₀NI So, the magnetic field of a solenoid is given as: *B = μ₀NI/L == μ₀nI* Where: : n = N/L For a solenoid, n = N/2πr

Conductors in Electrostatic Equilibrium (ch 24)

*ELECTROSTATIC EQUILIBRIUM* occurs when there is no net motion of charge within a conductor. This conductor in equilibrium has the following properties: 1) The electric field is zero EVERYWHERE INSIDE the conductor, whether the conductor is solid or hollow. 2) If the conductor is isolated and carries a charge, the charge resides on its surface. 3) The electric field at a point just outside a charged conductor is perpendicular to the surface of the conductor and has a magnitude ρ/ε₀, where 'ρ' is the surface charge density at that point. 4) On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature of the surface is smallest. Due to these properties, the electric field immediately outside a charged conductor is given as: *E = σA / ε₀*

Ohm's Law, Electric Resistance, and Electric Power (ch 27)

== RESISTANCE ================ When a material's current density is proportional to the electric field, it is said to follow Ohm's Law: *J = σE* Where: : J = I / A == nqvᵈ, with units A/m² : σ = Conductivity of the conductor : E = Electric field strength Substitutions can be made to obtain the more popular version of the equation: ∆V = EL == JL/σ == IL / σA == RI Which cleans up to: *R = ∆V / I* Resistance is measured as 1 Ohm (Ω) = 1 Volt (V) / Ampere (A) Resistivity is the inverse of conductivity: ρ = 1 / σ Resistance of a uniform material along its length: *R = ρL / A* The resistivity of a conductor varies approximately linearly with temperature according to the expression given as: *ρ = ρ₀ [1 + α(T - T₀) ]* Where: : ρ₀ = Resistivity at some reference temperature T₀ : *α =* Temperature coefficient of resistivity == *∆ρ/ρ₀∆T* Because resistance is proportional to resistivity, the variation of resistance of a sample is given as: R = R₀ [1 + α(T - T₀) ] Where: : R₀ = Resistance at some reference temperature T₀ == POWER ================ The rate at which energy is delivered to a resistance is given as: P = dU/dt == d/dt(Q∆V) == dQ/dt(∆V) == I∆V Which cleans up to: *P = I∆V == I²R == (∆V)² / R*

Series and Parallel Capacitance (ch 26)

== SERIES CAPACITANCE ================ It is reciprocal because the electric potential (V) must contribute to each capacitor before traveling onto the next: *1/C = 1/C₁ + 1/C₂ + ... + 1/Cᵢ* == PARALLEL CAPACITANCE ================ It is additive because each parallel portion of the circuit receives the same amount of potential difference (∆V): *C = C₁ + C₂ + ... + Cᵢ*

Resistors in Series and Parallel (ch 28)

== SERIES RESISTANCE ================ The same amount of charge (q) passes through series resistors in a given time interval, and the currents are the same in all series resistors: *I = I₁ = I₂ = ... = Iᵢ* The potential difference applied across the series combination is additive: *∆V = ∆V₁ + ∆V₂ + ... + ∆Vᵢ == I₁R₁ + I₂R₂ + ... + IᵢRᵢ* or ∆V = IR[eq] Resistance in series is additive, so an equivalent resistance can be found by combining all series resistances: *R[eq] = R₁ + R₂ + ... + Rᵢ* == PARALLEL RESISTANCE ================ Charge must split at parallel junctions, so the current also must split: *I = I₁ + I₂ + ... + Iᵢ = ∆V₁/R₁ + ∆V₂/R₂ + ... + ∆Vᵢ/Rᵢ* or I = ∆V/R[eq] The potential difference applied across a parallel combination is all the same: *∆V = ∆V₁ = ∆V₂ = ... = ∆Vᵢ* Using the definition of current for a parallel resistance, the expression can be divided by ∆V in order to get the equivalent resistance for the parallel combination: ∆V/R[eq] = ∆V₁/R₁ + ∆V₂/R₂ + ... + ∆Vᵢ/Rᵢ —> *1/R[eq] = 1/R₁ + 1/R₂ + ... + 1/Rᵢ*

Motion of a Charged Particle in a Uniform Magnetic Field (ch 29)

