Magnetism and Moving Charges: Study Guide (Physics 2 12.2)

Moving charges make magnetic fields

In Topic 12.1 you saw that magnetic dipoles come from charges moving inside atoms. The same idea works for a single charged object. A charged object at rest makes only an electric field. A moving charged object also makes a magnetic field.

The magnetic field at a point near a moving charge depends on:

• the velocity of the charge: a faster charge makes a stronger field, and
• the distance from the charge to the point: the field is weaker further away.

Its direction is at right angles to two things at once: the velocity v of the charge, and the position vector r from the charge to the point. So the field circles round the line along which the charge moves.

Right-hand rule for the field. For a positive charge, point your right thumb along v. Your fingers curl the way B circles round the line of motion. For a negative charge, the field is the other way round.

Example. A positive charge moves to the right across the page. At a point directly above it, the field points out of the page. At a point directly below it, the field points into the page.

The field is largest when v and r are at right angles, as at those two points. At a point straight ahead of the charge, or straight behind it, v and r are parallel, and the charge makes no magnetic field there.

Magnetic forces between moving charges

Magnetic forces describe interactions between moving charges. One moving charge makes a magnetic field. A second charge moving through that field feels a force. Two positive charges moving side by side in the same direction, for example, pull towards each other magnetically. They still repel electrically. For charges moving much more slowly than light, the electric repulsion is far larger, but the magnetic part is real, and in a wire full of moving charges it is the part you notice (Topic 12.3).

The force on a moving charge

A charge q moving with velocity v through a magnetic field B feels a magnetic force of size

F_B = |q| v B sin θ

• |q| is the size of the charge (C), v the speed (m/s), B the field (T);
• θ is the angle between the velocity and the field.

So the force is proportional to the charge, the speed and the field, and it depends on the angle:

Angle between v and B	sin θ	Force
0° (v parallel to B)	0	F = 0
90° (v at right angles to B)	1	F = qvB, the largest value
180° (v antiparallel to B)	0	F = 0

For angles in between, the force lies between 0 and |q|vB. It grows as the angle moves towards 90°. In this course you calculate the force only at 0°, 90° and 180°, and reason about other angles without numbers.

Units: 1 T = 1 N/(A·m) = 1 N·s/(C·m), so C × m/s × T gives N.

Direction: the right-hand rule

The magnetic force is at right angles to both v and B. To find which way:

1. Point the fingers of your right hand along v.
2. Curl them towards the direction of B.
3. Your thumb points along the force on a positive charge.
4. For a negative charge, the force is in the opposite direction.

Diagrams usually show fields into the page as crosses (×, the tail of an arrow) and out of the page as dots (·, the tip of an arrow).

Circular motion

The magnetic force is always at right angles to the velocity. So it does no work on the charge, and the charge’s speed and kinetic energy do not change. The force only turns the velocity. If v is at right angles to a uniform field, the force has a constant size and always points sideways: that is exactly the condition for uniform circular motion (Figure 1).

The magnetic force provides the centripetal force. Writing q for the size of the charge:

qvB = mv²/r, so r = mv / (qB)

• Faster or heavier charges move in larger circles.
• Bigger charges or stronger fields give smaller circles.
• The time for one orbit is T = 2πr/v = 2πm/(|q|B). It does not depend on the speed: a faster charge goes round a bigger circle in the same time.

If v is parallel to B, there is no force at all, and the charge moves in a straight line at constant speed.

Electric and magnetic fields together

When a region contains both an electric field and a magnetic field, a moving charge feels two independent forces:

• the electric force qE, which does not depend on the motion, and
• the magnetic force, of size |q|vB for v at right angles to B.

You find each one separately and add them as vectors. A useful case is crossed fields, arranged so that the two forces point in opposite directions. They balance when (with q the size of the charge)

qE = qvB, so v = E/B

The charge cancels. So only charges with exactly this speed pass straight through, whatever their charge or mass. Faster charges feel a bigger magnetic force and bend towards the magnetic-force side. Slower charges bend towards the electric-force side. This device is called a velocity selector.

The Hall effect

A current in a conductor is made of charge carriers drifting along it. Put the conductor in a magnetic field that has a part at right angles to the current, and the field pushes the carriers sideways (Figure 2). They pile up on one edge, leaving the opposite charge on the other edge. This separation of charge creates a potential difference across the width of the conductor: the Hall potential difference ΔV_H.

The build-up stops when the electric field E_H made by the separated charges pushes back on each carrier as hard as the magnetic field pushes it sideways. For a strip of width w, E_H = ΔV_H/w, and balance gives

q(ΔV_H/w) = qvB, so ΔV_H = vBw

where v is the drift speed of the carriers. Two uses follow:

• Sign of the carriers. Positive carriers moving with the current and negative carriers moving against it are pushed to the same edge. So the sign of the charge on that edge tells you the sign of the carriers. In Figure 2 the top edge is negative, so the carriers are negative (electrons).
• Measuring B. If you know v and w, measuring ΔV_H gives B. Small Hall sensors are used this way to measure magnetic fields.

Worked example 1: a proton in a uniform field

Question. A proton (charge +1.60 × 10⁻¹⁹ C, mass 1.67 × 10⁻²⁷ kg) moves to the right at 3.0 × 10⁵ m/s through a uniform field of 0.25 T directed into the page, as in Figure 1. (a) Find the size and direction of the magnetic force. (b) Find the radius of its path and the time for one orbit. (c) An electron enters the same field with the same velocity. Compare its force and its path.

