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
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.
1Formulas and methods
Series & parallel resistors
- Resistors in series
- Req = R1 + R2 + …
- Resistors in parallel
- 1/Req = 1/R1 + 1/R2 + …
- Two in parallel
- Req = 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; vC cannot jump
- Inductor at DC
- short circuit; iL 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, Erms = 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
- ZR = R, ZL = jωL, ZC = 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: XL = ωL and XC = 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
Two identical small metal spheres carry charges of +10 µC and +4 µC. They are touched together and then put back 15 cm apart. Find the magnitude of the force F between them now. Use k = 8.99 × 10⁹ N·m²/C².
q1 = +10 µCq2 = +4 µCr = 15 cm
F = N
2.Coulomb’s law & electric fieldsHard
Two identical small balls, each of mass 3 g, hang from the same point on insulating threads 50 cm long. They are given equal charges and each thread settles at 20° from the vertical. Use g = 9.81 m/s². Find the charge q on each ball in nC. Use k = 8.99 × 10⁹ N·m²/C².
m = 3 gL = 50 cmθ = 20°g = 9.81 m/s²
q = nC
3.Series & parallel resistorsMedium
Find the current IR1 delivered by the source, which flows through R1.
V1 = 12 VR1 = 20 ΩR2 = 5 ΩR3 = 8 ΩR4 = 15 Ω
IR1 = A
4.Series & parallel resistorsHard
Find the current IR2 through R2.
V1 = 10 VR1 = 5 ΩR2 = 20 ΩR3 = 15 ΩR4 = 30 ΩR5 = 10 ΩR6 = 4 Ω
IR2 = A
5.RC & RL transientsMedium
The inductor carries no current before the switch closes at t = 0. Find the voltage vL across the inductor at t = 1 ms.
Vs = 20 VR = 47 ΩL = 68 mHt = 1 ms
vL = V
6.RC & RL transientsHard
Just before the switch closes at t = 0, the capacitor holds 30 V with the polarity shown. How long does it take for vC to reach 13.2 V? Find t.
Vs = 6 VR = 10 kΩC = 10 µFvC(0) = 30 V
t = ms
7.Magnetic fields & inductionMedium
A metal rod 20 cm long slides at 1 m/s along two rails in a uniform field of 0.1 T into the page. The rails are joined by a resistor of 1 Ω; the rod and rails have negligible resistance. Find the current I.
B = 0.1 TL = 20 cmv = 1 m/sR = 1 Ω
I = mA
8.Magnetic fields & inductionHard
The current in a 5 mH inductor changes steadily from 1 A to 1.5 A in 10 ms. Find the magnitude of the self-induced emf ε.
L = 5 mH1 A → 1.5 AΔt = 10 ms
ε = V
9.AC impedance & phasorsMedium
A series RLC circuit is connected to a 12 V (rms), 60 Hz supply. Find the power factor cos φ of the circuit.
V = 12 V (rms)f = 60 HzR = 80 ΩL = 200 mHC = 4.7 µF
cos φ =
10.AC impedance & phasorsHard
A series RLC circuit is connected to a 24 V (rms), 200 Hz supply. Find the resonant frequency f0 of the circuit.
V = 24 V (rms)f = 200 HzR = 100 ΩL = 200 mHC = 47 µF
f0 = Hz
3Solutions
1.Coulomb’s law & electric fields
Answer: F = 19.58 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.
Charge after touching
q′ = (10 + 4)/2 = 7 µC
New force
F = 8.99 × 10⁹·(7 × 10⁻⁶)²/0.15² = 19.58 N
Answer
F = 19.58 N
Both spheres now carry the same sign, so they repel.
2.Coulomb’s law & electric fields
Answer: q = 373.3 nC
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.
Electric force from the equilibrium
The tension's horizontal part balances F, its vertical part balances m·g.
