Moving Charges & Magnetism; Magnetism & Matter

How currents make fields, how fields push on currents, and what makes a magnet · Class 12 Physics

1The big idea: magnetism is moving charge

The whole of this chapter grows from one sentence: a magnetic field is produced by moving charge, and it pushes only on moving charge. A stationary electron feels an electric field but is completely blind to a magnetic one. Set that same electron moving and a magnetic field suddenly grips it. A current in a wire is just charge in motion, so everything about currents and magnets is really a story about charges that are going somewhere.

Two questions run through the chapter, and they are mirror images of each other. First, given a magnetic field B, what force does it exert on a moving charge or a current? Second, turning it around, what magnetic field does a current itself create in the space around it? Keep those two questions separate in your mind — students who blur them together get lost — and the chapter falls into place.

2The magnetic force on a moving charge

Put a charge q moving with velocity v into a magnetic field B. The force on it is

Magnetic (Lorentz) force F = q v B sin θ

where θ is the angle between the velocity and the field. Three features of this little formula do almost all the work in the chapter:

No motion, no force. If v = 0 the force is zero. A magnet does nothing to a charge at rest.
Move along the field and still nothing. If the charge travels parallel to B then θ = 0, sin θ = 0, and the force vanishes. The force is largest when the charge cuts across the field at 90°.
The force is sideways. It points at right angles to both v and B — the direction your right palm pushes when the fingers point along v and curl toward B (for a positive charge; reverse it for a negative one).

Because the force is always perpendicular to the velocity, it can never speed the charge up or slow it down — it can only bend its path. This is the single most important consequence, worth stating on its own:

Remember this

The magnetic force does no work. It is always perpendicular to the motion, so it changes a charge's direction but never its speed or kinetic energy. A magnetic field can steer a particle forever without adding a single joule.

If an electric field E is also present, the two forces simply add, giving the full Lorentz force F = q E + q v B sin θ (with the magnetic part sideways as above). The electric part can do work; the magnetic part never does.

A positive charge moving upward across a field pointing into the page feels a force to the right — always toward the centre — so it wheels round in a circle of radius r = mv/qB.

3A charge going in a circle — the cyclotron

Send a charge straight across a uniform field (so θ = 90° and sin θ = 1). The force qvB is perpendicular to the motion, constant in size, and always turned toward one side — exactly the recipe for uniform circular motion. The magnetic force supplies the centripetal force:

Balance the forces q v B = m v²r

Cancel one factor of v and rearrange for the radius:

Radius of the circle r = m vq B

Read it off: a faster particle sweeps a wider circle, and a stronger field winds it into a tighter one. Now find the time for one loop by dividing the circumference by the speed, T = 2πr / v, and substitute r:

Period — notice what drops out T = 2π rv = 2π mq B

The speed has cancelled. Every charge of a given type takes the same time to go round, whether it crawls in a small circle or races in a large one — a fast particle simply has more distance to cover, and the two effects exactly balance. This surprising fact is the whole principle of the cyclotron: because the period is fixed, an alternating voltage tuned to that one cyclotron frequency f = qB / 2πm can kick the particle every half-turn, spiralling it out to ever higher energy. (If the field is not perpendicular, the along-field velocity component is untouched and the particle traces a helix instead of a flat circle.)

◆ Activity — a charge in a magnetic field

A proton is fired across a field that points into the screen (⊗). Change the field strength B and the speed v, and watch the orbit. A stronger field pulls the circle tighter; a faster proton flings it wider — exactly what r = mv/qB promises. Watch too how the time per loop follows the field only, never the speed.

4Force on a current-carrying wire

A wire carrying a current is a stream of charges on the move, so a magnetic field pushes on the whole wire. Add up the little qvB forces on all the drifting charges and the tidy result is

Force on a straight wire in a field F = B I L sin θ

with I the current, L the length of wire in the field, and θ the angle between the wire and B. It is the same physics as qvB sin θ, just bookkept per unit of wire rather than per charge. The direction is again given by the right hand: point the fingers along the current and curl them toward the field, and the thumb (or palm-push) gives the force. This is the force that spins every electric motor.

5Two parallel currents — and the ampere

Here two ideas of the chapter meet. Wire 1 creates a field; wire 2, sitting in that field, feels a force. Wire 1 with current I₁ makes a field B₁ = μ₀ I₁ / 2π d at the location of wire 2, a distance d away. Wire 2, carrying I₂, then feels a force per unit length B₁ I₂:

Force per metre between two wires FL = μ₀ I₁ I₂2π d

The rule of thumb is worth memorising: currents in the same direction attract; opposite currents repel — the reverse of what you might guess from charges. This mutual force is exactly how the ampere used to be defined: one ampere is the current which, in two infinitely long parallel wires one metre apart, produces a force of 2 × 10⁻⁷ newton on every metre of wire. Put I₁ = I₂ = 1 A and d = 1 m into the formula and, since μ₀ = 4π × 10⁻⁷ T m A⁻¹, you get precisely μ₀ / 2π = 2 × 10⁻⁷ N m⁻¹ — the definition drops straight out.

6The fields that currents make

Now the second question: what field does a current itself set up? Two standard results carry most exam problems.

A long straight wire. The field circles the wire in closed loops, its strength falling off with distance r:

Field around a long straight wire B = μ₀ I2π r

The direction is the right-hand grip rule: point your right thumb along the current and your fingers curl the way the field goes. This is a direct fruit of Ampère's circuital law, ∮ B · dl = μ₀ I, which says the field summed around any closed loop equals μ₀ times the current threading it.

