Magnetism was known to ancient civilisations through natural magnets, but it was only in 1820 that Hans Christian Oersted discovered that an electric current produces a magnetic field. This chapter studies the magnetic effects of moving charges and currents. We begin with the magnetic force on a moving charge, the Lorentz force, and the motion of charged particles in magnetic fields.
We then derive the magnetic field produced by currents. The Biot-Savart law gives the field due to a current element, and Ampere's circuital law provides a powerful tool for symmetric current distributions. We calculate the fields of a straight wire, a circular loop, and a solenoid. We also study the torque on a current loop in a magnetic field, which is the basis of the galvanometer and electric motors.
The chapter ends with the force between two parallel current-carrying wires, which is used to define the ampere. These ideas unify electricity and magnetism and lead naturally to the study of electromagnetic induction in a later chapter.
A charge q moving with velocity v in a magnetic field B experiences a force:
F = q (v x B)
The magnitude of the force is F = q v B sin(theta), where theta is the angle between v and B. The force is always perpendicular to both v and B, so it does no work on the charge; it changes only the direction of motion, not the speed.
When both electric and magnetic fields are present, the total force is the Lorentz force:
F = q (E + v x B)
The Lorentz force explains how charged particles move in combined fields, and it is the basis of many practical devices. Since the magnetic force is perpendicular to velocity, a magnetic field cannot increase the kinetic energy of a particle, only deflect it.
If a charged particle enters a uniform magnetic field perpendicular to its velocity, the magnetic force acts as a centripetal force, producing circular motion. Equating the centripetal force to the magnetic force:
q v B = m v^2 / r
The radius of the circular path is:
r = m v / (q B)
The time period and angular frequency are independent of the speed:
T = 2 pi m / (q B), omega = q B / m
If the velocity has a component parallel to the field, the particle moves in a helical path: the component parallel to B is unaffected, while the perpendicular component causes circular motion. Helical motion occurs in devices like the cyclotron, which accelerates charged particles to high energies.
A current-carrying wire placed in a magnetic field experiences a force because the field acts on the moving charges inside it. For a straight wire of length L carrying current I at an angle theta to a uniform field B:
F = B I L sin(theta)
In vector form, F = I (L x B). The force is zero when the wire is parallel to the field and maximum when it is perpendicular. The direction of the force is given by Fleming's left-hand rule: the forefinger points along the field, the middle finger along the current, and the thumb gives the direction of force.
The force per unit length between two long parallel wires carrying currents I1 and I2 separated by distance d is:
F/L = mu_0 I1 I2 / (2 pi d)
where mu_0 = 4 pi x 10^-7 T m/A is the permeability of free space. Parallel currents attract, and antiparallel currents repel. This force defines the ampere: one ampere is the current that produces a force of 2 x 10^-7 N per metre between two parallel wires 1 metre apart.
A rectangular loop of area A carrying current I, with N turns, placed in a uniform magnetic field experiences a torque that tends to rotate it. The magnitude of the torque is:
tau = N I A B sin(theta)
where theta is the angle between the magnetic field and the normal to the loop. Defining the magnetic moment of the loop as m = N I A, the torque becomes:
tau = m B sin(theta), or tau = m x B
The direction of the magnetic moment is given by the right-hand rule: curl the fingers along the current, and the thumb points along m. The loop behaves like a magnetic dipole whose moment is m = NIA. The potential energy of the loop in the field is U = -m B cos(theta), minimum when the loop is aligned with the field.
A moving coil galvanometer detects small currents. It consists of a coil of N turns and area A suspended in a radial magnetic field. When a current I flows, the deflecting torque N I A B is balanced by the restoring torque of the suspension spring, which is proportional to the deflection angle:
N I A B = k theta, so theta = (N A B / k) I
Thus the deflection is proportional to the current, giving a linear scale. The radial field ensures that the torque is independent of the angle of rotation, keeping the scale uniform. The current sensitivity is N A B/k, and the voltage sensitivity is N A B/(k R). A galvanometer is converted into an ammeter by connecting a low resistance shunt in parallel, and into a voltmeter by connecting a high resistance in series.
The Biot-Savart law gives the magnetic field produced by a current element. For a small current element I dl at a distance r, the magnetic field is:
dB = (mu_0 / 4 pi) * I dl sin(theta) / r^2
In vector form, dB = (mu_0 / 4 pi) I (dl x r) / r^3. The field due to any current configuration is obtained by integrating this law. For an infinitely long straight wire carrying current I, the field at perpendicular distance a is:
B = mu_0 I / (2 pi a)
The direction of the field lines around the wire is given by the right-hand thumb rule, and the field lines form concentric circles around the wire.
Ampere's circuital law states that the line integral of the magnetic field around any closed path equals mu_0 times the current enclosed:
B dot dl = mu_0 I_enclosed
This law is to magnetism what Gauss's law is to electrostatics. It gives the field of a long solenoid with n turns per unit length:
B = mu_0 n I
The field inside an ideal solenoid is uniform, and outside it is nearly zero. A toroid is a solenoid bent into a ring; its field inside is B = mu_0 n I, and outside it is zero. Ampere's law also gives the field inside a current-carrying wire and provides the definition of the ampere through the force between parallel wires.
| Quantity | Formula | Remarks |
|---|---|---|
| Force on moving charge | F = q v B sin theta | Perpendicular to v and B |
| Lorentz force | F = q(E + v x B) | No work done by magnetic part |
| Radius of circular path | r = m v / (q B) | Independent of speed for T |
| Time period | T = 2 pi m / (q B) | Independent of speed |
| Force on a wire | F = B I L sin theta | Fleming's left-hand rule |
| Force between parallel wires | F/L = mu_0 I1 I2/(2 pi d) | Parallel currents attract |
| Torque on a loop | tau = N I A B sin theta | Basis of galvanometer |
| Magnetic moment | m = N I A | Unit A m^2 |
| Field of straight wire | B = mu_0 I/(2 pi a) | Right-hand thumb rule |
| Field of solenoid | B = mu_0 n I | Uniform inside |
| Biot-Savart law | dB = (mu_0/4 pi) I dl sin theta/r^2 | Integration over circuit |
| Device | Conversion | Connecting element |
|---|---|---|
| Galvanometer to ammeter | Measures large current | Low resistance shunt in parallel |
| Galvanometer to voltmeter | Measures voltage | High resistance in series |
This chapter developed the magnetic effects of moving charges and currents. The Lorentz force F = q(E + v x B) described the force on a moving charge, and we saw that charged particles move in circles (r = mv/qB) or helixes in magnetic fields. We found the force on a current-carrying wire, the force between parallel wires that defines the ampere, and the torque on a current loop with magnetic moment m = NIA, which underlies the galvanometer. The Biot-Savart law and Ampere's circuital law gave the fields of wires and solenoids, B = mu_0 I/(2 pi a) and B = mu_0 n I. These results connect electricity and magnetism and set the stage for electromagnetic induction, alternating current, and electromagnetic waves.