F = m·g·tan θ = 0.003·9.81·tan 20° = 0.01071 N
Separation
r = 2·L·sin θ = 2·0.5·sin 20° = 0.342 m
Charge
F = k·q²/r² → q = r·√(F/k) = 0.342·√(0.01071/8.99 × 10⁹) = 3.733 × 10⁻⁷ C = 373.3 nC
Answer
q = 373.3 nC
3.Series & parallel resistors
Answer: IR1 = 0.4978 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 = 8 + 15 = 23 Ω
R2 ∥ 23 = 5·23/(5+23) = 4.107 Ω
Req = R1 + 4.107 = 20 + 4.107 = 24.11 Ω
Current from the source
The source sees only Req, so Ohm’s law gives the current it delivers. That current flows through R₁.
IR1 = V1/Req = 12/24.11 = 0.4978 A
Answer
IR1 = 0.4978 A
4.Series & parallel resistors
Answer: IR2 = 0.3439 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.
R5 + R6 = 10 + 4 = 14 Ω
R4 ∥ 14 = 9.545 Ω
R3 + 9.545 = 24.55 Ω
R2 ∥ 24.55 = 11.02 Ω
Req = R1 + 11.02 = 5 + 11.02 = 16.02 Ω
Current from the source
The source sees only Req, so Ohm’s law gives the current it delivers. That current flows through R₁.
IR1 = V1/Req = 10/16.02 = 0.6242 A
Current through R₂
Subtract the drop across R₁ to get the voltage at node A, which is the voltage across R₂.
VA = V1 − IR1·R1 = 10 − 0.6242·5 = 6.879 V
IR2 = VA/R2 = 6.879/20 = 0.3439 A
Answer
IR2 = 0.3439 A
5.RC & RL transients
Answer: vL = 10.02 V
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 = 68 mH / 47 Ω = 1.447 ms
i(∞) = Vs/R = 20/47 = 0.4255 A
Inductor voltage
At t = 0 the whole supply voltage appears across the inductor; it then decays to zero.
vL(t) = Vs·e−t/τ = 20·e−1/1.447 = 10.02 V
Answer
vL = 10.02 V
6.RC & RL transients
Answer: t = 120.4 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.
Initial value, final value, time constant
The capacitor voltage cannot jump, so it starts at its stored value and heads for the source voltage.
vC(0) = 30 V, vC(∞) = Vs = 6 V
τ = R·C = 10 kΩ · 10 µF = 100 ms
vC(t) = 6 + (30 − 6)·e−t/τ
Solve for the time
13.2 = 6 + (24)·e−t/τ → e−t/τ = (13.2 − 6)/(30 − 6) = 0.3
t = τ·ln((30 − 6)/(13.2 − 6)) = 100·ln(3.333) = 120.4 ms
Answer
t = 120.4 ms
7.Magnetic fields & induction
Answer: I = 20 mA
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).
Motional emf
ε = BLv = 0.1·0.2·1 = 0.02 V
Current
I = ε/R = 0.02/1 = 0.02 A
Answer
I = 20 mA
8.Magnetic fields & induction
Answer: ε = 0.25 V
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).
Self-induced emf
The emf opposes the change of current (Lenz), so its sign is opposite to ΔI; the size is L·|ΔI|/Δt.
ε = L·ΔI/Δt = 0.005·0.5/0.01 = 0.25 V
Answer
ε = 0.25 V
9.AC impedance & phasors
Answer: cos φ = 0.1615
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
XL = 75.4 Ω, XC = 564.4 Ω
Impedance
|Z| = √(80² + (−489)²) = 495.5 Ω
Power factor
cos φ = R/|Z| = 80/495.5 = 0.1615
The circuit is capacitive, so the power factor is leading.
Answer
cos φ = 0.1615
10.AC impedance & phasors
Answer: f0 = 51.91 Hz
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
At resonance the inductive and capacitive reactances cancel: ωL = 1/(ωC). The supply frequency does not matter.
f0 = 1/(2π√(LC)) = 1/(2π·√(0.2·47·10⁻⁶)) = 51.91 Hz
Answer
f0 = 51.91 Hz
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