Physics 2: Electricity and Magnetism: cram pack

Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.

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1Formulas and methods

Coulomb’s law & electric fields

Coulomb’s law: F = k·|q₁q₂|/r²
Field of a point charge: E = k·|q|/r²
Constant: k = 8.99 × 10⁹ N·m²/C²
Force on a charge: F = q·E

Series & parallel resistors

Resistors in series: R_eq = R₁ + R₂ + …
Resistors in parallel: 1/R_eq = 1/R₁ + 1/R₂ + …
Two in parallel: R_eq = R₁R₂ / (R₁ + R₂)
Ohm's law: V = I·R
Power: P = V·I = I²R = V²/R

Method

Start at the end of the network furthest from the terminals.
Combine resistors in series by adding them, and resistors in parallel with 1/R = 1/R₁ + 1/R₂ + …
Redraw the circuit after each step.
Repeat until one resistor is left between the terminals.

RC & RL transients

Time constant: τ = RC | τ = L/R
Response: x(t) = x(∞) + [x(0) − x(∞)]·e^−t/τ
Capacitor at DC: open circuit; v_C cannot jump
Inductor at DC: short circuit; i_L cannot jump
Settled: t ≈ 5τ

Method

Find the initial value just after switching: a capacitor voltage and an inductor current cannot jump.
Find the final value long after switching: a capacitor acts as an open circuit, an inductor as a short circuit.
Find the time constant: τ = RC or τ = L/R, using the resistance seen by the capacitor or inductor.
Use x(t) = x(∞) + [x(0) − x(∞)]·e^−t/τ.

Magnetic fields & induction

Long straight wire: B = μ₀I/(2πr)
Centre of a flat coil: B = μ₀NI/(2R)
Long solenoid: B = μ₀nI, n = N/L
Constant: μ₀ = 4π × 10⁻⁷ T·m/A
Force on a wire: F = BIL·sin θ
Force on a charge: F = |q|vB·sin θ
Circular motion: r = mv/(|q|B), T = 2πm/(|q|B)
Parallel wires: F/L = μ₀I₁I₂/(2πd)
Flux: Φ = BA·cos θ
Faraday’s law: ε = −N·ΔΦ/Δt
Moving rod: ε = BLv
Generator: E₀ = NBAω, E_rms = E₀/√2
Inductance: ε = L·ΔI/Δt, L = μ₀N²A/l, W = ½LI²
Mutual inductance: ε₂ = M·di₁/dt, M = k√(L₁L₂)
Velocity selector and spectrometer: v = E/B, r = √(2mV/|q|)/B

Method

Convert every quantity to SI units: cm to m, cm² to m², mT and µT to T.
Pick the right formula: B = μ₀I/(2πr) for a wire, μ₀NI/(2R) for the centre of a coil, μ₀nI inside a solenoid; F = BIL·sin θ or |q|vB·sin θ for forces.
For induction, find the change of flux ΔΦ = Δ(BA·cos θ) and use ε = N·ΔΦ/Δt, or ε = BLv for a moving rod and NBAω for a rotating coil.
Find directions separately with the right-hand rules, and Lenz’s law for induced currents.

AC impedance & phasors

Impedances: Z_R = R, Z_L = jωL, Z_C = 1/(jωC)
Angular frequency: ω = 2πf
Magnitude and angle: |Z| = √(R² + X²), φ = atan(X/R)
Phasor Ohm’s law: V = I·Z

Method

Convert the frequency: ω = 2πf.
Find each reactance: X_L = ωL and X_C = 1/(ωC).
Add impedances like resistors, with Z = R + jX.
Find the magnitude |Z| = √(R² + X²) and the angle φ = atan(X/R), then use Ohm’s law with phasors.

2Practice problems

1.Coulomb’s law & electric fieldsMedium

Charges of +8 µC and +4 µC are 100 cm apart. At what distance x from the +8 µC charge, on the line between them, is the electric field zero?

q₁ = +8 µCq₂ = +4 µCd = 100 cm

x = cm

2.Coulomb’s law & electric fieldsHard

Three positive point charges sit at the corners of a right triangle: q₃ = 2 µC is at the right-angle corner, q₁ = 5 µC is 3 cm from it along one side and q₂ = 2 µC is 10 cm from it along the other side. Find the magnitude of the net electric force F on q₃.

q₁ = 5 µCq₂ = 2 µCq₃ = 2 µCr₁₃ = 3 cmr₂₃ = 10 cm

F = N

3.Series & parallel resistorsMedium

Find the current I_R1 delivered by the source, which flows through R₁.

