PHY132H1/Unit 5

Unit 5 · Introduction to Physics II

Magnetic Fields

A second field, with a strange rule: it acts only on moving charges, and the force is perpendicular to both the velocity and the field. That perpendicularity has a striking consequence — a magnetic force can never change a particle's speed.

By the end you can

• Apply the right-hand rule reliably.
• Compute the force on a moving charge and on a current-carrying wire.
• Explain why magnetic forces do no work.
• Find the field produced by currents.

Why it matters & where it comes from

Why it exists

Moving charges both feel and produce magnetic fields. That reciprocity is what makes motors and generators possible, and Unit 6 shows it is the same phenomenon viewed two ways.

In the real world

Every electric motor, every hard disk, every MRI scanner. The Earth's field deflects charged particles from the Sun, which is what produces the aurora and what keeps the atmosphere from being stripped away.

Story & open questions

Ørsted noticed in 1820 that a compass needle moved when a nearby current was switched on — the first evidence that electricity and magnetism were connected. Magnetic monopoles have never been found; unlike charge, magnetism seems to come only in pairs.

Force on a moving charge

Rule The magnetic force

$F = qvB\sin\theta$, with $\theta$ between the velocity and the field.

Direction: perpendicular to both $v$ and $B$, given by the right-hand rule.

Three consequences follow immediately:

A stationary charge feels nothing. $v = 0$ gives $F = 0$.

A charge moving along the field feels nothing. $\sin 0 = 0$.

The force is maximum when the motion is perpendicular to the field.

Rule The right-hand rule

Point your fingers along $v$, curl them toward $B$; your thumb gives the force on a positive charge. For a negative charge, reverse it.

Practise physically rather than mentally. Direction errors, not arithmetic, are what lose marks in this unit.

Example Why magnetic forces do no work

Work is $Fd\cos\theta$ with $\theta$ between force and displacement.

The magnetic force is always perpendicular to the velocity, hence to the displacement. So $\cos\theta = 0$ and the work is always zero.

A magnetic field can change a particle's direction but never its speed or kinetic energy. Motors do work, but the energy comes from the electrical supply, not from the magnet.

Example Circular motion in a field

A charge moving perpendicular to a uniform field feels a constant-magnitude force always perpendicular to its motion — exactly the condition for a circle.

Setting $qvB = mv^2/r$ gives $r = \dfrac{mv}{qB}$.

Heavier or faster means a wider circle; stronger field means tighter. This is how a mass spectrometer separates isotopes and how a cyclotron confines particles.

Force on a current

Rule A wire is many moving charges

$F = BIL\sin\theta$ for a straight wire of length $L$ carrying current $I$.

Same right-hand rule, with the current direction playing the role of $v$.

Example How a motor works

A current loop in a magnetic field feels opposite forces on its two sides, since the current runs in opposite directions there.

Those forces make a couple — zero net force, nonzero torque — so the loop rotates.

Half a turn later the torque would reverse and stall it, so a commutator flips the current direction each half turn. That is the entire principle of a direct-current motor.

Sources of magnetic field

Rule Currents make fields

Long straight wire: $B = \dfrac{\mu_0 I}{2\pi r}$, in circles around the wire. Grip the wire with your right hand, thumb along the current, and your fingers curl the field direction.

Centre of a loop of radius $R$: $B = \dfrac{\mu_0 I}{2R}$.

Inside a long solenoid: $B = \mu_0 n I$ with $n$ turns per metre — uniform, and independent of position.

Here $\mu_0 = 4\pi \times 10^{-7}$ T·m/A.

Example Two parallel wires

Each wire sits in the other's field, so each feels a force.

Currents in the same direction: the wires attract.

Currents in opposite directions: they repel.

This is the reverse of the electrostatic intuition, where like repels like, and it is worth checking with the right-hand rule twice rather than trusting memory.

No magnetic monopoles

Electric field lines start and end on charges. Magnetic field lines always form closed loops, because there is no magnetic charge to begin or end on. Cutting a bar magnet in half gives two magnets, not a north and a south pole.

Common pitfalls

• Forgetting the force is zero for a stationary charge, or for motion along the field.
• Using the right-hand rule for a negative charge without reversing.
• Thinking a magnetic force changes speed. It never does.
• Confusing the field produced by a current with the force on it.
• Assuming parallel currents repel. They attract.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Zero force

Name two situations where a charge in a magnetic field feels no force.

Answer

When it is at rest, and when it moves parallel or antiparallel to the field.

Practice 2 · Magnitude

A proton at $2 \times 10^6$ m/s crosses a $0.5$ T field perpendicularly. Find the force.

Answer

$F = (1.6 \times 10^{-19})(2 \times 10^6)(0.5) = 1.6 \times 10^{-13}$ N.

Practice 3 · Radius

Find that proton's orbital radius. Take $m_p = 1.67 \times 10^{-27}$ kg.

Answer

$r = mv/(qB) = (1.67 \times 10^{-27})(2 \times 10^6)/((1.6 \times 10^{-19})(0.5)) = 0.042$ m.

Practice 4 · No work

Why can a magnetic field not speed up a particle?

Answer

The force is always perpendicular to the velocity, so it does no work and the kinetic energy cannot change.

Practice 5 · Wire field

Find the field $0.05$ m from a wire carrying $10$ A.

Answer

$B = (4\pi \times 10^{-7})(10)/(2\pi \times 0.05) = 4 \times 10^{-5}$ T — comparable to the Earth's field.

Practice 6 · Parallel wires

Two wires carry current in the same direction. Attract or repel?

Solution

Attract. Apply the right-hand rule twice: find the field of wire one at wire two, then the force on wire two in that field. The result points toward wire one.

Practice 7 · Solenoid

Find the field inside a solenoid with $500$ turns per metre carrying $2$ A.

Answer

$B = \mu_0 nI = (4\pi \times 10^{-7})(500)(2) = 1.26 \times 10^{-3}$ T, uniform throughout.

Practice 8 · Monopoles

What happens if you cut a bar magnet in half?

Answer

You get two complete magnets, each with a north and a south pole. Isolated poles have never been observed, which is why magnetic field lines always close.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY132H1 topic map. It should cover the following items before you move on:

• The magnetic force. Its formula, its direction, and the two cases where it vanishes.
• No work. Why speed cannot change, and what that implies for motors.
• Forces on currents. Wires, loops, and how a motor turns.
• Sources. Wire, loop and solenoid fields, and the absence of monopoles.

Textbook/source alignment

The calendar names "magnetism". Check the syllabus on Quercus for the assigned problems; right-hand-rule questions are standard on term tests.

Core idea map

Moving charges feel magnetic forces and also create magnetic fields. The force is perpendicular to the motion, so it steers without speeding up. Unit 6 closes the loop by showing changing magnetic fields create electric ones.

Study routine. Use your actual right hand every time, and say aloud which quantity each finger represents. Mental right-hand rules go wrong far more often than physical ones.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · Requirement

What must a charge be doing to feel a magnetic force?

Guide

Moving, and not parallel to the field.

Extension 2 · Work

How much work does a magnetic force do?

Guide

None, ever. It is always perpendicular to the displacement.

Extension 3 · Circular motion

Why does a charge move in a circle in a uniform field?

Guide

The force has constant magnitude and stays perpendicular to the velocity — the exact condition for uniform circular motion.

Extension 4 · Closed lines

Why do magnetic field lines form loops?

Guide

There are no magnetic charges for them to start or end on, so they have nowhere to terminate except back on themselves.

← Unit 04Current & circuits Unit 06 →Induction