A charged particle in a constant/uniform magnetic field (B) will move in a circular motion, because the magnetic force (Fᴮ) created on that particle by the magnetic field is ALWAYS perpendicular to the magnetic field. Since the particle is moving in a circular motion, the [PARTICLE IN UNIFORM CIRCULAR MOTION] model can be used here. The angular speed (ie. the rate) and the period (ie. the time) of circular motion DO NOT DEPEND ON THE RADIUS OF THE ORBIT OR THE DIRECTIONAL MOTION (ie. forward, backward, up, down) OF THE PARTICLE. Recall that Fᴮ = qvB for a perpendicular magnetic field. This means that the direction of spin is COUNTERCLOCKWISE if it is a +q, and CLOCKWISE if it is a -q. Since this is a [PARTICLE IN UNIFORM CIRCULAR MOTION], as well as a [PARTICLE UNDER A NET FORCE], the magnetic force for the particle can be written as such: *Fᴮ = qvB == ma == mv²/r* This implies that the radius of this circular motion is as follows: *r = mv/qB* The angular speed (aka. the "cyclotron frequency"): *ω = v/r == qB/m* The period: *T = 2πr/v == 2π/ω == 2πm/qB* If a charged particle moves in a uniform magnetic field with its velocity at some arbitrary angle with respect to ⟨B⟩, its path is a helix. For example, if the field is directed in the x-direction, there is no component of force in the x-direction. As a result, aₓ = 0, and the x-component of velocity remains constant. This means that the particle is a [PARTICLE IN EQUILIBRIUM] in this direction. However, the magnetic force q⟨v⟩ × ⟨B⟩ causes the components vʸ and vᶻ to change in time, and the resulting motion is a helix with its axis parallel to the magnetic field. The projection of the path onto the yz-plane is a circle, and the projections onto the xy-plane and xz-plane are sinusoids. THE ABOVE EQUATIONS STILL APPLY PROVIDED v IS REPLACED WITH v⊥ = √v̅[̅y̅]̅²̅ ̅+̅ ̅v̅[̅z̅]̅²̅ A charged particle moving in the presence of both an electric field and a magnetic field is described by two [PARTICLE IN A FIELD] models. The total force (called the Lorentz force) is as such: *⟨F⟩ = q⟨E⟩ + q⟨v⟩ × ⟨B⟩*

RC Circuits (ch 28)

A circuit containing a series combination of a resistor and a capacitor. τ = RC == CHARGING A CAPACITOR ================ Charge as a function of time: *q(t) = C(emf)(1 - e⁽⁻ᵗ/ʳᶜ⁾) == Qᵐᵃˣ(1 - e⁽⁻ᵗ/ʳᶜ⁾)* Current as a function of time: *i(t) = emf/R e⁽⁻ᵗ/ʳᶜ⁾* Where: : Qᵐᵃˣ = C(emf) is the max charge on the capacitor == DISCHARGING A CAPACITOR ================ Charge as a function of time: *q(t) = Qᵢ e⁽⁻ᵗ/ʳᶜ⁾* Current as a function of time: *i(t) = -Qᵢ/RC e⁽⁻ᵗ/ʳᶜ⁾* Where: : Qᵢ = Initial charge on the capacitor : Qᵢ/RC = Initial current in the circuit

Capacitors with Dielectrics (ch 26)

A dielectric is a nonconducting material. They are rated by a dimensionless factor called the dielectric constant (κ). If a dielectric is inserted in between the plates of a parallel-plate capacitor, there is a change in electric potential (ie. there is a potential difference) given as: *∆V = ∆V₀ / κ* Because the charge Q on the capacitor does not change, the capacitance must change: *C = Q/∆V == Q/[∆V₀ / κ] == κ Q/∆V₀ == κC₀* Because C₀ = ε₀A/d, the capacitance of a parallel-plate capacitor with an inserted dielectric is given as: C = κ ε₀A/d The net electric field in a dielectric has a magnitude of E = E₀ - Eᶦⁿᵈ, and therefore the dielectric has a net charge density of: σᶦⁿᵈ = (κ-1 / κ) σ

[Analysis Model] Particle in an Electric Field (ch 23)

An *electric field* exists in the region of space around a charged object, the *source charge.* The presence of an electric field can be detected by placing a *test charge* in the field and noting the electric force on it. The electric field due to the source charge at the location of the test charge is the electric force on the test charge per unit charge, or rather the *electric field vector* at a point in space is defined as: *⟨E⟩ = ⟨Fᴱ⟩ / q₀* Where: : *⟨E⟩ = Newtons per Coulomb* The direction of ⟨E⟩ is the direction of the force a test charge experiences when placed in the electric field. Just like the "Particle in a Gravitation Field" model ― where ⟨Fᶢ⟩ = m⟨g⟩ ― the Particle in an Electric Field model is given as: *⟨Fᴱ⟩ = q⟨E⟩* → ⟨Fᴱ⟩ = kₑ (qq₀/r²) ⟨r⟩ → *⟨E⟩ =* ⟨Fᴱ⟩ / q₀ == *kₑq/r² ⟨r⟩* The magnitude of the electric field at ANY POINT around the source charge can be found by: *E = kₑ q/r²* If ⟨Fᴱ⟩ is the only force acting upon a particle, it must be the net force, and it causes the particle to accelerate according to Newton's laws: *⟨a⟩ = q⟨E⟩/m*