V₁ = 15 VR₁ = 4 ΩR₂ = 4 ΩR₃ = 15 ΩR₄ = 3 Ω

I_R1 = A

4.Series & parallel resistorsHard

Find the equivalent resistance R_eq seen by the source V₁.

V₁ = 6 VR₁ = 6 ΩR₂ = 8 ΩR₃ = 8 ΩR₄ = 3 ΩR₅ = 6 ΩR₆ = 2 Ω

R_eq = Ω

5.RC & RL transientsMedium

The inductor carries a steady current of 1.5 A. At t = 0 the switch closes the loop through R and the source is removed, so the inductor current decays through R. Find the current i at t = 1 ms.

i(0) = 1.5 AR = 33 ΩL = 47 mHt = 1 ms

i = A

6.RC & RL transientsHard

Just before the switch closes at t = 0, the capacitor holds 4 V with the polarity shown. How long does it take for v_C to reach 11.7 V? Find t.

V_s = 15 VR = 3.3 kΩC = 33 µFv_C(0) = 4 V

t = ms

7.Magnetic fields & inductionMedium

A coil of 50 turns has an area of 50 cm². A uniform field perpendicular to the coil rises steadily from 0.05 T to 0.65 T in 500 ms. Find the average induced emf ε.

N = 50A = 50 cm²B: 0.05 T → 0.65 TΔt = 500 ms

ε = mV

8.Magnetic fields & inductionHard

A coil of 150 turns, 8 cm × 5 cm, rotates in a uniform field of 0.2 T about an axis perpendicular to the field. At what speed n (in revolutions per minute) must it turn to give an rms emf of 5.331 V?

N = 150A = 8 cm × 5 cmB = 0.2 TE_rms = 5.331 V

n = rpm

9.AC impedance & phasorsMedium

A series RLC circuit is connected to a 120 V (rms), 60 Hz supply. Find the phase angle φ of the impedance, in degrees (positive when the current lags the voltage).

V = 120 V (rms)f = 60 HzR = 50 ΩL = 250 mHC = 10 µF

φ = °

10.AC impedance & phasorsHard

A series RLC circuit has R = 20 Ω, L = 20 mH and C = 0.47 µF. Find its bandwidth B (the width of the band between the half-power frequencies), in Hz.

R = 20 ΩL = 20 mHC = 0.47 µF

B = Hz

3Solutions

1.Coulomb’s law & electric fields

Answer: x = 58.58 cm

Coulomb’s law: two point charges attract or repel with F = k·|q₁·q₂|/r², with k = 8.99 × 10⁹ N·m²/C². The field of a point charge is E = k·|q|/r², pointing away from a positive charge and towards a negative one. Forces and fields from several charges add as vectors.

Equal fields

8/x² = 4/(100 − x)²

(100 − x)/x = √(4/8) = 0.7071

Position

x = 100/(1 + 0.7071) = 58.58 cm

Answer

x = 58.58 cm

The zero point is always closer to the smaller charge.

2.Coulomb’s law & electric fields

Answer: F = 99.95 N

Each force on q₃

F₁ = k·5·2 × 10⁻¹²/0.03² = 99.89 N

F₂ = k·2·2 × 10⁻¹²/0.1² = 3.596 N

Vector sum

F = √(99.89² + 3.596²) = 99.95 N

Answer

F = 99.95 N

3.Series & parallel resistors

Answer: I_R1 = 2.063 A

Series and parallel resistors: in series the same current flows through each resistor and R = R₁ + R₂ + …. In parallel each resistor has the same voltage and 1/R = 1/R₁ + 1/R₂ + …. A ladder network is reduced one pair at a time until a single equivalent resistance remains.

Combine resistors from the far end

Start at the end farthest from the source and work back towards it. Resistors carrying the same current are in series; resistors across the same two nodes are in parallel.

R₃ + R₄ = 15 + 3 = 18 Ω

R₂ ∥ 18 = 4·18/(4+18) = 3.273 Ω

R_eq = R₁ + 3.273 = 4 + 3.273 = 7.273 Ω

Current from the source

The source sees only R_eq, so Ohm’s law gives the current it delivers. That current flows through R₁.