1. Angle. v is to the right and B is into the page, so θ = 90° and sin θ = 1.
2. Size of force. F = |q|vB = (1.60 × 10⁻¹⁹ C)(3.0 × 10⁵ m/s)(0.25 T) = 1.2 × 10⁻¹⁴ N.
3. Direction. Fingers to the right, curl into the page: thumb points to the top of the page. The proton is positive, so that is the force direction.
4. Radius. r = mv/(|q|B) = (1.67 × 10⁻²⁷ kg)(3.0 × 10⁵ m/s) ÷ [(1.60 × 10⁻¹⁹ C)(0.25 T)] = 1.25 × 10⁻² m, about 1.3 cm.
5. Period. T = 2πm/(qB) = 2π(1.67 × 10⁻²⁷) ÷ [(1.60 × 10⁻¹⁹)(0.25)] = 2.62 × 10⁻⁷ s. (Check: 2πr/v = 2π(0.012525 m) ÷ (3.0 × 10⁵ m/s) = 2.62 × 10⁻⁷ s.)
6. Electron. Same |q|, v and B, so the force has the same size, 1.2 × 10⁻¹⁴ N, but points to the bottom of the page. Its mass is much smaller, so r = (9.11 × 10⁻³¹)(3.0 × 10⁵) ÷ [(1.60 × 10⁻¹⁹)(0.25)] = 6.83 × 10⁻⁶ m. It circles clockwise in a circle about 1830 times smaller.

Answer. (a) 1.2 × 10⁻¹⁴ N towards the top of the page. (b) r = 1.3 cm, T = 2.6 × 10⁻⁷ s. (c) Same size of force, opposite direction; a far smaller circle, travelled the other way round.

Interpretation. The proton’s speed stays 3.0 × 10⁵ m/s the whole time, because the force does no work. If the proton had been moving along the field lines instead (θ = 0°), the force would have been zero and it would have gone straight on.

Worked example 2: a velocity selector

Question. Two parallel plates 2.0 cm apart have a potential difference of 600 V, with the top plate positive. Positive ions move to the right between the plates. A uniform magnetic field of 0.15 T fills the same region. (a) Which direction must B point so that the magnetic force opposes the electric force? (b) At what speed do ions pass through undeflected? (c) Which way are ions moving at 3.0 × 10⁵ m/s deflected?

1. Electric field. E = ΔV/d = 600 V ÷ 0.020 m = 3.0 × 10⁴ V/m, pointing from the positive top plate down to the bottom plate.
2. Electric force. The ions are positive, so qE points down.
3. Magnetic force needed. It must point up. Fingers to the right (v), thumb up (F): the fingers must curl into the page. So B points into the page.
4. Balance. qE = qvB gives v = E/B = (3.0 × 10⁴ V/m) ÷ (0.15 T) = 2.0 × 10⁵ m/s.
5. Faster ions. At 3.0 × 10⁵ m/s, the magnetic force per unit charge is vB = 4.5 × 10⁴ N/C, larger than E = 3.0 × 10⁴ N/C. The upward magnetic force wins, so these ions are deflected upwards.

Answer. (a) Into the page. (b) 2.0 × 10⁵ m/s. (c) Upwards, towards the positive plate.

Check. The selected speed does not depend on q or m. An electron at 2.0 × 10⁵ m/s also goes straight: both of its forces reverse, so they still cancel.

Worked example 3: reading a Hall sensor

Question. A strip of semiconductor 3.0 mm wide carries a current to the right, as in Figure 2, in a field of 0.40 T into the page. The Hall potential difference across its width is 0.36 mV, and the top edge is at the higher potential. (a) What is the sign of the charge carriers? (b) Find their drift speed.

1. Sign. For current to the right and B into the page, carriers of either sign are pushed to the top edge (Figure 2 and the right-hand rule). The top edge is at the higher potential, so the charge piling up there is positive. The carriers are positive.
2. Drift speed. v = ΔV_H/(Bw) = (0.36 × 10⁻³ V) ÷ [(0.40 T)(3.0 × 10⁻³ m)] = 0.30 m/s.

Answer. (a) Positive. (b) 0.30 m/s.

Interpretation. This is the opposite result to the strip in Figure 2, where the carriers are electrons and the top edge is negative. A Hall measurement is one of the few direct ways to tell the sign of the charges that carry a current.

Common misconceptions

• “A magnetic field pushes a charge along the field lines.” The force is at right angles to B, and also to v. A charge moving along B feels no force at all.
• “The magnetic force speeds charges up.” It is always at right angles to v, so it does no work. Speed and kinetic energy stay constant; only the direction changes.
• “Use the same hand rule for electrons.” Find the direction for a positive charge, then reverse it for a negative one.
• “A charge at rest in a magnetic field feels a force.” F_B is proportional to v. With v = 0, F_B = 0.
• “The field of a moving charge points along its motion.” It circles round the line of motion, at right angles to both v and r, and is zero straight ahead and straight behind.
• “A velocity selector selects by charge or mass.” The balance speed is v = E/B; the charge cancels and mass does not appear.
• “Saying ‘right-hand rule’ is a justification.” In a written answer, state the directions of v and B, say the force is at right angles to both, and give the result of the rule, including the reversal for a negative charge.

Where this leads

Topic 12.3, Magnetism and Current-Carrying Wires, adds up the forces on all the moving charges in a wire to give the force on a current, and the fields of all those charges to give the field of a wire. Test yourself with the practice questions, then use the revision notes and the topic checklist to consolidate.