RLC Circuits (ch 32)

An RLC circuit contains a resistor, capacitor, and inductor connected in series. The RLC circuit is analogous to the damped harmonic oscillator. For a block-spring system, this was given as: m (d²x/dt²) + b (dx/dt) + kx = 0 Comparing equations, q corresponds to the position x, L to the mass m, R to the damping coefficient b, and C to 1/k where k is the force constant of the spring. When resistance (R) is small, a situation that is analogous to light damping in the mechanical version of the oscillator occurs, given as: *q = Q e⁻ᴿᵗ/²ᴸ cos (ωᵈt)* Where: : ωᵈ = Angular frequency at which the slightly damped circuit oscillates == √[ (1/LC) - (R/2L)² ]

Electric Dipole in an Electric Field (ch 26)

An electric dipole consists of 2 charges of equal magnitude and opposite sign separated by a distance of 2a. The electric dipole moment is defined as the vector ⟨p⟩ directed from -q toward +q along the line between the charges and having a magnitude of: *⟨p⟩ = 2aq* Suppose an electric dipole is placed in a uniform electric field ⟨E⟩ that makes an angle Θ with the field. The forces acting upon both charges are equal in magnitude but opposite in direction! This creates a net torque at the center of the dipole, and therefore it begins to rotate as a Rigid Object Under Net Torque model. *τ = 2Fa sin Θ == 2aqE sin Θ == pE sin Θ* *⟨τ⟩ = ⟨p⟩ × ⟨E⟩* The potential energy of the system of an electric dipole in an external electric field is given as: *Uᴱ = -pE cos Θ == -⟨p⟩ ∙ ⟨E⟩*

Faraday's Law of Induction (ch 31)

An emf is induced in a loop when the magnetic flux through the loop changes with time. In general, this emf is directly proportional to the time rate of change of the magnetic flux through the loop. This is expressed as: *emf = -dΦᴮ/dt* In a coil, just multiply by the number of loops, N. So emf for a coil then becomes: -NdΦᴮ/dt

Charging by Induction (ch 23)

Causing a charge within a 'destination' object by bringing a charged 'source' object close to the destination object, which is connected to grounding. This happens because the overabundance of electrons within the source object repel the electrons within the initially-neutral destination object, causing those electrons to flow into the grounding. Then, the grounding is severed so that electrons can not flow back into the destination object, thereby leaving the destination object charged without ever having physically touched it.

Momentum and Radiation Pressure (ch 34)

EM waves transport linear momentum as well as energy. As this momentum is absorbed by some surface, pressure is exerted on the surface. If an EM wave is COMPLETELY ABSORBED, then the momentum ⟨p⟩ has a magnitude: *p = Tᴱᵂ / c* Where: : p = Momentum : Tᴱᵂ = Total energy of the EM wave : c = Eᵐᵃˣ / Bᵐᵃˣ == E/B The PRESSURE on that surface is defined as: *P = F/A = 1/A dp/dt* Substituting p in for d(p)/dt above arrives at the following pressure for complete EM wave absorption: *P = S/c* For COMPLETE REFLECTION, the momentum is: *p = 2Tᴱᵂ / c* Same with pressure for complete EM wave reflection: *P = 2S/c*

Properties of Electric Charge (ch 23)

Electric charge is *always conserved* in an isolated system. Charge is transferred from one object to another. Electric charge exists in discrete "packets" which can be written as: *q = Ne* Where: : q = The quantized electric charge : N = Some integer multiplier : e = Fundamental constant (1.6x10⁻¹⁹ Coulomb) The Coulomb is defined as: 1 C = 6.24x10¹⁸ e

RL Circuits (ch 32)

If a circuit contains a coil such as a solenoid, the inductance of the coil prevents the current in the circuit from increasing or decreasing instantaneously. A circuit is an "RL circuit" if the elements it consists of are a battery, a resistor (R), and an inductor (L). Using Kirchhoff's Loop Rule, the circuit would give the expression: emf - IR - L(dI/dt) = 0 Solving for the differential equation: *i = emf/R (1 - e⁻ᴿᵗ/ᴸ)* == emf/R (1 - e⁻ᵗ/ᵀ) Where: ᵀ = ᴸ/ᴿ

Resistors in an AC Circuit (ch 33)

In a simple R-circuit, Kirchhoff's loop rule is written ΔV - IR = 0, or ΔV = IR Replacing ΔV with Vᵐᵃˣ sin(ωt) bears the Ohm's Law for AC circuits: *Vᵐᵃˣ sin(ωt) = IR* Therefore, maximum current in a resistor is: *Iᵐᵃˣ = (Vᵐᵃˣ/R) sin(ωt)* And voltage across the resistor is: *ΔV = IᵐᵃˣR sin(ωt)* And power delivered to a resistor is: *Pᴬⱽᴳ = Iᵐᵃˣ²R* Current and voltage in an AC resistor are *IN PHASE* with each other.