I_R1 = V₁/R_eq = 15/7.273 = 2.063 A

Answer

I_R1 = 2.063 A

4.Series & parallel resistors

Answer: R_eq = 10.48 Ω

Combine resistors from the far end

Start at the end farthest from the source and work back towards it. Resistors carrying the same current are in series; resistors across the same two nodes are in parallel.

R₅ + R₆ = 6 + 2 = 8 Ω

R₄ ∥ 8 = 2.182 Ω

R₃ + 2.182 = 10.18 Ω

R₂ ∥ 10.18 = 4.48 Ω

R_eq = R₁ + 4.48 = 6 + 4.48 = 10.48 Ω

Answer

R_eq = 10.48 Ω

5.RC & RL transients

Answer: i = 0.7433 A

First-order transients: in a circuit with one capacitor or one inductor, every voltage and current moves exponentially from its initial value to its final value: x(t) = x(∞) + [x(0) − x(∞)]·e^−t/τ, with τ = RC or τ = L/R. After about 5τ the circuit has settled.

Time constant

τ = L/R = 47 mH / 33 Ω = 1.424 ms

Decay of the current

i(t) = 1.5·e^−1/1.424 = 0.7433 A

Answer

i = 0.7433 A

6.RC & RL transients

Answer: t = 131.1 ms

Initial value, final value, time constant

The capacitor voltage cannot jump, so it starts at its stored value and heads for the source voltage.

v_C(0) = 4 V, v_C(∞) = V_s = 15 V

τ = R·C = 3.3 kΩ · 33 µF = 108.9 ms

v_C(t) = 15 + (4 − 15)·e^−t/τ

Solve for the time

11.7 = 15 + (−11)·e^−t/τ → e^−t/τ = (11.7 − 15)/(4 − 15) = 0.3

t = τ·ln((4 − 15)/(11.7 − 15)) = 108.9·ln(3.333) = 131.1 ms

Answer

t = 131.1 ms

7.Magnetic fields & induction

Answer: ε = 300 mV

Electromagnetic induction: the flux through a loop is Φ = BA·cos θ, with θ between B and the normal of the loop. Faraday's law: a change of flux induces emf = −N·dΦ/dt in a coil of N turns, and Lenz's law says the induced current flows so that its own field opposes the change. A rod moving across a field gives emf = BLv. A coil rotating at ω gives e = NBAω·sin ωt, peak NBAω. Self-inductance: emf = −L·di/dt, with L = μ₀N²A/l for a long solenoid, and the energy stored is W = ½LI². Two coupled coils: emf₂ = −M·di₁/dt with M = k√(L₁L₂).

Change of flux per turn

ΔΦ = (0.65 − 0.05)·0.005 = 0.003 Wb

Average emf

ε = NΔΦ/Δt = 50·0.003/0.5 = 0.3 V

Answer

ε = 300 mV

8.Magnetic fields & induction

Answer: n = 599.9 rpm

Peak emf

E₀ = √2·E_rms = 7.539 V

Angular speed

ω = E₀/(NBA) = 7.539/(150·0.2·0.004) = 62.83 rad/s

Revolutions per minute

n = 60·62.83/(2π) = 599.9 rpm

Answer

n = 599.9 rpm

9.AC impedance & phasors

Answer: φ = −73.7 °

AC circuits with phasors: resistors, inductors and capacitors have impedances R, jωL and 1/(jωC), with ω = 2πf. In series, impedances add like resistances, but as complex numbers. Rms current and voltage follow Ohm’s law with |Z|.

Reactances

ω = 2πf = 2π·60 = 377 rad/s

X_L = ωL = 377·0.25 = 94.25 Ω

X_C = 1/(ωC) = 1/(377·10·10⁻⁶) = 265.3 Ω

Impedance

In series the impedances add as complex numbers: Z = R + j(X_L − X_C).

|Z| = √(R² + (X_L − X_C)²) = √(50² + (−171)²) = 178.2 Ω

Phase angle

φ = arctan((X_L − X_C)/R) = arctan(−171/50) = −73.7°

Answer

φ = −73.7°

10.AC impedance & phasors

Answer: B = 159.2 Hz

Resonant frequency

f₀ = 1/(2π·√(0.02·0.47·10⁻⁶)) = 1642 Hz

Quality factor

Q compares the reactance at resonance with the resistance: Q = ω₀L/R.

Q = (1/R)·√(L/C) = (1/20)·√(0.02/(0.47·10⁻⁶)) = 10.31

Bandwidth

B = f₀/Q = 1642/10.31 = 159.2 Hz

Answer

B = 159.2 Hz

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