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

Three point charges lie on a straight line: −10 µC on the left, −2 µC on the right 40 cm away, and −4 µC between them, 10 cm from the left charge. Find the magnitude of the net electric force F on the middle charge. Use k = 8.99 × 10⁹ N·m²/C².

q_L = −10 µCq_R = −2 µCq_M = −4 µCr_L = 10 cmr_R = 30 cm

F = N

2.Coulomb’s law & electric fieldsHard

Three positive point charges sit at the corners of a right triangle: q₃ = 8 µC is at the right-angle corner, q₁ = 8 µC is 6 cm from it along one side and q₂ = 10 µC is 5 cm from it along the other side. Find the magnitude of the net electric force F on q₃.

q₁ = 8 µCq₂ = 10 µCq₃ = 8 µCr₁₃ = 6 cmr₂₃ = 5 cm

F = N

3.Series & parallel resistorsMedium

Find the current I_R2 through R₂.

V₁ = 30 VR₁ = 8 ΩR₂ = 8 ΩR₃ = 6 ΩR₄ = 20 Ω

I_R2 = A

4.Series & parallel resistorsHard

Find the equivalent resistance R_eq seen by the source V₁.

V₁ = 10 VR₁ = 2 ΩR₂ = 4 ΩR₃ = 3 ΩR₄ = 2 ΩR₅ = 2 ΩR₆ = 15 Ω

R_eq = Ω

5.RC & RL transientsMedium

The source Vₛ charges the capacitor through R₁, and R₂ is connected in parallel with C. The capacitor is uncharged when the switch closes at t = 0. Find the time constant τ of the circuit after the switch closes.

V_s = 12 VR₁ = 4.7 kΩR₂ = 10 kΩC = 100 µF

τ = ms

6.RC & RL transientsHard

The switch across R₂ has been open for a long time, so the inductor current is steady. At t = 0 the switch closes and shorts out R₂. How long after t = 0 does the inductor current reach 0.109 A? Find t.

V_s = 6 VR₁ = 47 ΩR₂ = 47 ΩL = 22 mH

t = ms

7.Magnetic fields & inductionMedium

Two long parallel wires 3 cm apart carry currents of 8 A and 30 A in opposite directions. Find the force per metre of length F/L between them.

I₁ = 8 AI₂ = 30 Ad = 3 cmμ₀ = 4π × 10⁻⁷ T·m/A

F/L = µN/m

8.Magnetic fields & inductionHard

A coil of 500 turns, 8 cm × 20 cm, rotates in a uniform field of 0.3 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 213.3 V?

N = 500A = 8 cm × 20 cmB = 0.3 TE_rms = 213.3 V

n = rpm

9.AC impedance & phasorsMedium

A resistor and an inductor are connected in parallel across a 24 V (rms), 50 Hz supply. Find the rms current I.

V = 24 V (rms)f = 50 HzR = 25 ΩL = 200 mH

I = A

10.AC impedance & phasorsHard

A series RLC circuit is driven by a 1 V (rms) source whose frequency is set to the resonant frequency of the circuit. Find the rms voltage V_L across the inductor.

V = 1 V (rms)R = 50 ΩL = 20 mHC = 0.47 µF

V_L = V

3Solutions

1.Coulomb’s law & electric fields

Answer: F = 35.16 N

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.

Force from each charge

Right is positive.

F_L = +k·10·4 × 10⁻¹²/0.1² = 35.96 N

F_R = −k·2·4 × 10⁻¹²/0.3² = −0.7991 N

Net force

F = 35.96 + (−0.7991) = 35.16 N → |F| = 35.16 N

Answer

F = 35.16 N

The net force points to the right.

