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

A charge of +8 µC and a charge of +5 µC are 20 cm apart. Point P lies on the line between them, 12 cm from the first charge. Find the magnitude of the electric field E at P.

q₁ = +8 µCq₂ = +5 µCr₁ = 12 cmr₂ = 8 cm

E = N/C

2.Coulomb’s law & electric fieldsHard

Three positive point charges sit at the corners of a right triangle: q₃ = 3 µC is at the right-angle corner, q₁ = 2 µC is 5 cm from it along one side and q₂ = 4 µC is 3 cm from it along the other side. Find the magnitude of the net electric force F on q₃.

q₁ = 2 µCq₂ = 4 µCq₃ = 3 µCr₁₃ = 5 cmr₂₃ = 3 cm

F = N

3.Series & parallel resistorsMedium

Find the current I_R2 through R₂.

V₁ = 18 VR₁ = 20 ΩR₂ = 8 ΩR₃ = 4 ΩR₄ = 8 Ω

I_R2 = A

4.Series & parallel resistorsHard

Find the current I_R2 through R₂.

V₁ = 5 VR₁ = 10 ΩR₂ = 30 ΩR₃ = 25 ΩR₄ = 30 ΩR₅ = 8 ΩR₆ = 20 Ω

I_R2 = A

5.RC & RL transientsMedium

The inductor carries no current before the switch closes at t = 0. Find the energy W(∞) stored in the inductor once the circuit has settled, in mJ.

V_s = 5 VR = 33 ΩL = 150 mH

W(∞) = mJ

6.RC & RL transientsHard

A timer starts charging an uncharged capacitor through a resistor from a 12 V supply at t = 0. It must trigger when the capacitor voltage reaches 9.6 V, exactly 25 ms after the start. With R = 47 kΩ, find the capacitance C needed.

V_s = 12 VV_th = 9.6 VT = 25 msR = 47 kΩ

C = µF

7.Magnetic fields & inductionMedium

A proton moves at 2.5 × 10⁶ m/s perpendicular to a uniform magnetic field of 0.5 T. Find the radius r of its circular path.

m = 1.673 × 10⁻²⁷ kg|q| = 1.602 × 10⁻¹⁹ Cv = 2.5 × 10⁶ m/sB = 0.5 T

r = cm

8.Magnetic fields & inductionHard

A coil of 100 turns, 5 cm × 20 cm, rotates in a uniform field of 0.25 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 33.32 V?

N = 100A = 5 cm × 20 cmB = 0.25 TE_rms = 33.32 V

n = rpm

9.AC impedance & phasorsMedium

A series RLC circuit is connected to a 10 V (rms), 60 Hz supply. Find the phase angle φ of the impedance, in degrees (positive when the current lags the voltage).

V = 10 V (rms)f = 60 HzR = 20 ΩL = 20 mHC = 47 µF

φ = °

10.AC impedance & phasorsHard

A series RLC circuit has R = 50 Ω, L = 5 mH and C = 0.022 µF. Find its bandwidth B (the width of the band between the half-power frequencies), in Hz.

R = 50 ΩL = 5 mHC = 0.022 µF

B = Hz

3Solutions

1.Coulomb’s law & electric fields

Answer: E = 2.029 × 10⁶ N/C

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.

Field of each charge

E₁ = k·8 × 10⁻⁶/0.12² = 4.994 × 10⁶ N/C

E₂ = k·5 × 10⁻⁶/0.08² = 7.023 × 10⁶ N/C

Directions and sum

Both charges are positive: at P their fields point away from each charge, so in opposite directions.

E = |4.994 × 10⁶ − 7.023 × 10⁶| = 2.029 × 10⁶ N/C

Answer

E = 2.029 × 10⁶ N/C

2.Coulomb’s law & electric fields

Answer: F = 121.8 N

Each force on q₃

F₁ = k·2·3 × 10⁻¹²/0.05² = 21.58 N

F₂ = k·4·3 × 10⁻¹²/0.03² = 119.9 N

Vector sum

F = √(21.58² + 119.9²) = 121.8 N

Answer

F = 121.8 N

3.Series & parallel resistors

Answer: I_R2 = 0.4355 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 = 4 + 8 = 12 Ω

R₂ ∥ 12 = 8·12/(8+12) = 4.8 Ω

R_eq = R₁ + 4.8 = 20 + 4.8 = 24.8 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 = 18/24.8 = 0.7258 A

Current through R₂

Subtract the drop across R₁ to get the voltage at node A, which is the voltage across R₂.