A solenoid. Wind the wire into a long tight coil and the loops' fields reinforce inside and cancel outside, giving a nearly uniform field along the axis that depends only on the current and how tightly it is wound — n being the number of turns per metre:

Field inside a long solenoid B = μ₀ n I

Notice the radius of the coil does not appear — a fat solenoid and a thin one give the same field if wound equally tightly and fed the same current. A solenoid is, in effect, a bar magnet you can switch on and off, which is the heart of every electromagnet and relay.

Inside a long solenoid the loops' fields add to a uniform field B = μ₀ n I; from outside it behaves just like a bar magnet with a north and a south pole.

7Torque on a loop & the galvanometer

Put a rectangular current loop into a field. Opposite sides carry current in opposite directions, so the forces on them are equal and opposite — the loop feels no net push, but the two forces are offset and form a couple that twists it. Working through the geometry for a coil of N turns, area A, carrying current I, the turning effect is

Torque on a current loop τ = N B I A sin θ = m B sin θ

Here θ is the angle between the field and the loop's normal (the axle direction, perpendicular to its face), and m = N I A is the loop's magnetic moment — the same quantity that makes the loop behave like a tiny bar magnet. The torque is largest when the loop lies along the field (face parallel to B, so the normal is at 90°) and zero when the loop has swung round so its normal lines up with B. That is why a compass needle, itself a little magnetic moment, twists until it points along the field and then sits still.

The moving-coil galvanometer — the instrument behind every analogue ammeter and voltmeter — is this torque put to work. A coil hangs in the field of a specially shaped (radial) magnet, so that θ stays 90° whatever angle the coil turns to, and sin θ = 1 always. The magnetic twist N B I A is then balanced by a spring that pushes back in proportion to the deflection φ, i.e. k φ = N B I A. Solving, φ = (N B A / k) I — the needle's deflection is directly proportional to the current, giving an evenly spaced, linear scale you can simply read off.

8Magnetism & matter

Why is iron magnetic and copper not? Every atom is a swarm of orbiting and spinning electrons — tiny current loops, each with its own magnetic moment. How those atomic moments respond to an applied field sorts all materials into three families.

Diamagnetic (copper, water, bismuth). The atoms have no permanent moment. An applied field induces a feeble moment that, by Lenz's law, opposes the field, so these materials are weakly repelled. Their susceptibility χ is small and negative, and independent of temperature.
Paramagnetic (aluminium, oxygen, sodium). The atoms do carry permanent moments, but thermal jostling keeps them pointing every which way. A field nudges them into partial alignment, so the material is weakly attracted; χ is small and positive. Heat fights the alignment, which gives the Curie law: χ = CT — the magnetisation weakens in inverse proportion to the absolute temperature.
Ferromagnetic (iron, cobalt, nickel). Here neighbouring atomic moments lock together into large aligned regions called domains. A modest field swings whole domains into line, producing a strong magnetisation that can remain even after the field is removed — a permanent magnet. Heat it past its Curie temperature T_c and the domains collapse into disorder: the iron abruptly loses its ferromagnetism and becomes an ordinary paramagnet.

The magnetic susceptibility χ ties the magnetisation M a material develops to the field H that drives it, M = χ H. Its sign and size — tiny-negative, tiny-positive, or huge-positive — are the fingerprints that tell the three families apart.

Worked example

A proton (q = 1.6 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg) enters a uniform field B = 0.50 T at right angles, moving at v = 2.0 × 10⁶ m s⁻¹. Find (a) the radius of its circular path and (b) the time for one full revolution.

(a) Radius. Use r = mv / qB:

r = (1.67 × 10⁻²⁷)(2.0 × 10⁶)(1.6 × 10⁻¹⁹)(0.50) = 3.34 × 10⁻²¹8.0 × 10⁻²⁰ ≈ 0.042 m (about 4.2 cm).

(b) Period. Use T = 2π m / qB — note the speed does not enter:

T = 2π (1.67 × 10⁻²⁷)(1.6 × 10⁻¹⁹)(0.50) = 1.049 × 10⁻²⁶8.0 × 10⁻²⁰ ≈ 1.3 × 10⁻⁷ s.

Double it to v = 4.0 × 10⁶ m s⁻¹ and the radius would double to 8.4 cm, but the period would stay exactly the same — the cyclotron's defining trick.

9Common confusions to clear up

The magnetic force does no work, ever. It bends the path but never changes the speed. If a charge speeds up in a region, an electric field (or something else) is responsible — not the magnetic field.
A charge at rest feels no magnetic force. The v in qvB sin θ is not optional. Only moving charge — and hence only current — responds to a magnetic field.
Parallel currents attract; anti-parallel repel. This is the opposite of electric charges, and it is a favourite trap. Like currents pull together.
In τ = NBIA sin θ, the angle is measured from the loop's normal, not its face. Torque is maximum when the loop lies flat along the field (normal at 90°) and zero when the normal is aligned with the field.
Diamagnets are repelled, paramagnets attracted. Diamagnetic susceptibility is negative; paramagnetic is small-positive and falls off with temperature (Curie law); ferromagnetic is large-positive and vanishes above the Curie temperature.
The solenoid field μ₀ n I does not depend on its radius — only on current and turns per metre. And n is turns per unit length, not the total number of turns.

10Check yourself

Class 12 Physics · Moving Charges & Magnetism; Magnetism & Matter · aligned to NCERT Chapters 4–5 · SmartStudy.School