Generators & Motors (ch 31)

In an AC generator, the magnetic flux through the rotating coil (of N turns) between the magnets at any given time is: *Φᴮ = NBAcosΘ == NBAcos(ωt)* Where: : ω = Angular speed of the rotating coil The induced emf in the coil is given as: *emf = -NdΦᴮ/dt == -NBAd[cos(ωt)]/dt == ωNBAcos(ωt)* With a max emf of: ωNBA

RLC Series Circuits (ch 33)

In an RLC circuit, all these components are connected in series. This means that each wave function affects the other wave functions. When the circuit elements are all connected together to an AC source, the voltages across each component at any given time are expressed as: *ΔVᴿ = IᵐᵃˣR sin(ωt) == ΔVᴿ sin(ωt)* *ΔVᴸ = IᵐᵃˣXᴸ sin(ωt + π/2) == ΔVᴸ cos(ωt)* *ΔVᶜ = IᵐᵃˣXᶜ sin(ωt - π/2) == -ΔVᶜ cos(ωt)* Representing these functions as phasors, they can be summed together using vector addition. The resultant vector gives Vᵐᵃˣ, which the value can be determined by finding the magnitude. This results in a modified Ohm's Law for this type of circuit: *Iᵐᵃˣ = Vᵐᵃˣ / √[R² + (Xᴸ - Xᶜ)²]* == Vᵐᵃˣ / Z Since √[R² + (Xᴸ - Xᶜ)²] must have resisting properties, it is known as the *IMPEDANCE* of the circuit: *Z = √[R² + (Xᴸ - Xᶜ)²]* The angle that the resultant vector of Vᵐᵃˣ makes with the positive x-axis is therefore given by: *Θ = arctan[(Xᴸ - Xᶜ) / R]* Average power delivered to an RLC circuit is given as: *Pᴬⱽᴳ = 1/2 IᵐᵃˣVᵐᵃˣ cos Φ == IʳᵐˢVʳᵐˢ cos Φ == IʳᵐˢVʳᵐˢ == Iʳᵐˢ²R* Where: cos Φ = Power Factor

Induced emf & Electric Fields (ch 31)

In the same way that changing magnetic flux induces an emf and a current in a conducting loop, we can relate an induced current in a conducting loop to an electric field by claiming that an electric field is created in the conductor as a result of the changing magnetic flux. A changing magnetic field also generates an electric field in empty space, much like the presence of an electric field is independent of any test charge. However, the induced electric field by a change in magnetic flux is NONCONSERVATIVE, unlike the electrostatic field generated by stationary charges. Faraday's Law of Induction can then be written as such: *emf = ∮⟨E⟩ ∙ d⟨s⟩ == -dΦᴮ/dt*

Kirchhoff's Rules (ch 28)

Often in real-world circuitry, it is not possible to reduce a circuit to a single loop and compute its equivalent properties (potential difference, capacitance, resistance). Kirchhoff's Rules make it possible to analyze these more complex circuits. == JUNCTION RULE ================ At any junction, the sum of the currents must equal zero: *Σ I = 0* In general, the number of times you can use the Junction Rule is one fewer than the number of junction points in a circuit. == LOOP RULE ================ The sum of the potential differences across all elements around any closed circuit loop must be zero: *Σ ΔV = 0* In general, the Loop Rule can be used as often as needed, as long as a new circuit element (resistor or battery) or a new current appears in each new equation. Potential energy of a system DECREASES whenever a charge moves through a potential drop -IR across a resistor, or whenever it moves in the reverse direction through a source of emf. Potential energy INCREASES whenever the charge passes through a battery from the negative terminal to the positive terminal. When applying the Loop Rule, remember these sign conventions: 1) Charges move from the high-potential end of a resistor to the low-potential end, so if a resistor is traversed in the direction of the current, then *ΔV = -IR* 2) If a resistor is traversed in the direction opposite the current, then *ΔV = +IR* 3) If a source of emf (with zero internal resistance) is traversed in the direction of the emf (negative to positive), then *ΔV = +emf* 4) If a source of emf (with zero internal resistance) is traversed in the opposite direction of the emf (positive to negative), then *ΔV = -emf*

Ampère's Law & The Magnetic Force Between Two Parallel Conductors (ch 30)

