Electromagnetism

Module 6 is worth about a quarter of the paper. Electromagnetic induction is the most heavily examined part (nearly 10 marks a year on average), followed by charged particles in fields and the applications of the motor effect.

• Charged particles, conductors and fields: uniform electric fields between parallel plates, force and work on a charge, motion in electric fields (parabolic) and magnetic fields (circular).
• The motor effect: force on a current-carrying wire in a magnetic field, force between parallel wires and its link to the ampere.
• Electromagnetic induction: magnetic flux, Faraday's law, Lenz's law, transformers (ideal and real) and high-voltage transmission.
• Applications of the motor effect: torque on a coil, DC motors, generators, back emf, AC induction motors and magnetic braking.

Uniform electric fields

Two parallel plates with a potential difference $V$ across them produce a uniform electric field between them (away from the edges). The field points from the positive plate to the negative plate. A positive charge feels a force along the field; a negative charge (such as an electron) feels a force against it.

• $E$: electric field strength (${\text{V m}}^{-1}$ or ${\text{N C}}^{-1}$); $d$: plate separation (m).
• $F$: force (N); $q$: charge (C), with $e=1.602\times 10^{-19}$ C.
• $W$: work done on the charge by the field (J), which becomes kinetic energy if it starts from rest: $\frac{1}{2}m{v}^2=qV$.

A particle that enters the field moving parallel to the plates is like a projectile: constant velocity along the plates, constant acceleration $a=qE/m$ across them, so it follows a parabola. Gravity on electrons and protons is negligible compared with the electric force.

Worked example: deflecting an electron

An electron enters midway between plates 3.0 cm apart and 5.0 cm long, moving parallel to them at $2.0\times 10^7{\text{m s}}^{-1}$. The potential difference is 150 V. ($m_e=9.109\times 10^{-31}$ kg)

The force is towards the positive plate. The time spent between the plates and the sideways deflection are

The deflection of 2.7 mm is less than 15 mm, so the electron clears the plate.

Magnetic fields

A charge moving through a magnetic field feels a force

where $B$ is the magnetic field strength (T) and $\theta$ is the angle between $v$ and $B$. A charge moving parallel to the field feels no force. Find the direction with the right-hand palm rule: fingers along $B$, thumb along the velocity of a positive charge, and the palm pushes in the direction of the force. For an electron, the force is the opposite way.

Because the magnetic force is always perpendicular to the velocity, it does no work: it changes the direction but not the speed. A particle moving perpendicular to a uniform field therefore moves in a circle, and setting $qvB=m{v}^2/r$ gives

Faster or heavier particles curve less; stronger fields or larger charges curve more. Mass spectrometers and cyclotrons use this.

If an electric field and a magnetic field are set at right angles so their forces balance, only particles with $v=E/B$ pass straight through. This is a velocity selector, used in Thomson's experiment (Module 8).

A current-carrying wire in a magnetic field feels a force because the moving charges in it do:

$l$ is the length of wire in the field (m), $I$ is the current (A), and $\theta$ is the angle between the wire and the field. Use the right-hand palm rule with the thumb along conventional current.

Example. A 0.45 m wire carrying 3.2 A sits at $30^{\circ }$ to a 0.065 T field:

Parallel wires

Each wire sits in the magnetic field of the other, so they exert forces on each other:

with ${\mu }_0=4\pi \times 10^{-7}{\text{T m A}}^{-1}$ and $r$ the separation (m). Currents in the same direction attract; opposite directions repel. By Newton's third law, the two forces are equal in size even when the currents are different.

Example. Two wires 2.5 cm apart carry 8.0 A and 4.0 A in opposite directions.

So a 1.2 m length of either wire feels $3.1\times 10^{-4}$ N of repulsion.

This force was historically used to define the ampere (1 A in two wires 1 m apart gives $2\times 10^{-7}{\text{N m}}^{-1}$). Since 2019 the ampere has been defined by fixing the value of the elementary charge $e$, but the syllabus still expects you to relate the parallel-wire force to the ampere.

Magnetic flux

Magnetic flux $\Phi$ measures how much magnetic field passes through an area:

$\Phi$ is in webers (Wb), $A$ is the area (${\text{m}}^2$) and $\theta$ is the angle between the field and the normal (perpendicular) to the area. Flux is greatest when the field passes straight through the loop and zero when the field lies in the plane of the loop.

Faraday's law and Lenz's law

An emf $\epsilon$ (V) is induced in a coil of $N$ turns whenever the flux through it changes. The flux can change because $B$ changes, the area changes, or the angle changes (rotating coil). A bigger rate of change gives a bigger emf.

The minus sign is Lenz's law: the induced current flows in the direction that produces a magnetic field opposing the change in flux that caused it. This is required by conservation of energy. If the induced current helped the change, the motion would speed itself up and create energy from nothing. Work must be done against the opposing force, and that work becomes electrical energy.

Applying Lenz's law. Suppose the south pole of a magnet is pulled away from a coil. The flux from that south pole through the coil is decreasing, so the induced current tries to keep it: the near end of the coil becomes a north pole, attracting the departing south pole and opposing its motion. Always state (1) what is changing, (2) what the induced field does, (3) the resulting force or current direction.

Worked example: Faraday's law

A square coil of 150 turns, 12 cm on each side, sits with its plane perpendicular to a 0.45 T field. The field is switched off in 0.030 s.

This is the average emf while the field is switched off.

Eddy currents

Changing flux through a solid conductor induces circulating eddy currents. By Lenz's law they oppose the motion that causes them, which is the basis of magnetic braking in trains and roller coasters. They also heat transformer cores, wasting energy.

