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
Three point charges lie on a straight line: +2 µC on the left, +6 µC on the right 40 cm away, and −10 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.
qL = +2 µCqR = +6 µCqM = −10 µCrL = 10 cmrR = 30 cm
F = N
2.Coulomb’s law & electric fieldsHard
Charges of +10 µC and +10 µC are 6 cm apart. Point P lies 3 cm from the midpoint between them, on the perpendicular bisector. Find the magnitude of the electric field E at P. Use k = 8.99 × 10⁹ N·m²/C².
q1 = +10 µCq2 = +10 µCd = 6 cmh = 3 cm
E = N/C
3.Series & parallel resistorsMedium
Find the equivalent resistance Req seen by the source V1.
V1 = 12 VR1 = 6 ΩR2 = 10 ΩR3 = 15 ΩR4 = 12 Ω
Req = Ω
4.Series & parallel resistorsHard
Find the equivalent resistance Req seen by the source V1.
V1 = 9 VR1 = 15 ΩR2 = 2 ΩR3 = 30 ΩR4 = 6 ΩR5 = 20 ΩR6 = 2 Ω
Req = Ω
5.RC & RL transientsMedium
The inductor carries no current before the switch closes at t = 0. How long after the switch closes does the current reach 75% of its final value? Find t.
Vs = 9 VR = 100 ΩL = 220 mH
t = ms
6.RC & RL transientsHard
Just before the switch closes at t = 0, the capacitor holds 2 V with the polarity shown. Find the capacitor voltage vC at t = 1000 ms.
Vs = 15 VR = 10 kΩC = 220 µFvC(0) = 2 Vt = 1000 ms
vC = V
7.Magnetic fields & inductionMedium
An alpha particle moves at 1 × 10⁶ m/s perpendicular to a uniform magnetic field of 1 T. Find the radius r of its circular path.
m = 6.645 × 10⁻²⁷ kg|q| = 3.204 × 10⁻¹⁹ Cv = 1 × 10⁶ m/sB = 1 T
r = cm
8.Magnetic fields & inductionHard
Two coils with self-inductances of 10 mH and 50 mH have a coupling coefficient of 0.8. The current in the first coil changes at 10 A/s. Find the emf ε induced in the second coil.
L1 = 10 mHL2 = 50 mHk = 0.8di/dt = 10 A/s
ε = V
9.AC impedance & phasorsMedium
A series RLC circuit with L = 10 mH must resonate at 5000 Hz. Find the capacitance C needed.
f0 = 5000 HzL = 10 mH
C = µF
10.AC impedance & phasorsHard
A series RLC circuit has R = 50 Ω, L = 20 mH and C = 0.22 µF. Find its quality factor Q at resonance.
R = 50 ΩL = 20 mHC = 0.22 µF
Q =
3Solutions
1.Coulomb’s law & electric fields
Answer: F = 11.99 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.
FL = −k·2·10 × 10⁻¹²/0.1² = −17.98 N
FR = +k·6·10 × 10⁻¹²/0.3² = 5.993 N
Net force
F = −17.98 + (5.993) = −11.99 N → |F| = 11.99 N
Answer
F = 11.99 N
The net force points to the left.
2.Coulomb’s law & electric fields
Answer: E = 7.063 × 10⁷ 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.
Distance and field of one charge
r = √(3² + 3²) = 4.243 cm
E1 = k·10 × 10⁻⁶/0.04243² = 4.994 × 10⁷ N/C
Components
The components along the line joining the charges cancel.
E = 2·E1·(h/r) = 2·4.994 × 10⁷·0.7071 = 7.063 × 10⁷ N/C
Answer
E = 7.063 × 10⁷ N/C
The field points along the bisector, away from the charges.
3.Series & parallel resistors
Answer: Req = 13.3 Ω
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 + 12 = 27 Ω
R2 ∥ 27 = 10·27/(10+27) = 7.297 Ω
Req = R1 + 7.297 = 6 + 7.297 = 13.3 Ω
Answer
Req = 13.3 Ω
4.Series & parallel resistors
Answer: Req = 16.89 Ω
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 = 20 + 2 = 22 Ω
R4 ∥ 22 = 4.714 Ω
R3 + 4.714 = 34.71 Ω
R2 ∥ 34.71 = 1.891 Ω
Req = R1 + 1.891 = 15 + 1.891 = 16.89 Ω
Answer
Req = 16.89 Ω
5.RC & RL transients
Answer: t = 3.05 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.
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 = 220 mH / 100 Ω = 2.2 ms
i(∞) = Vs/R = 9/100 = 0.09 A
Solve for the time
0.75 = 1 − e−t/τ → t = −τ·ln(1 − 0.75)
t = −2.2·ln(0.25) = 3.05 ms
Answer
t = 3.05 ms
6.RC & RL transients
Answer: vC = 6.748 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.
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) = 2 V, vC(∞) = Vs = 15 V
τ = R·C = 10 kΩ · 220 µF = 2200 ms
vC(t) = 15 + (2 − 15)·e−t/τ
Voltage at the given time
vC(1000) = 15 + (−13)·e−1000/2200 = 6.748 V
Answer
vC = 6.748 V
7.Magnetic fields & induction
Answer: r = 2.074 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 = 6.645 × 10⁻²⁷·1 × 10⁶/(3.204 × 10⁻¹⁹·1) = 0.02074 m = 2.074 cm
Answer
r = 2.074 cm
8.Magnetic fields & induction
Answer: ε = 0.1789 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).
Mutual inductance
M = 0.8·√(10·50) = 17.89 mH
Induced emf
ε = M·di/dt = 0.01789·10 = 0.1789 V
Answer
ε = 0.1789 V
9.AC impedance & phasors
Answer: C = 0.1013 µF
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·5000 = 31416 rad/s
C = 1/(ω₀²·L) = 1/(31416²·0.01) = 0.1013 µF
Answer
C = 0.1013 µF
10.AC impedance & phasors
Answer: Q = 6.03
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|.
Resonant frequency
f0 = 1/(2π·√(0.02·0.22·10⁻⁶)) = 2399 Hz
Quality factor
Q compares the reactance at resonance with the resistance: Q = ω₀L/R.
Q = (1/R)·√(L/C) = (1/50)·√(0.02/(0.22·10⁻⁶)) = 6.03
Answer
Q = 6.03
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