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 +1 µC and a charge of −3 µC are 30 cm apart. Point P lies on the line between them, 18 cm from the first charge. Find the magnitude of the electric field E at P.

q₁ = +1 µCq₂ = −3 µCr₁ = 18 cmr₂ = 12 cm

E = N/C

2.Coulomb’s law & electric fieldsHard

An electron enters the uniform field between two parallel plates moving parallel to them at 3 × 10⁷ m/s. The plates are 10 cm long and the field between them is 700 N/C. Electron charge 1.602 × 10⁻¹⁹ C, mass 9.109 × 10⁻³¹ kg; ignore gravity. How far y is it deflected sideways when it leaves the plates? Give the answer in mm.

v₀ = 3 × 10⁷ m/sL = 10 cmE = 700 N/Ce = 1.602 × 10⁻¹⁹ Cm_e = 9.109 × 10⁻³¹ kg

y = mm

3.Series & parallel resistorsMedium

Find the current I_R2 through R₂.

V₁ = 15 VR₁ = 4 ΩR₂ = 8 ΩR₃ = 12 ΩR₄ = 20 Ω

I_R2 = A

4.Series & parallel resistorsHard

Find the current I_R2 through R₂.

V₁ = 36 VR₁ = 4 ΩR₂ = 20 ΩR₃ = 2 ΩR₄ = 2 ΩR₅ = 20 ΩR₆ = 3 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 at t = 1 ms, in mJ.

V_s = 5 VR = 33 ΩL = 47 mHt = 1 ms

W = mJ

6.RC & RL transientsHard

A timer starts charging an uncharged capacitor through a resistor from a 15 V supply at t = 0. It must trigger when the capacitor voltage reaches 7.5 V, exactly 250 ms after the start. With R = 2.2 kΩ, find the capacitance C needed.

V_s = 15 VV_th = 7.5 VT = 250 msR = 2.2 kΩ

C = µF

7.Magnetic fields & inductionMedium

A circular loop lies in a uniform magnetic field pointing out of the page. The field strength is increasing. Click the direction of the induced current in the loop as seen in the drawing.

B: out of the pagedB/dt > 0

1clockwise
2counterclockwise

8.Magnetic fields & inductionHard

In a velocity selector, crossed fields E = 5 × 10⁴ V/m and B = 0.5 T let only ions of one speed pass straight through. Find that speed v.

E = 5 × 10⁴ V/mB₁ = 0.5 T

v = m/s

9.AC impedance & phasorsMedium

A series RLC circuit with C = 22 µF must resonate at 50 Hz. Find the inductance L needed.

f₀ = 50 HzC = 22 µF

L = mH

10.AC impedance & phasorsHard

A series RLC circuit is driven by a 12 V (rms) source whose frequency is set to the resonant frequency of the circuit. Find the rms voltage V_L across the inductor.

V = 12 V (rms)R = 2 ΩL = 50 mHC = 10 µF

V_L = V

3Solutions

1.Coulomb’s law & electric fields

Answer: E = 2.15 × 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·1 × 10⁻⁶/0.18² = 277469 N/C

E₂ = k·3 × 10⁻⁶/0.12² = 1.873 × 10⁶ N/C

Directions and sum

The first field points away from the positive charge and the second towards the negative charge: both point the same way.

E = |277469 + 1.873 × 10⁶| = 2.15 × 10⁶ N/C

Answer

E = 2.15 × 10⁶ N/C

2.Coulomb’s law & electric fields

Answer: y = 0.6839 mm

Time between the plates

t = L/v₀ = 0.1/3 × 10⁷ = 3.333 × 10⁻⁹ s

Sideways acceleration

a = e·E/m = 1.602 × 10⁻¹⁹·700/9.109 × 10⁻³¹ = 1.231 × 10¹⁴ m/s²

Deflection

y = ½·a·t² = ½·1.231 × 10¹⁴·(3.333 × 10⁻⁹)² = 0.0006839 m = 0.6839 mm

Answer

y = 0.6839 mm

The electron is pushed towards the positive plate, against the field direction.

3.Series & parallel resistors

Answer: I_R2 = 1.154 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 = 12 + 20 = 32 Ω

R₂ ∥ 32 = 8·32/(8+32) = 6.4 Ω

R_eq = R₁ + 6.4 = 4 + 6.4 = 10.4 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 = 15/10.4 = 1.442 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 = 15 − 1.442·4 = 9.231 V

I_R2 = V_A/R₂ = 9.231/8 = 1.154 A

Answer

I_R2 = 1.154 A

4.Series & parallel resistors

Answer: I_R2 = 0.803 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 = 20 + 3 = 23 Ω

R₄ ∥ 23 = 1.84 Ω

R₃ + 1.84 = 3.84 Ω

R₂ ∥ 3.84 = 3.221 Ω

R_eq = R₁ + 3.221 = 4 + 3.221 = 7.221 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 = 36/7.221 = 4.985 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 = 36 − 4.985·4 = 16.06 V

I_R2 = V_A/R₂ = 16.06/20 = 0.803 A

Answer

I_R2 = 0.803 A

5.RC & RL transients

Answer: W = 0.1373 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 = 47 mH / 33 Ω = 1.424 ms

i(∞) = V_s/R = 5/33 = 0.1515 A

Current at the given time

i(t) = 0.1515·(1 − e^−1/1.424) = 0.07643 A

Stored energy

W = ½·L·i² = ½·47·0.07643² = 0.1373 mJ

Answer

W = 0.1373 mJ

6.RC & RL transients

Answer: C = 163.9 µF

Required time constant

7.5 = 15·(1 − e^−250/τ) → τ = 250/ln(15/(15 − 7.5))

τ = 250/0.6931 = 360.7 ms

Capacitance

kΩ times µF gives milliseconds.

C = τ/R = 360.7/2.2 = 163.9 µF

Answer

C = 163.9 µF

7.Magnetic fields & induction

Answer: clockwise

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

The flux out of the page increases, or the flux into the page decreases.

Field of the induced current

The induced current opposes the change.

Induced field inside the loop: into the page.

Direction of the current

Right-hand grip rule: a counterclockwise current gives a field out of the page inside the loop.

clockwise

Answer

clockwise

8.Magnetic fields & induction

Answer: v = 100000 m/s

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).

Selected speed

Undeflected: the electric force qE balances the magnetic force qvB.

v = E/B₁ = 5 × 10⁴/0.5 = 100000 m/s

Answer

v = 100000 m/s

9.AC impedance & phasors

Answer: L = 460.6 mH

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|.

Resonance condition

At resonance the inductive and capacitive reactances cancel: ωL = 1/(ωC).

ω₀ = 2π·50 = 314.2 rad/s

L = 1/(ω₀²·C) = 1/(314.2²·22·10⁻⁶) = 460.6 mH

Answer

L = 460.6 mH

10.AC impedance & phasors

Answer: V_L = 424.3 V

Resonance

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

ω₀ = 1/√(LC) = 1/√(0.05·10·10⁻⁶) = 1414 rad/s

I = V/R = 12/2 = 6 A

Voltage across the inductor

X_L = ω₀L = 1414·0.05 = 70.71 Ω

V_L = I·X_L = 6·70.71 = 424.3 V

Answer

V_L = 424.3 V

This is Q = 35.36 times the supply voltage.

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