Transformers

A transformer has primary and secondary coils wound on a shared iron core. An alternating current in the primary produces a changing flux in the core, which induces an alternating emf in the secondary. Transformers do not work on steady DC, because there is no change in flux.

The second equation applies to an ideal transformer (100% efficient). Real transformers lose energy through:

• Flux leakage: not all primary flux passes through the secondary. Reduced by winding coils on top of each other on a closed core.
• Eddy currents in the core: reduced by laminating the core (thin insulated sheets) or using ferrite.
• Resistive heating in the coils: reduced by using thick, low-resistance copper wire.

Power transmission

Power lost as heat in transmission lines is $P_{loss}=I^{2}R$. For a fixed power $P=VI$, raising the voltage lowers the current, and the loss falls with the square of the current. Step-up transformers raise voltage at the power station; step-down transformers lower it near homes for safety.

Example. A substation sends 500 kW through lines of total resistance 4.0 $\Omega$. At 11 kV, $I=500000/11000=45.5$ A and $P_{loss}=45.5^2\times 4.0=8.3$ kW (1.7%). At 22 kV, $I=22.7$ A and $P_{loss}=2.1$ kW, a quarter of the loss. A step-down transformer from 11 kV to 415 V needs a turns ratio $N_p/N_s=11000/415=26.5$.

Torque on a coil

A current-carrying coil in a magnetic field has forces on its two sides acting in opposite directions, producing a torque:

$n$ is the number of turns, $A$ the coil area (${\text{m}}^2$), and $\theta$ the angle between the magnetic field and the normal to the coil. Torque is greatest when the plane of the coil is parallel to the field and zero when the plane is perpendicular to the field.

Example. A 120-turn coil measuring 5.0 cm by 3.0 cm carries 1.5 A in a 0.35 T field, with the normal to the coil at $50^{\circ }$ to the field. The area is $0.050\times 0.030=0.0015{\text{m}}^2$, so

The maximum possible torque for this coil is 0.095 N m.

DC motors

A DC motor has a coil on an axle in a magnetic field, connected to the supply through a split-ring commutator and brushes. Every half turn, as the coil passes the position where its plane is perpendicular to the field, the commutator reverses the current in the coil. This keeps the torque acting in the same rotational direction, so rotation is continuous. Real motors use several coils and curved pole pieces (radial field) to give a steadier torque.

Generators and back emf

A generator is a motor run backwards: rotating a coil in a magnetic field changes the flux through it, inducing an emf. With slip rings, the output is AC (the emf reverses every half turn). With a split-ring commutator, the output is DC that pulses in one direction.

A spinning motor coil is also a generator. It induces a back emf that opposes the supply voltage (Lenz's law), so the current in the coil is

Back emf grows with speed. That is why a motor draws a large current when starting or when heavily loaded (slow), and why motors can overheat if jammed.

AC induction motors

The stator coils carry AC arranged to produce a rotating magnetic field. The rotor is a "squirrel cage" of conducting bars, not connected to any supply. The rotating field changes the flux through the rotor, inducing currents in the bars; these currents then feel a motor-effect force that drags the rotor around after the field. The rotor always turns slightly slower than the field, because if it matched the field's speed there would be no change in flux and no induced current.

Magnetic braking

A moving conductor in a magnetic field has eddy currents induced in it; by Lenz's law these produce a force opposing the motion. The kinetic energy becomes heat in the conductor. Braking force is proportional to speed, so the braking is smooth and needs no contact, but it cannot hold a vehicle stationary.

• Using the angle to the plane of the coil in $\Phi =BA\cos \theta$ or $\tau =nIAB\sin \theta$, when both use the angle to the normal.
• Applying the right-hand rule to an electron without reversing the force.
• Saying a magnetic force speeds up or slows down a charge. It only changes direction.
• Stating Lenz's law as "the induced current opposes the magnetic field". It opposes the change in flux.
• Saying transformers "create" power or work on DC. Power out is at most power in, and DC gives no changing flux.
• Saying high voltage reduces resistance. It reduces current, and therefore $I^{2}R$ losses.
• Confusing a commutator (DC motor, DC generator) with slip rings (AC generator).

• Lenz's law answers need a causal chain: change in flux, induced current, induced field, opposing force, energy conservation. Each link is usually worth a mark.
• Explain how a motor or generator works: name the parts and say what each does and why (for example, "the commutator reverses the current every half turn so the torque stays in the same direction").
• Compare DC motors with AC induction motors, or ideal with real transformers, using similarities and differences side by side.
• Evaluate questions (for example on energy losses or transmission) need a judgement supported by a calculation or specific features.
• Direction questions: draw a small diagram with axes and say which hand rule you used. Use "into the page" and "out of the page" correctly.
• Convert cm to m and ${\text{cm}}^2$ to ${\text{m}}^2$ before substituting. Round to the significant figures of the data and include units (Wb, T, V, N m).

• Module 5: charges in uniform electric fields move like projectiles; charges in magnetic fields move in uniform circular motion; torque on a coil uses $\tau =rF\sin \theta$.
• Module 7: Maxwell unified electricity and magnetism, predicting electromagnetic waves. Induced emf from changing magnetic fields is half of that story.
• Module 8: Thomson's charge-to-mass experiment combines electric and magnetic deflection; particle accelerators use the same principles.
• Year 11 Electricity and Magnetism: electric fields, current, $P=VI$ and magnetic fields of wires and solenoids are assumed knowledge.