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Flashcards · AP® Physics C: Electricity and Magnetism · Unit 13

Unit 13: Electromagnetic Induction flashcards

45 cards on topics 13.1 to 13.6. Pick a topic or study the whole unit.

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13.1 Magnetic Flux

What is magnetic flux?
A scalar measuring how much field passes through a surface: Φ_B = B·A = BA cos θ for uniform B on a flat surface, in webers (1 Wb = 1 T·m²). θ is measured from the area vector, the normal.
How do you find magnetic flux when B varies over a surface?
Φ_B = ∫B·dA. Split the surface into strips or rings where B is constant. For B depending only on r, Φ_B = ∫B(r) 2πr dr.
What is the flux through a rectangle beside a long straight wire?
With near side at d, width b and length ℓ parallel to the wire: Φ_B = (μ₀Iℓ/2π) ln[(d + b)/d], from integrating μ₀I/(2πx) ℓ dx.
What is the flux through one turn of a long solenoid?
Φ_B = μ₀nI πR², using only the solenoid's own cross-section, since B is nearly zero outside.
How does flux change when a loop makes a half turn in a uniform field?
It goes from +BA to −BA, so the change is 2BA, not zero.
What angle do you use if B makes 25° with the plane of a loop?
θ is from the normal, so θ = 90° − 25° = 65° and Φ_B = BA cos 65°.
What is the net magnetic flux through any closed surface?
Zero, since every field line that enters also leaves. This does not mean the field is zero.

13.2 Electromagnetic Induction

What does Faraday's law state?
A changing magnetic flux through a loop induces an emf: ℰ = −N dΦ_B/dt for N turns. A steady flux, however large, induces nothing.
What does Lenz's law state?
The induced current creates a magnetic field that opposes the change in flux. Increasing flux: induced field opposite to the external field. Decreasing flux: induced field in the same direction.
How do you find the direction of an induced current?
Find the external field through the loop, decide whether the flux is increasing or decreasing, choose the induced field to oppose that change, then use the right-hand rule: thumb along the induced field, fingers give the current.
What is the emf of a rod sliding on rails in a magnetic field?
ℰ = BLv for a rod of length L moving at v perpendicular to B, because the enclosed area changes at rate Lv.
What is the emf of a coil rotating in a uniform field?
ℰ = NBAω sin ωt, peaking at NBAω. The emf is greatest when the flux is zero and zero when the flux is greatest.
What is the field form of Faraday's law?
∮E·dl = −dΦ_B/dt. A changing magnetic field creates a circulating induced electric field, even in empty space. This field is not conservative.
What induced electric field does a changing solenoid field produce?
Inside, at radius r: E = (r/2) dB/dt. Outside a solenoid of radius R: E = (R²/2r) dB/dt.
The flux through each turn of a 100-turn coil changes at 2.0 × 10⁻³ Wb/s. What is the emf?
|ℰ| = N dΦ_B/dt = 100 × 2.0 × 10⁻³ = 0.20 V.

13.3 Induced Currents and Magnetic Forces

What force acts on a rod moving on rails in a magnetic field?
The induced current I = BLv/R feels F = ILB = B²L²v/R, opposing the velocity. Only segments inside the field feel a force.
Why does an induced magnetic force always oppose motion?
By Lenz's law the induced current opposes the change in flux, so the force on it opposes the relative motion, whether a loop enters or leaves the field. This is magnetic braking.
Where does the energy go when you pull a rod at constant speed through a field?
Into thermal energy in the resistance: P = I²R = B²L²v²/R = Fv, equal to the power the agent supplies. Doubling v quadruples the power.
How does a rod coast to rest under magnetic braking?
m dv/dt = −B²L²v/R gives v = v₀e^(−t/τ) with τ = mR/(B²L²). The stopping distance is v₀τ.
What is the terminal speed of a loop or rod under a steady force with magnetic braking?
v_T = F₀R/(B²L²), with F₀ = mg for a falling loop. From rest, v = v_T(1 − e^(−t/τ)). Doubling B reduces v_T to a quarter.
Is there a magnetic force on a loop moving entirely inside a uniform field?
No. The flux through it is constant, so there is no induced current and no force.
Does a larger resistance give stronger magnetic braking?
No. Larger R gives a smaller induced current, so the braking force B²L²v/R is weaker.
Why can't constant-acceleration equations be used for magnetic braking?
The force depends on v, so the acceleration changes as the rod slows. Solve Newton's second law as a differential equation.

