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
Charges of +5 µC and +10 µC are 40 cm apart. At what distance x from the +5 µC charge, on the line between them, is the electric field zero?
q1 = +5 µCq2 = +10 µCd = 40 cm
x = cm
2.Coulomb’s law & electric fieldsHard
Charges of +1 µC and +1 µC are 20 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 = +1 µCq2 = +1 µCd = 20 cmh = 3 cm
E = N/C
3.Series & parallel resistorsMedium
Find the current IR1 delivered by the source, which flows through R1.
V1 = 12 VR1 = 3 ΩR2 = 4 ΩR3 = 15 ΩR4 = 4 Ω
IR1 = A
4.Series & parallel resistorsHard
Find the equivalent resistance Req seen by the source V1.
V1 = 30 VR1 = 4 ΩR2 = 30 ΩR3 = 6 ΩR4 = 15 ΩR5 = 5 ΩR6 = 2 Ω
Req = 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 capacitor voltage vC at t = 150 ms.
Vs = 24 VR1 = 4.7 kΩR2 = 3.3 kΩC = 100 µFt = 150 ms
vC = V
6.RC & RL transientsHard
Just before the switch closes at t = 0, the capacitor holds 24 V with the polarity shown. How long does it take for vC to reach 19.2 V? Find t.
Vs = 12 VR = 10 kΩC = 22 µFvC(0) = 24 V
t = ms
7.Magnetic fields & inductionMedium
An electron moves to the left through a uniform magnetic field pointing out of the page. Click the direction of the magnetic force on it.
v: to the leftB: out of the page
- 1up the page
- 2to the right
- 3down the page
- 4to the left
8.Magnetic fields & inductionHard
A proton passes undeflected through a velocity selector with crossed fields E = 5 × 10⁵ V/m and B₁ = 0.5 T. It then enters a second region with only a magnetic field B₂ = 1 T, at right angles. Find the radius r of its path there.
E = 5 × 10⁵ V/mB1 = 0.5 TB2 = 1 Tm = 1.673 × 10⁻²⁷ kg|q| = 1.602 × 10⁻¹⁹ C
r = cm
9.AC impedance & phasorsMedium
A series RLC circuit is connected to a 24 V (rms), 60 Hz supply. Find the real power P taken from the supply.
V = 24 V (rms)f = 60 HzR = 50 ΩL = 250 mHC = 47 µF
P = W
10.AC impedance & phasorsHard
A resistor R1 is connected in series with a parallel combination of R2 and a capacitor C, across a 12 V (rms), 500 Hz supply. Find the rms current I.
V = 12 V (rms)f = 500 HzR1 = 15 ΩR2 = 80 ΩC = 10 µF
I = A
3Solutions
1.Coulomb’s law & electric fields
Answer: x = 16.57 cm
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.
Equal fields
5/x² = 10/(40 − x)²
(40 − x)/x = √(10/5) = 1.414
Position
x = 40/(1 + 1.414) = 16.57 cm
Answer
x = 16.57 cm
The zero point is always closer to the smaller charge.
2.Coulomb’s law & electric fields
Answer: E = 473992 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 = √(10² + 3²) = 10.44 cm
E1 = k·1 × 10⁻⁶/0.1044² = 824771 N/C
Components
The components along the line joining the charges cancel.
E = 2·E1·(h/r) = 2·824771·0.2873 = 473992 N/C
Answer
E = 473992 N/C
The field points along the bisector, away from the charges.
3.Series & parallel resistors
Answer: IR1 = 1.903 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 = 15 + 4 = 19 Ω
R2 ∥ 19 = 4·19/(4+19) = 3.304 Ω
Req = R1 + 3.304 = 3 + 3.304 = 6.304 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 = 12/6.304 = 1.903 A
Answer
IR1 = 1.903 A
4.Series & parallel resistors
Answer: Req = 11.93 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 = 5 + 2 = 7 Ω
R4 ∥ 7 = 4.773 Ω
R3 + 4.773 = 10.77 Ω
R2 ∥ 10.77 = 7.926 Ω
Req = R1 + 7.926 = 4 + 7.926 = 11.93 Ω
Answer
Req = 11.93 Ω
5.RC & RL transients
Answer: vC = 5.333 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.
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 = 4.7·3.3/(4.7 + 3.3) = 1.939 kΩ
vC(∞) = 24·3.3/(4.7 + 3.3) = 9.9 V
Time constant
τ = Rth·C = 1.939·100 = 193.9 ms
Charging curve
vC(t) = 9.9·(1 − e−150/193.9) = 5.333 V
Answer
vC = 5.333 V
6.RC & RL transients
Answer: t = 112.4 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) = 24 V, vC(∞) = Vs = 12 V
τ = R·C = 10 kΩ · 22 µF = 220 ms
vC(t) = 12 + (24 − 12)·e−t/τ
Solve for the time
19.2 = 12 + (12)·e−t/τ → e−t/τ = (19.2 − 12)/(24 − 12) = 0.6
t = τ·ln((24 − 12)/(19.2 − 12)) = 220·ln(1.667) = 112.4 ms
Answer
t = 112.4 ms
7.Magnetic fields & induction
Answer: down the page
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).
Right-hand rule
Point the fingers along the current or velocity and curl them towards B; the thumb gives the force on a positive charge.
v: to the left, B: out of the page
Negative charge
The force on an electron is opposite to that on a positive charge.
Direction of the force
down the page
Answer
F: down the page
8.Magnetic fields & induction
Answer: r = 1.044 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).
Selected speed
Undeflected: the electric force qE balances the magnetic force qvB.
v = E/B1 = 5 × 10⁵/0.5 = 1 × 10⁶ m/s
Radius in the second field
r = mv/(|q|B2) = 0.01044 m = 1.044 cm
Answer
r = 1.044 cm
9.AC impedance & phasors
Answer: P = 7.329 W
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 = 94.25 Ω, XC = 56.44 Ω
Impedance
|Z| = √(50² + (37.81)²) = 62.69 Ω
Real power
Inductors and capacitors only store and return energy; the resistor absorbs all of the real power.
I = 24/62.69 = 0.3829 A
P = I²·R = 0.3829²·50 = 7.329 W
Answer
P = 7.329 W
10.AC impedance & phasors
Answer: I = 0.3176 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
XC = 1/(2π·500·10·10⁻⁶) = 31.83 Ω
Parallel combination
Multiply top and bottom by the complex conjugate to split Zp into real and imaginary parts.
Zp = R₂·(−jXC)/(R₂ − jXC) = (R₂·XC² − jR₂²·XC)/(R₂² + XC²)
Zp = 10.93 − j27.48 Ω
Total impedance
Z = 15 + 10.93 − j27.48 = 25.93 − j27.48 Ω
|Z| = √(25.93² + 27.48²) = 37.79 Ω
Current
I = V/|Z| = 12/37.79 = 0.3176 A
Answer
I = 0.3176 A
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