VCE Physics electromagnetism deep dive: fields, induction and transformers

What this guide is for

Electromagnetism spans most of Unit 3, Areas of Study 2 and 3, and it carries a heavy weight in the end-of-year exam. The topic rewards a clean mental model: charges feel forces from electric and magnetic fields, moving charges create magnetic fields, and changing magnetic flux induces an EMF. Build that model, then layer the formulas onto it.

Magnetic fields and how they arise

A magnetic field $B$ (units: tesla, T) is produced by moving charges. Two routine cases at VCE level:

Long straight wire. Concentric circular field lines around the wire. The right-hand grip rule sets the direction: thumb along the conventional current, fingers curl in the direction of the field. The field magnitude decreases with distance from the wire.

Solenoid. A long coil with many turns. Inside the coil the field is approximately uniform and parallel to the axis; outside, the field is weak. A solenoid behaves like a bar magnet with a north and south pole. The right-hand rule applies: curl the fingers in the direction of current, the thumb points to the north pole.

VCE Physics does not require students to derive field magnitudes from first principles. Conceptual fluency with field shapes, direction, and superposition is what is tested.

Force on a moving charge

A charge $q$ moving with velocity $v$ in a magnetic field $B$ feels a force

where $\theta$ is the angle between $v$ and $B$. When $v$ is perpendicular to $B$, the magnitude is $F = qvB$ and the force is always perpendicular to the velocity, so the charge moves in a circle of radius $r = mv/(qB)$.

Worked example. A proton enters a uniform magnetic field $B = 0.20$ T with velocity $5.0 \times 10^6$ m/s perpendicular to the field. Find the radius of its circular path. ($m_p = 1.67 \times 10^{-27}$ kg, $q = 1.6 \times 10^{-19}$ C.)

$r = mv/(qB) = (1.67 \times 10^{-27})(5.0 \times 10^6) / (1.6 \times 10^{-19} \times 0.20) = 2.6 \times 10^{-1}$ m = 26 cm.

Force on a current-carrying conductor

A wire of length $L$ carrying current $I$ in a field $B$ feels

where $n$ is the number of conductors (for a coil with $N$ turns, $n = N$) and $\theta$ is the angle between current direction and $B$. When current is perpendicular to the field, the force has magnitude $F = nBIL$. This is the principle behind the DC motor.

The DC motor

A rectangular current-carrying coil in a magnetic field. Forces on the two sides of the coil parallel to the rotation axis produce a torque that turns the coil. A split-ring commutator reverses the current direction at every half rotation, so the torque continues in the same rotational sense.

VCE exam answers should mention: forces on opposite sides of the coil are in opposite directions, producing a couple; the commutator reverses current at the dead point; without the commutator, the coil would oscillate rather than rotate.

Magnetic flux

Magnetic flux through a loop of area $A$ is

where $\theta$ is the angle between the field and the normal to the loop. Units: weber (Wb = T m$^2$). Flux is a scalar; flux density $B$ is a vector.

For a coil with $N$ turns, the total flux linkage is $N\Phi$.

Faraday's law

When the flux through a coil changes, an EMF is induced:

At VCE level the calculation is usually average EMF over a time interval:

For a coil of area $A$ in a changing field, $\Delta\Phi = A\,\Delta B$ (if orientation is fixed). For a coil moving in a fixed field, $\Delta\Phi$ comes from the changing $\cos\theta$ or changing area.

Worked example. A coil of 300 turns and area $0.020$ m$^2$ sits in a magnetic field that drops from $0.40$ T to zero in $0.10$ s. Average induced EMF?

$\Delta\Phi = 0.40 \times 0.020 = 8.0 \times 10^{-3}$ Wb per turn.

$\varepsilon = 300 \times (8.0 \times 10^{-3}) / 0.10 = 24$ V.

Lenz's law

The induced current flows in a direction such that its own magnetic field opposes the change in flux that produced it. Two consequences:

• Energy conservation: pushing a magnet into a coil takes work because the induced current's field opposes the push.
• Sign in $\varepsilon = -N\,\mathrm{d}\Phi/\mathrm{d}t$: the minus sign encodes Lenz's law.

VCE markers expect a clear chain: state how flux is changing, state Lenz's law, state the direction of the induced current consistent with that.

Generators

A rotating coil in a magnetic field produces a sinusoidally varying EMF:

with peak EMF $\varepsilon_{\text{peak}} = NBA\omega$ and angular frequency $\omega = 2\pi f$.

AC generator. Slip rings deliver the alternating EMF unchanged. Australian mains is 50 Hz.

DC generator. A split-ring commutator reverses the connections every half period. The output is unidirectional but pulses. Multi-segment commutators smooth the output.

RMS voltage of a sinusoid: $V_{\text{rms}} = V_{\text{peak}}/\sqrt{2}$. Power dissipated in a resistor: $P = V_{\text{rms}}^2/R$, not $V_{\text{peak}}^2/R$.