13.4 Inductance

What is inductance?
A conductor's tendency to oppose a change in its own current, by self-induction. L = NΦ_B/I, the flux linkage per ampere, in henries (1 H = 1 V·s/A = 1 Wb/A).
What is the self-induced emf in an inductor?
ℰ = −L dI/dt. It depends on the rate of change of current, not the current itself. It opposes the current while it rises and acts along it while it falls.
What is the inductance of a long solenoid?
L = μ_core N²A/ℓ, with μ_core = μ₀ for air. L ∝ N², so tripling the turns multiplies L by 9.
How much energy does an inductor store?
U = ½LI², from integrating the power P = LI dI/dt. A 0.50 H inductor at 4.0 A stores 4.0 J.
How much energy is released when an inductor's current falls from I₁ to I₂?
½L(I₁² − I₂²). Subtract the two stored energies; it is not ½L(I₁ − I₂)².
Does an inductor oppose current?
No. It opposes a change in current. A steady current passes through an ideal inductor with no potential difference across it.
Where is an inductor's energy stored?
In its magnetic field. It can later be dissipated in a resistor or passed to a capacitor, with total energy conserved.

13.5 Circuits with Resistors and Inductors (LR Circuits)

What is the loop equation for a series circuit of a battery, resistor and inductor?
ℰ = IR + L dI/dt. The time constant is τ = L/R, in seconds.
How does current grow in an LR circuit?
I = (ℰ/R)(1 − e^(−t/τ)), with the inductor voltage V_L = ℰe^(−t/τ). The initial rate is dI/dt = ℰ/L. After one τ the current is 63% of its final value ℰ/R.
How does an inductor behave just after a switch closes and long after?
Just after, its current cannot jump, so an inductor with zero current acts like an open branch and the full emf appears across it. Long after, it acts like a wire with zero resistance.
How does current decay in an LR circuit with no battery?
I = I₀e^(−t/τ), with τ = L/R using the resistance the inductor's current flows through. After one τ, 37% remains; 2.0 A falls to about 0.27 A after 2τ.
How fast does the stored energy decay in an LR circuit?
U_L = ½LI² ∝ e^(−2t/τ), so energy decays with time constant τ/2. All of ½LI₀² is dissipated in the resistor during the decay.
What is the energy balance in a growing LR circuit?
ℰI = I²R + d/dt(½LI²): battery power splits between heating the resistor and building the inductor's stored energy.
What happens to a parallel resistor's current when the switch to an inductor opens?
The inductor's current continues and flows through the resistor, which can reverse the current direction in that resistor.

13.6 Circuits with Capacitors and Inductors (LC Circuits)

What happens when a charged capacitor is connected across an inductor?
The circuit oscillates: energy moves back and forth between the capacitor's electric field and the inductor's magnetic field, with total energy q²/(2C) + ½LI² constant.
What is the differential equation for an LC circuit?
The loop rule q/C + L dI/dt = 0 with I = dq/dt gives d²q/dt² = −q/(LC), simple harmonic motion with ω = 1/√(LC).
What are the period and frequency of an LC circuit?
T = 2π√(LC) and f = 1/(2π√(LC)). They depend only on L and C, not on the starting charge.
What is the maximum current in an LC circuit?
From energy conservation, Q₀²/(2C) = ½LI_max², so I_max = Q₀/√(LC) = ωQ₀. It occurs when the capacitor's charge is zero.
How do charge and current vary in an LC circuit that starts fully charged?
q = Q₀ cos ωt and I = −ωQ₀ sin ωt. All the energy is first in the inductor at T/4; at T/8 it is shared equally.
What is the mass–spring analogy for an LC circuit?
Charge q ↔ position x, current I ↔ velocity v, inductance L ↔ mass m, and 1/C ↔ spring constant k.
How often do the capacitor and inductor energies repeat in an LC circuit?
Every T/2. U_C = (Q₀²/2C)cos² ωt and U_L = (Q₀²/2C)sin² ωt.
L = 0.10 H and C = 10 μF. What are ω and T?
ω = 1/√(LC) = 1/√(1.0 × 10⁻⁶) = 1.0 × 10³ rad/s; T = 2π/ω ≈ 6.3 ms.

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