V_A = V₁ − I_R1·R₁ = 18 − 0.7258·20 = 3.484 V

I_R2 = V_A/R₂ = 3.484/8 = 0.4355 A

Answer

I_R2 = 0.4355 A

4.Series & parallel resistors

Answer: I_R2 = 0.105 A

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 = 8 + 20 = 28 Ω

R₄ ∥ 28 = 14.48 Ω

R₃ + 14.48 = 39.48 Ω

R₂ ∥ 39.48 = 17.05 Ω

R_eq = R₁ + 17.05 = 10 + 17.05 = 27.05 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 = 5/27.05 = 0.1849 A

Current through R₂

Subtract the drop across R₁ to get the voltage at node A, which is the voltage across R₂.

V_A = V₁ − I_R1·R₁ = 5 − 0.1849·10 = 3.151 V

I_R2 = V_A/R₂ = 3.151/30 = 0.105 A

Answer

I_R2 = 0.105 A

5.RC & RL transients

Answer: W(∞) = 1.722 mJ

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 and final current

Long after the switch closes the inductor behaves like a short circuit, so the final current is set by R alone.

τ = L/R = 150 mH / 33 Ω = 4.545 ms

i(∞) = V_s/R = 5/33 = 0.1515 A

Stored energy

W(∞) = ½·L·i(∞)² = ½·150·0.1515² = 1.722 mJ

Answer

W(∞) = 1.722 mJ

6.RC & RL transients

Answer: C = 0.3305 µF

Required time constant

9.6 = 12·(1 − e^−25/τ) → τ = 25/ln(12/(12 − 9.6))

τ = 25/1.609 = 15.53 ms

Capacitance

kΩ times µF gives milliseconds.

C = τ/R = 15.53/47 = 0.3305 µF

Answer

C = 0.3305 µF

7.Magnetic fields & induction

Answer: r = 5.222 cm

Magnetic forces: a straight wire carrying I in a field B feels F = BIL·sin θ, and a moving charge F = |q|vB·sin θ, where θ is the angle between the current (or velocity) and B. The force is perpendicular to both: point the fingers of the right hand along I or v, curl them towards B, and the thumb gives F for a positive charge; reverse it for a negative charge. Because F is perpendicular to v, a charge moving across a uniform field moves on a circle: |q|vB = mv²/r gives r = mv/(|q|B) and the period T = 2πm/(|q|B), which does not depend on the speed. Two parallel wires attract for currents in the same direction: F/L = μ₀I₁I₂/(2πd).

Force balance

The magnetic force is always perpendicular to the velocity, so it bends the path into a circle.

|q|vB = mv²/r → r = mv/(|q|B)

Radius

r = 1.673 × 10⁻²⁷·2.5 × 10⁶/(1.602 × 10⁻¹⁹·0.5) = 0.05222 m = 5.222 cm

Answer

r = 5.222 cm

8.Magnetic fields & induction

Answer: n = 1800 rpm

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).

Peak emf

E₀ = √2·E_rms = 47.12 V

Angular speed

ω = E₀/(NBA) = 47.12/(100·0.25·0.01) = 188.5 rad/s

Revolutions per minute

n = 60·188.5/(2π) = 1800 rpm

Answer

n = 1800 rpm

9.AC impedance & phasors

Answer: φ = −67.75 °

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 = 377·0.02 = 7.54 Ω

X_C = 1/(ωC) = 1/(377·47·10⁻⁶) = 56.44 Ω

Impedance

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

|Z| = √(R² + (X_L − X_C)²) = √(20² + (−48.9)²) = 52.83 Ω

Phase angle

φ = arctan((X_L − X_C)/R) = arctan(−48.9/20) = −67.75°

Answer

φ = −67.75°

10.AC impedance & phasors

Answer: B = 1592 Hz

Resonant frequency

f₀ = 1/(2π·√(0.005·0.022·10⁻⁶)) = 15175 Hz

Quality factor

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

Q = (1/R)·√(L/C) = (1/50)·√(0.005/(0.022·10⁻⁶)) = 9.535

Bandwidth

B = f₀/Q = 15175/9.535 = 1592 Hz

Answer

B = 1592 Hz

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