Transformers

Two coils sharing an iron core. AC in the primary creates a changing flux that links the secondary, inducing an EMF.

For an ideal transformer:

Worked example. A power station feeds a transformer at 10 kV. The transformer has 200 primary turns and 5000 secondary turns. Output voltage and current if primary draws 50 A?

$V_s = V_p (N_s/N_p) = 10000 \times 25 = 2.5 \times 10^5$ V = 250 kV.

$I_s = I_p (N_p/N_s) = 50 / 25 = 2.0$ A.

Real transformers have losses (resistive heating in windings, eddy currents and hysteresis in the core), so output power is slightly less than input.

Power transmission

Transmission line loss is $P_{\text{loss}} = I^2 R$. For a fixed transmitted power $P = VI$, the current $I = P/V$ falls inversely with $V$, so the loss falls as $1/V^2$.

Doubling the transmission voltage cuts losses to a quarter. This is why national grids use 220-500 kV transmission lines, stepped down to 415 V three-phase or 240 V single-phase at the consumer.

Cross-links to dot points

This guide draws on the following Unit 3 dot points:

• Magnetic fields (around current-carrying wires and solenoids).
• Magnetic force on charges and on current-carrying conductors.
• DC motor.
• Electromagnetic induction and EMF.
• Generators and transformers.

For numerical practice, see the worked-problems guide. For the bigger-picture exam plan, see the Units 3 and 4 exam structure guide.

Check your knowledge

A focused set on Unit 3 AoS 2 and AoS 3 in the VCAA Section A / B style. Attempt under exam conditions before checking the solutions block. Take $g = 9.80 \ \text{m s}^{-2}$ where required; other constants are on the VCAA data sheet.

1. State Lenz's law and explain how it follows from conservation of energy. (2 marks)
2. A 2.0 m long straight wire carrying 5.0 A is placed perpendicular to a uniform magnetic field of 0.20 T. Calculate (a) the force on the wire and (b) the new force if the wire is rotated to lie at 30 degrees to the field. (3 marks)
3. (a, 3) A circular loop of radius 0.10 m sits in a uniform magnetic field that is initially 0.40 T and falls linearly to zero over 0.25 s. The loop has 50 turns. Calculate the magnitude of the induced EMF. (b, 2) State the direction of the induced current relative to the field, and justify with Lenz's law. (5 marks)
4. A satellite of mass 1500 kg is placed in a circular orbit 400 km above the surface of the Earth ($R_E = 6.37 \times 10^{6} \ \text{m}$, $M_E = 5.98 \times 10^{24} \ \text{kg}$, $G = 6.67 \times 10^{-11} \ \text{N m}^{2} \text{kg}^{-2}$). (a) Calculate the orbital speed. (b) Calculate the period in minutes. (c) Calculate the gravitational potential energy of the satellite at this altitude (taking infinity as zero). (7 marks)
5. A 240 V, 50 Hz domestic supply feeds a step-down transformer that delivers 12 V to a halogen lighting circuit drawing 5.0 A. (a) Calculate the turns ratio. (b) Calculate the current in the primary (assume ideal transformer). (c) Calculate the resistive power loss in 50 m of $1.5 \ \text{mm}^{2}$ copper wire on the secondary side with resistivity $\rho = 1.7 \times 10^{-8} \ \Omega \text{ m}$. (7 marks)
6. The Loy Yang to Melbourne 500 kV transmission line carries 2.0 GW of power over 150 km of cable of total resistance $5.0 \ \Omega$. (a) Calculate the line current. (b) Calculate the $I^2R$ power loss as a percentage of transmitted power. (c) Compare with the same line operated at 110 kV instead of 500 kV. (6 marks)
7. (a, 2) An electron is accelerated from rest through a potential difference of 250 V. Calculate its final speed ($e = 1.60 \times 10^{-19} \ \text{C}$, $m_e = 9.11 \times 10^{-31} \ \text{kg}$). (b, 3) The same electron enters a uniform 5.0 mT magnetic field perpendicular to its velocity. Calculate the radius of its circular path. (5 marks)
8. A practical investigation measures the peak-to-peak EMF of a generator's coil as it spins in a fixed magnetic field. The student records $V_{pp}$ at six rotation rates: 5.0 Hz $\rightarrow$ 0.84 V, 10.0 Hz $\rightarrow$ 1.65 V, 15.0 Hz $\rightarrow$ 2.55 V, 20.0 Hz $\rightarrow$ 3.30 V, 25.0 Hz $\rightarrow$ 4.20 V, 30.0 Hz $\rightarrow$ 5.05 V. (a) State the expected relationship between peak EMF and frequency and explain physically. (b) Calculate the gradient of a $V_{pp}$ versus $f$ plot from the first and last points and state its physical meaning, with uncertainty estimated from the spread of the data. (c) Identify one systematic and one random source of uncertainty in the measurement. (7 marks)