2.Coulomb’s law & electric fields

Answer: F = 329.1 N

Each force on q₃

F₁ = k·8·8 × 10⁻¹²/0.06² = 159.8 N

F₂ = k·10·8 × 10⁻¹²/0.05² = 287.7 N

Vector sum

F = √(159.8² + 287.7²) = 329.1 N

Answer

F = 329.1 N

3.Series & parallel resistors

Answer: I_R2 = 1.625 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 = 6 + 20 = 26 Ω

R₂ ∥ 26 = 8·26/(8+26) = 6.118 Ω

R_eq = R₁ + 6.118 = 8 + 6.118 = 14.12 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 = 30/14.12 = 2.125 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 = 30 − 2.125·8 = 13 V

I_R2 = V_A/R₂ = 13/8 = 1.625 A

Answer

I_R2 = 1.625 A

4.Series & parallel resistors

Answer: R_eq = 4.18 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 = 2 + 15 = 17 Ω

R₄ ∥ 17 = 1.789 Ω

R₃ + 1.789 = 4.789 Ω

R₂ ∥ 4.789 = 2.18 Ω

R_eq = R₁ + 2.18 = 2 + 2.18 = 4.18 Ω

Answer

R_eq = 4.18 Ω

5.RC & RL transients

Answer: τ = 319.7 ms

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.

Thévenin equivalent seen by the capacitor

Short the source to find R_th; open the capacitor to find the voltage it will settle at.

R_th = R₁ ∥ R₂ = 4.7·10/(4.7 + 10) = 3.197 kΩ

v_C(∞) = 12·10/(4.7 + 10) = 8.163 V

Time constant

τ = R_th·C = 3.197·100 = 319.7 ms

Answer

τ = 319.7 ms

6.RC & RL transients

Answer: t = 0.5757 ms

Initial value, final value, time constant

Before t = 0 both resistors are in the loop; afterwards the closed switch bypasses R₂.

i(0) = 6/(47 + 47) = 0.06383 A

i(∞) = 6/47 = 0.1277 A

τ = L/R = 22/47 = 0.4681 ms

i(t) = 0.1277 + (0.06383 − 0.1277)·e^−t/τ

Solve for the time

e^−t/τ = (0.109 − 0.1277)/(0.06383 − 0.1277) = 0.2923

t = 0.4681·ln(3.421) = 0.5757 ms

Answer

t = 0.5757 ms

7.Magnetic fields & induction

Answer: F/L = 1600 µN/m

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 per metre

F/L = μ₀I₁I₂/(2πd) = 2 × 10⁻⁷·8·30/0.03 = 1600 µN/m

Answer

F/L = 1600 µN/m

Currents in opposite directions repel.

8.Magnetic fields & induction

Answer: n = 1200 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 = 301.7 V

Angular speed

ω = E₀/(NBA) = 301.7/(500·0.3·0.016) = 125.7 rad/s

Revolutions per minute

n = 60·125.7/(2π) = 1200 rpm

Answer

n = 1200 rpm

9.AC impedance & phasors

Answer: I = 1.033 A

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|.

Reactance

X_L = ωL = 2π·50·0.2 = 62.83 Ω

Branch currents

Both branches are connected straight across the supply, so each sees the full voltage V.

I_R = V/R = 24/25 = 0.96 A

I_X = V/X_L = 24/62.83 = 0.382 A

Total current

The resistor current is in phase with V and the reactive current is 90° out of phase, so they add like the sides of a right triangle.

I = √(I_R² + I_X²) = √(0.96² + 0.382²) = 1.033 A

Answer

I = 1.033 A

10.AC impedance & phasors

Answer: V_L = 4.126 V

Resonance

At resonance the reactances cancel, so the impedance is just R.

ω₀ = 1/√(LC) = 1/√(0.02·0.47·10⁻⁶) = 10314 rad/s

I = V/R = 1/50 = 0.02 A

Voltage across the inductor

X_L = ω₀L = 10314·0.02 = 206.3 Ω

V_L = I·X_L = 0.02·206.3 = 4.126 V

Answer

V_L = 4.126 V

This is Q = 4.126 times the supply voltage.

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