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: +5 µC on the left, +5 µC on the right 10 cm away, and −10 µC between them, 2.5 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 = +5 µCqR = +5 µCqM = −10 µCrL = 2.5 cmrR = 7.5 cm
F = N
2.Coulomb’s law & electric fieldsHard
Charges of +3 µC and +3 µC are 8 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 = +3 µCq2 = +3 µCd = 8 cmh = 3 cm
E = N/C
3.Series & parallel resistorsMedium
Find the current IR1 delivered by the source, which flows through R1.
V1 = 15 VR1 = 8 ΩR2 = 12 ΩR3 = 12 ΩR4 = 2 Ω
IR1 = A
4.Series & parallel resistorsHard
Find the current IR1 delivered by the source, which flows through R1.
V1 = 12 VR1 = 12 ΩR2 = 3 ΩR3 = 4 ΩR4 = 25 ΩR5 = 4 ΩR6 = 25 Ω
IR1 = A
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.
Vs = 10 VR1 = 10 kΩR2 = 3.3 kΩC = 22 µF
τ = ms
6.RC & RL transientsHard
Just before the switch closes at t = 0, the capacitor holds 12 V with the polarity shown. How long does it take for vC to reach 14.4 V? Find t.
Vs = 15 VR = 1 kΩC = 22 µFvC(0) = 12 V
t = ms
7.Magnetic fields & inductionMedium
An alpha particle moving at 2 × 10⁶ m/s enters a uniform magnetic field at right angles and moves on a circle of radius 10.37 cm. Find the magnetic flux density B.
m = 6.645 × 10⁻²⁷ kg|q| = 3.204 × 10⁻¹⁹ Cv = 2 × 10⁶ m/sr = 10.37 cm
B = T
8.Magnetic fields & inductionHard
A coil of 20 turns, 15 cm × 5 cm, rotates at 3000 rpm in a uniform field of 0.05 T about an axis perpendicular to the field. Find the rms value Erms of the emf.
N = 20A = 15 cm × 5 cmB = 0.05 Tn = 3000 rpm
Erms = V
9.AC impedance & phasorsMedium
A series RLC circuit with L = 20 mH must resonate at 5000 Hz. Find the capacitance C needed.
f0 = 5000 HzL = 20 mH
C = µF
10.AC impedance & phasorsHard
A series RLC circuit has R = 2 Ω, L = 5 mH and C = 2.2 µF. Find its quality factor Q at resonance.
R = 2 ΩL = 5 mHC = 2.2 µF
Q =
3Solutions
1.Coulomb’s law & electric fields
Answer: F = 639.3 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·5·10 × 10⁻¹²/0.025² = −719.2 N
FR = +k·5·10 × 10⁻¹²/0.075² = 79.91 N
Net force
F = −719.2 + (79.91) = −639.3 N → |F| = 639.3 N
Answer
F = 639.3 N
The net force points to the left.
2.Coulomb’s law & electric fields
Answer: E = 1.295 × 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 = √(4² + 3²) = 5 cm
E1 = k·3 × 10⁻⁶/0.05² = 1.079 × 10⁷ N/C
Components
The components along the line joining the charges cancel.
E = 2·E1·(h/r) = 2·1.079 × 10⁷·0.6 = 1.295 × 10⁷ N/C
Answer
E = 1.295 × 10⁷ N/C
The field points along the bisector, away from the charges.
3.Series & parallel resistors
Answer: IR1 = 1.037 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 + 2 = 14 Ω
R2 ∥ 14 = 12·14/(12+14) = 6.462 Ω
Req = R1 + 6.462 = 8 + 6.462 = 14.46 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 = 15/14.46 = 1.037 A
Answer
IR1 = 1.037 A
4.Series & parallel resistors
Answer: IR1 = 0.8242 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 = 4 + 25 = 29 Ω
R4 ∥ 29 = 13.43 Ω
R3 + 13.43 = 17.43 Ω
R2 ∥ 17.43 = 2.559 Ω
Req = R1 + 2.559 = 12 + 2.559 = 14.56 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 = 12/14.56 = 0.8242 A
Answer
IR1 = 0.8242 A
5.RC & RL transients
Answer: τ = 54.59 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 Rth; open the capacitor to find the voltage it will settle at.
Rth = R1 ∥ R2 = 10·3.3/(10 + 3.3) = 2.481 kΩ
vC(∞) = 10·3.3/(10 + 3.3) = 2.481 V
Time constant
τ = Rth·C = 2.481·22 = 54.59 ms
Answer
τ = 54.59 ms
6.RC & RL transients
Answer: t = 35.41 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) = 12 V, vC(∞) = Vs = 15 V
τ = R·C = 1 kΩ · 22 µF = 22 ms
vC(t) = 15 + (12 − 15)·e−t/τ
Solve for the time
14.4 = 15 + (−3)·e−t/τ → e−t/τ = (14.4 − 15)/(12 − 15) = 0.2
t = τ·ln((12 − 15)/(14.4 − 15)) = 22·ln(5) = 35.41 ms
Answer
t = 35.41 ms
7.Magnetic fields & induction
Answer: B = 0.4 T
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).
Field from the radius
B = mv/(|q|r) = 6.645 × 10⁻²⁷·2 × 10⁶/(3.204 × 10⁻¹⁹·0.1037) = 0.4 T
Answer
B = 0.4 T
8.Magnetic fields & induction
Answer: Erms = 1.666 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).
Angular speed and area
ω = 2π·3000/60 = 314.2 rad/s
A = 0.0075 m²
Peak emf
E₀ = NBAω = 20·0.05·0.0075·314.2 = 2.356 V
rms value
Erms = 2.356/√2 = 1.666 V
Answer
Erms = 1.666 V
9.AC impedance & phasors
Answer: C = 0.05066 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/(31416²·0.02) = 0.05066 µF
Answer
C = 0.05066 µF
10.AC impedance & phasors
Answer: Q = 23.84
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·√(0.005·2.2·10⁻⁶)) = 1517 Hz
Quality factor
Q compares the reactance at resonance with the resistance: Q = ω₀L/R.
Q = (1/R)·√(L/C) = (1/2)·√(0.005/(2.2·10⁻⁶)) = 23.84
Answer
Q = 23.84
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