One wire establishes the magnetic field, while the other wire is modeled as a collection of particles in a magnetic field. The magnetic force (Fᴮ) per unit length of wire (L) is expressed as: Fᴮ / L = μ₀ I₁I₂/2πa The force between two parallel wires is used to define the AMPERE, which is: When the magnitude of the force per unit length between two long, parallel wires that carry identical currents and are separated by 1m is 2x10⁻⁷ N/m, the current in each wire is defined as 1 Amp. To determine the direction of the magnetic field surrounding a current-carrying conductor, point your thumb in the direction of the current, and curl your fingers inward. The direction your fingers curl is the direction of the magnetic field. *AMPERE'S LAW* The line integral of ⟨B⟩ ∙ d⟨s⟩ around any closed path equals μ₀I, where I is the total steady current passing through any surface bounded by the closed path: *∮⟨B⟩ ∙ d⟨s⟩ = μ₀I*

Electric Potential & Electric Potential Energy due to Point Charges (ch 25)

The ELECTRIC POTENTIAL (V) due to a point charge at any distance is given as: *V = kₑ q/r* Careful! This is very similar to the magnitude of the electric field at any given point, which was E = kₑ q/r². For a continuous distribution of charges, the electric potential at any given distance away from that distribution can be found by: *V = kₑ ∫ 1/r dq* The ELECTRIC POTENTIAL ENERGY (EPE) of a pair of point charges can be found as follows: *U = kₑ q₁q₂ / r₁₂* If the system contains more than a pair of charged particles, the total EPE can be calculated by adding up the U for every PAIR of charges. For example, in a 3-charge system, the EPE would be given as: U = kₑ [ (q₁q₂ / r₁₂) + (q₁q₃ / r₁₃) + (q₂q₃ / r₂₃) ]

Energy Carried by Electromagnetic Waves (ch 34)

The amount of energy transferred by EM waves is described as a POYNTING VECTOR, given as: *⟨S⟩ = 1/μ₀ ⟨E⟩×⟨B⟩* The magnitude of ⟨S⟩ represents the power per unit area. The direction is along the direction of wave propagation. The time average of S over one or more cycles is the WAVE INTENSITY, given as: *I = Sᴬⱽᴳ == EᵐᵃˣBᵐᵃˣ / 2μ₀ == Eᵐᵃˣ²/2μ₀c == cBᵐᵃˣ²/2μ₀* Or more simply: *I = ε₀CE²* And average wave intensity: *Iᴬⱽᴳ = 1/2 ε₀CE₀² == C/2μ₀ B₀² == 1/μ₀ E₀B₀* Total instantaneous energy density is then given as: *u = uᴱ + uᴮ == ε₀E² == B²/μ₀* And average energy density: *uᴬⱽᴳ = 1/2 ε₀Eᵐᵃˣ² == Bᵐᵃˣ²/2μ₀*

The Biot-Savart Law (ch 30)

The expression that describes the magnetic field at some point (P) in space in terms of the current that produces the field is given as: *d⟨B⟩ = μ₀/4π [(I d⟨s⟩ × ⟨r⟩) / r²]* Where: : mu0 = Permeability of Free Space = 4π x 10⁻⁷ Tm/A The expression is based on the following observations: * The vector d⟨B⟩ is perpendicular to both d⟨s⟩ (which points in the direction of the current), and to the unit vector ⟨r⟩ directed from d⟨s⟩ toward P. * The magnitude of d⟨B⟩ is inversely proportional to r^2, where r is the distance between d⟨s⟩ and P. * The magnitude of d⟨B⟩ is proportional to the current I and to the magnitude 'ds' of the length element d⟨s⟩. * The magnitude of d⟨B⟩ is proportional to sinθ, where θ is the angle between the vectors d⟨s⟩ and ⟨r⟩. The total magnetic field can be found by integration: *⟨B⟩ = μ₀I/4π ∫[d⟨s⟩ × ⟨r⟩ / r²]*

Lenz's Law (ch 31)

The induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop. That is, the induced current tends to keep the original magnetic flux through the loop from changing. If a magnetic field is directed into the screen, and a conductor in a circuit is moving to the right, then there is a current that moves in a counterclockwise direction, because the opposing magnetic field created by the current is directed out of the screen. If the conductor moves to the left, then the current would be in the clockwise direction.

Electric Potential Energy (EPE), Electric Potential (V), and Potential Difference (∆V) (ch 25)

