Magnetic fields

20.1 Concept of a magnetic field

A magnetic field is a field of force, produced either by moving charges (currents) or by permanent magnets. Field lines point from north to south outside a magnet. They show the direction of the force on a north pole, and closer lines mean a stronger field.

20.2 Force on a current-carrying conductor

A wire carrying a current in a magnetic field may feel a force:

$F = BIL\sin\theta$

where $\theta$ is the angle between the current and the field. The force is greatest at 90° and zero when the wire is parallel to the field.

Fleming's left-hand rule

Hold your left hand with the thumb, first finger and second finger all at right angles:

• First finger: Field (north to south).
• SeCond finger: Current (conventional, positive to negative).
• ThuMb: Motion, that is, the force.

Magnetic flux density

Magnetic flux density $B$ is the force acting per unit current per unit length on a wire placed at right angles to the magnetic field. Its unit is the tesla, where 1 T = 1 N A⁻¹ m⁻¹.

20.3 Force on a moving charge

$F = BQv\sin\theta$

Find the direction with the left-hand rule, with the second finger along the direction of conventional current. For a negative charge such as an electron, the current direction is opposite to its motion.

Circular motion in a magnetic field

For a charge moving at right angles to a uniform field, the force is always perpendicular to the velocity. So the speed stays constant and the path is a circle. The magnetic force provides the centripetal force:

$BQv = \frac{mv^2}{r} \quad\Rightarrow\quad r = \frac{mv}{BQ}$

A bigger momentum means a bigger circle, and a stronger field means a smaller circle. The force does no work, so the kinetic energy does not change.

Velocity selection

Put an electric field and a magnetic field at right angles, so that the forces on a charge are opposite. For an undeflected particle:

$qE = Bqv \quad\Rightarrow\quad v = \frac{E}{B}$

Only particles of this speed pass straight through, whatever their charge or mass.

The Hall effect

When a current flows through a thin slice of conductor with a magnetic field at right angles to it, the charge carriers are pushed to one side. Charge builds up until the electric force balances the magnetic force. This gives a p.d. across the slice, the Hall voltage.

Balance gives $qV_H/d = Bqv$, where $d$ is the width. Substituting $I = nAvq$ with $A = dt$ gives:

$V_H = \frac{BI}{ntq}$

where $t$ is the thickness and $n$ is the number density of carriers. Semiconductors give a much larger Hall voltage than metals, because their $n$ is much smaller. A Hall probe measures $B$: with a constant current, $V_H \propto B$, and the probe must be perpendicular to the field to read its full value.

20.4 Magnetic fields due to currents

• Long straight wire: concentric circles, more widely spaced further away. Use the right-hand grip rule: thumb along the current, and your fingers curl along the field.
• Flat circular coil: the field runs through the middle of the coil and loops round outside.
• Long solenoid: a strong, uniform field inside, and outside like a bar magnet's field. Putting a ferrous core (soft iron) inside greatly increases the field.

Forces between conductors

Each wire sits in the field of the other, so each feels a force. Currents in the same direction attract, and currents in opposite directions repel. The forces are equal and opposite, by Newton's third law.

20.5 Electromagnetic induction

Flux and flux linkage

Magnetic flux is the flux density × the cross-sectional area perpendicular to the flux density:

$\Phi = BA$

The unit is the weber (Wb). For a coil of $N$ turns, the flux linkage is $N\Phi$. If the field is at an angle $\theta$ to the normal, $\Phi = BA\cos\theta$.

Faraday's and Lenz's laws

$E = -\frac{\Delta(N\Phi)}{\Delta t}$

The minus sign is Lenz's law, which is a consequence of conservation of energy. If the induced current helped the change, you would get energy from nothing.

Experiments

• Moving a magnet into a coil connected to a sensitive meter gives a reading only while it moves.
• Reversing the direction reverses the reading.
• The e.m.f. is larger with a faster movement, a stronger magnet or more turns. This shows the factors that affect its size.
• A moving magnet's north pole approaching a coil makes the near end of the coil a north pole, which repels the magnet, so you must do work to push it in.