The internal work done by an electric field on a point charge (a charge-field system) is the negative change in potential energy of that charge. d⟨s⟩ represents an infinitesimal displacement that is oriented tangent to a path through space. WORK DONE by / ELECTRIC POTENTIAL ENERGY of... an electric field on a charge moving from point A to point B will take from the same principles as W = Fd by essentially replacing the variables with their meanings to electromagnetic principles: *∆U or W == F d == (Eq)(d) == -q ∫[A,B] ⟨E⟩ · d⟨s⟩* Where: : E = Magnitude of the electric field : q = Amount of charge (in Coulombs) : d = Distance traveled Notice that the final expression contains a negative. This is because ⟨E⟩ is Coulomb's Law, which includes a 1/r², or r⁻², and this converts into a -1r⁻¹ when integrated. So basically, the negative is there to maintain the integrity of the direction. ENERGY DENSITY in an electric field is given as: *uᴱ = 1/2 ε₀ E²* ELECTRIC POTENTIAL is the total amount of potential energy (U) relative to the charge (q) distribution: *V = U / q* The value of the electric field can be obtained from the electric potential as follows: *Eᵣ = - dV / dr* This can be also found using a gradient: ⟨E⟩ == -∇V == -( ⟨i⟩δ/δx + ⟨j⟩δ/δy + ⟨k⟩δ/δz ) V POTENTIAL DIFFERENCE is the difference of the electric potential between both points (∆V = V[a] - V[b]): *∆V == ∆U / q == -∫[A,B] ⟨E⟩ · d⟨s⟩ == Ed* Be careful... The potential is characteristic of the field only, independent of a charged particle that may be placed in the field. Potential energy is characteristic of the charge-field system due to an interaction between the field and a charged particle placed in the field. WORK DONE is therefore defined as: *W = q ∆V* Where: 1 Volt (V) = 1 Joule/Coulomb (J/C) Extra: 1 Electron Volt (eV) = 1.6x10⁻¹⁹ C·V = 1.6x10⁻¹⁹ J

Magnitude of Particular Magnetic Fields (ch 30)

The magnetic field... At a distance 'r' from a long straight wire carrying an electric current 'I' is: B = μ₀I/2πr Inside a toroid: B = μ₀NI/2πr Inside a solenoid: B = μ₀NI/L == μ₀nI

Magnetic Force Acting on a Current-Carrying Conductor (ch 29)

The magnetic force exerted on a charge moving with a drift velocity within a conductor is simply q⟨vᵈ⟩ × ⟨B⟩. Because the volume of a segment of wire is area times length (AL), the number of charges on that segment is nAL, where 'n' is the charge density (number of charges per unit volume). This means that the magnetic force on any given segment of conductor wire is as such: *⟨Fᴮ⟩ = (nAL) q⟨vᵈ⟩ × ⟨B⟩* Since current can be found by I = nqvᵈA (from Ch 27), the above can be simplified to: *⟨Fᴮ⟩ = I⟨L⟩ × ⟨B⟩* For an arbitrarily-shaped conductor, ⟨L⟩ can be replaced with d⟨s⟩. In which case the above equation becomes: *d⟨Fᴮ⟩ = I ∫[A,B] d⟨s⟩ × ⟨B⟩*

Coulomb's Law (ch 23)

The magnitude of Electric Force between two Point Charges is given by: *Fᴱ = kₑ ( |q₁ q₂| ) / r²* Where: : *kₑ (const) = 1 / 4πε₀ = 8.9876x10⁹ Nm²/C²* : *ε₀ (const) = Permittivity of Free Space = 8.8542x10⁻¹² C²m²/N* Since force is a vector, the vector form of Coulomb's Law (ie. the electric force exerted by charge q₁ onto charge q₂) is give as: *⟨F₁₂⟩ = [ kₑ (|q₁| |q₂|) / r² ] ⟨r₁₂⟩*

Plane Electromagnetic Waves (ch 34)

The properties of an electromagnetic wave can be deduced from Maxwell's equations. One approach is to solve the 2nd-order differential equation obtained from Maxwell's third and fourth equations. However, this is an extremely rigorous mathematical treatment. To get around this, we assume that the electric & magnetic field vectors have a specific space-time behavior that is simple. LINEARLY POLARIZED ELECTROMAGNETIC WAVES have an electric & magnetic fields that are perpendicular to the wave, and to each other. That is, if an EM wave ⟨c⟩ is traveling along the positive x-axis, there is an electric field ⟨E⟩ in the positive y-axis, and a magnetic field ⟨B⟩ in the positive z-axis. If we define a collection of rays along the direction of the traveling EM wave, the entire collection is called a PLANE WAVE. A surface connecting points of equal phase on all waves is a geometric plane called a WAVE FRONT. If the source was a point that radiated outward, then the EM wave would be a SPHERICAL WAVE. δE/δx = -δB/δt *δ²E/δx² = μ₀ε₀ δ²E/δt²* δB/δx = -μ₀ε₀ δE/δt *δ²B/δx² = μ₀ε₀ δ²B/δt²* The speed of an EM wave is given as: *c = 1 / √(μ₀ε₀)* In a vacuum, c = 3.00x10⁸ m/s The simplest solution to the above 2nd-order differential equations is a sinusoidal wave for which the field magnitudes E and B vary with x and t according to: E = Eᵐᵃˣ cos(kx - ωt) B = Bᵐᵃˣ cos(kx - ωt) At any instant: kEᵐᵃˣ = ωBᵐᵃˣ Therefore: *c = Eᵐᵃˣ / Bᵐᵃˣ == E/B* The wavelength and frequency of an EM wave are related by: *λ = c / f*

Electric Field of a Continuous Charge Distribution (ch 23)

The total electric field at a point P due to all elements in a charge distribution (such as a wire or a surface) is approximately: *⟨E⟩ ≈ kₑ ∑ᵢ ∆qᵢ/rᵢ² ⟨rᵢ⟩* → *⟨E⟩ = kₑ ∫ 1/r² dq ⟨r⟩* It is convenient to use the concept of *charge density* when charge is distributed through an object: *Volume Charge Density* (C/m³) ρ = Q/V → dq = ρ dV *Surface Charge Density* (C/m²) σ = Q/A → dq = σ dA *Linear Charge Density* (C/m) λ = Q/L → dq = λ dL FOR AN INFINITE UNIFORMLY-CHARGED PLATE, the electric field at any point above the surface is given as: *E = 2kπσ*

Maxwell's Equations and Hertz's Discoveries (ch 34)

These equations were fully developed by J. C. Maxwell and are fundamental to electromagnetic phenomena: GAUSS'S LAW — Total electric flux through any closed surface equals the net charge inside that surface divided by the permittivity of free space: *Φᴱ = ∮⟨E⟩ ∙ d⟨A⟩ == q / ε₀* GAUSS'S LAW IN MAGNETISM — The net magnetic flux through any closed surface is zero, because the same number of field lines that enter a closed volume must also leave it: *Φᴮ = ∮⟨B⟩ ∙ d⟨A⟩ == 0* FARADAY'S LAW OF INDUCTION — Describes the creation of an electric field by a changing magnetic flux: *emf = ∮⟨E⟩ ∙ d⟨s⟩ == -dΦᴮ/dt* AMPERE-MAXWELL LAW — Describes the creation of a magnetic field by a changing electric field & by current: *∮⟨B⟩ ∙ d⟨s⟩ = μ₀I + μ₀ε₀(dΦᴱ/dt)* LORENTZ FORCE LAW — Once the electric & magnetic fields are known at some point in space, the force acting on a charged particle (q) can be calculated from the electric & magnetic versions of the [PARTICLE IN A FIELD] model: *⟨F⟩ = q⟨E⟩ + q⟨v⟩×⟨B⟩*

Gauss's Law in Magnetism (ch 30)

This is similar to Gauss's Law for electric flux: *Φᴮ = ∫⟨B⟩ ∙ d⟨A⟩* In which this becomes BAcosΘ when an angle is involved. The magnetic flux through ANY CLOSED SURFACE IS ALWAYS ZERO. ∮⟨B⟩ ∙ d⟨A⟩ = 0

Electric Flux (ch 24)

This is the number of electric field lines that are penetrating a particular area in space, which is written as: *Φᴱ = E A* If the electric field lines are entering that area at an angle other than perpendicular, then you will have to account for that as usual: *Φᴱ = EA cos Θ* If the electric field varies over a surface, then we can use an integral to get the electric flux by unit area: *Φᴱ = ∮⟨E⟩ ∙ d⟨A⟩* This is a CLOSED SURFACE INTEGRAL, which means it must be evaluated over the entire surface in question, and must be 'closed' like a sphere.

LC Circuits & Oscillations (ch 32)

When a charged capacitor is connected in a loop to an inductor, it is known as an LC circuit. When the loop is closed, the current in the circuit and the charge on the capacitor begin to oscillate between maximum positive and negative values. As the capacitor begins to discharge, the energy stored in its electric field decreases. The capacitor's discharge represents a current in the circuit, and some energy is now stored in the magnetic field of the inductor. Therefore, energy is transferred from the electric field of the capacitor to the magnetic field of the inductor. Once the capacitor is fully discharged, all the energy is then stored in the magnetic field of the inductor. The energy then begins to move out of the magnetic field and back into an electric field on the opposite side of the capacitor. After the charge has been completely reversed and all energy converted back into an electric field across the capacitor, the capacitor then discharges in the opposite direction, back across the inductor which stores the energy again as a magnetic field. The energy heads back to the 'starting' plate of the capacitor as an electric field again, thus causing the oscillatory behavior. These oscillations are analogous to the [PARTICLE IN SIMPLE HARMONIC MOTION] model. Potential energy kx²/2 stored in a stretched block-spring system is analogous to the potential energy stored in the capacitor: *Q²/2C* Kinetic energy mv²/2 in a moving block-spring system is analogous to the magnetic energy stored in the inductor: *LI²/2* Total energy stored in an LC circuit is then given as the sum of the potential and kinetic energies: *U = q²/2C + LI²/2 == Q²/2C* Because of an LC circuit's oscillatory behavior, charge as a function of time is then described as a sinusoidal function: *q = Q cos (ωt + φ)* With an angular frequency of: *ω = 1 / √(LC)* Because q varies sinusoidally with time, the current in the circuit also varies sinusoidally: *I = dq/dt == -ωQ sin (ωt + φ)*

The Hall Effect (ch 29)

When a current-carrying conductor is placed in a magnetic field, a potential difference is generated on the conductor in a direction perpendicular to both the current and the magnetic field, which is KNOWN AS THE HALL EFFECT. The potential difference created on the conductor is called the Hall Voltage: *ΔVᴴ = Eᴴd == vᵈBd* Where: : Eᴴ = The Hall Field == vᵈB Since vᵈ = I / nqA, the Hall Voltage can also be expressed as: *ΔVᴴ = IBd / nqA* Since A = td, the Hall Voltage can also be expressed as: *ΔVᴴ = IB / nqt == RᴴIB / t* Where: : Rᴴ = The Hall Coefficient == 1/nq THIS RELATIONSHIP SHOWS THAT A PROPERLY CALIBRATED CONDUCTOR CAN BE USED TO MEASURE THE MAGNITUDE OF AN UNKNOWN MAGNETIC FIELD.

Torque on a Current Loop in a Uniform Magnetic Field, & The Magnetic Dipole Moment (ch 29)

When a current-carrying loop of wire is placed in a magnetic field, the parts of the loop that are perpendicular to the magnetic field will experience a Fᴮ, whereas the parts of the loop that are parallel to the magnetic field will experience Fᴮ = 0 (ie. ⟨L⟩ × ⟨B⟩ = 0). Since it is a loop of wire that carries current around in a circular/rectangular path, there is a Fᴮ in two different directions (each individually expressed as: IaB; where I is the current, a is the lengths of wire perpendicular to the magnetic field, and B is the magnitude of the magnetic field). The total torque on a current-carrying loop involves the entire area of that loop: IabB. Therefore, TORQUE ON THE LOOP IS EXPRESSED AS: *τ = IAB == IAB sinΘ == I⟨A⟩ × ⟨B⟩* Where: : A = Area of the loop, which is just (ab) for a rectangular loop : Θ = Angle of the moment arm The product I⟨A⟩ is also known as the MAGNETIC DIPOLE MOMENT (with units Ampere-meters²): *μ = I⟨A⟩* This allows us to form an expression of torque that is VALID AT ANY ORIENTATION, and is analogous to τ = ⟨p⟩ × ⟨E⟩ for the torque exerted on an electric dipole in the presence of an electric field. The expression is written as such: *τ = ⟨μ⟩ × ⟨B⟩* THE DIRECTION OF ⟨A⟩ AND ⟨μ⟩ ARE THE SAME and can be determined by another Right-Hand Rule: Curl fingers in the direction of the current around the loop. Your thumb points in the direction of both ⟨A⟩ and ⟨μ⟩. For coils of wire, the magnetic moment would be: *μ = NI⟨A⟩* Where: N = Number of loops of the same area The POTENTIAL ENERGY OF A SYSTEM OF A MAGNETIC DIPOLE IN A MAGNETIC FIELD depends on the orientation of the dipole int he magnetic field, and is analogous to the potential energy of a system of an electric dipole in an electric field Uᴱ = -⟨p⟩ ∙ ⟨E⟩. It is written as such: *Uᴮ = -⟨μ⟩ ∙ ⟨B⟩*

Self-Induction and Inductance (ch 32)

When the switch to a circuit is closed (ie. the circuit is turned on), the current does not immediately jump from zero to its maximum value emf/R. As current increases with time, the magnetic field lines surrounding the wires pass through the loop created by the circuit itself. This magnetic field passing through the loop causes a magnetic flux through the loop. The increasing flux then creates an induced *"back emf"* in the circuit. It is in reverse direction of the current that is being applied because the direction of the induced emf is such that it would cause an induced current in the loop (if it didn't already carry a current due to the battery or other source), which would establish a magnetic field opposing the change in the original magnetic field. Therefore, the direction of the induced emf is opposite the direction of the emf of the source (battery, etc), which results in a gradual rather than instantaneous increase in the current to its final equilibrium value. This "back emf" is also called *self-induction* because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. To describe this quantitatively, for any loop of wire the self-induced emf is always proportional to the time rate of change of the current, written as: *emfᴸ = -L dI/dt* Where: : L = Inductance proportionality constant (depends on the geometry of the loop and other physical characteristics) : I = Current Inductance of an N-turn coil is then given as: *L = NΦᴮ/I == -emfᴸ/(dI/dt)* *SI UNIT: 1 Henry (H) = 1 Volt-second/Ampere* Inductance of a solenoid: *L = Aμ₀ (N²/L)*