Theory

Magnetic field and Lorentz force

How a magnetic field acts on moving charges and on currents, from the Lorentz force to the circular and helical motion of particles, the velocity selector, the mass spectrometer and the Hall effect, up to the force on a wire and the torque on a loop, described by the magnetic moment.

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Magnetic field and field lines
Lorentz force

Magnetic field and field lines

Throughout this topic the magnetic field is given and constant in time, the reference frame is inertial, and the speeds of the particles are small compared with the speed of light. The currents that produce the field are treated in the topic on the sources of the magnetic field.

1. Magnetic field

There is a magnetic field in a region of space if a moving charge in it experiences, in addition to any electric force, a force that depends on its velocity. Experimentally, at each point there is a direction along which a moving charge does not experience this force; for a velocity perpendicular to that direction the force has its maximum magnitude $F_{m a x}$ , proportional to the charge $q$ and to the speed $v$ . The magnetic field $B$ is the vector that has that direction and magnitude

B = \frac{F _{m a x}}{∣ q ∣ v},

while its sense is fixed by the Lorentz force law. In the International System the magnetic field is measured in teslas, $1 T = 1 N \cdot s/ (C \cdot m) = 1 N/ (A \cdot m)$ ; the gauss is also used, $1 G = 1 0^{- 4} T$ . The Earth’s magnetic field has a magnitude of the order of $5 \cdot 1 0^{- 5} T$ , a laboratory magnet produces fields of the order of $0.1 T$ near its poles, and magnets for magnetic resonance imaging reach a few teslas.

2. Magnetic field lines

The magnetic field lines are oriented curves tangent at each point to $B$ and with the same sense; where the lines are denser the field is stronger. A small magnetic needle free to rotate aligns itself along the field line, and its north pole indicates the sense of the line. Outside a magnet the lines leave the north pole and enter the south pole, and inside they continue from the south pole to the north pole; the magnetic field lines are therefore closed.

The electrostatic field has lines that start on positive charges and end on negative ones. The closed lines of the magnetic field correspond instead to the absence of isolated magnetic charges; indeed, breaking a magnet yields two magnets, each with a north pole and a south pole. The quantitative form of this property, Gauss’s law for the magnetic field, is treated in the topic on the sources of the magnetic field.

Magnetic field lines of a bar magnet: they leave the north pole, enter the south pole and close inside the magnet

Field perpendicular to the page

In plane drawings a vector perpendicular to the page is represented by a symbol: $\otimes$ denotes a vector into the page, like the tail of an arrow moving away, and $⊙$ a vector out of the page, like the tip of an arrow approaching. A uniform field into the page is therefore drawn as a regular grid of $\otimes$ symbols. With the page in the $x y$ plane and the $z$ axis pointing toward the reader, a field into the page is $B = - B \hat{k}$ and a field out of the page is $B = B \hat{k}$ .

Lorentz force

A point charge $q$ moving with velocity $v$ in a magnetic field $B$ experiences the force

F = q v \times B, F = ∣ q ∣ v B sin φ,

where $φ$ is the angle between $v$ and $B$ . It is an experimental law, which completes the definition of the magnetic field by fixing the sense of $B$ . The force is perpendicular both to the velocity and to the field; it is zero for a charge at rest or moving parallel to $B$ , and it is maximum, $∣ q ∣ v B$ , for a velocity perpendicular to the field. The sense is found with the right-hand rule applied to $v \times B$ , and it is reversed for a negative charge.

For example, a proton with velocity $v = 2.0 \cdot 1 0^{6} \overset{}{^} m/s$ in a field $B = 0.50 \overset{}{^} T$ experiences the force $F = e v B (\overset{}{^} \times \overset{}{^}) = 1.6 \cdot 1 0^{- 13} \hat{k} N$ , while an electron with the same velocity experiences the opposite force.

The force

F = q v \times B

on a positive charge is perpendicular to the plane of

v

and

B

, with the sense given by the right-hand rule

Electric force and magnetic force together

If there are an electric field $E$ and a magnetic field $B$ in the region, the force on a point charge is the sum of the electric force and the magnetic force,

F = q (E + v \times B),

and in many texts it is this sum that is called the Lorentz force. The electric part acts also on a charge at rest, while the magnetic part acts only on moving charges and depends on the velocity. Comparing the two terms makes it possible to separate particles according to their speed, as in the velocity selector.

Work of the magnetic force

The magnetic force $q v \times B$ on a moving charge does zero work along any trajectory. Consequently, if only the magnetic force acts on the charge, the kinetic energy and the speed remain constant, and only the direction of the velocity changes. To change the energy of a charged particle an electric field is needed, as in the accelerator that precedes a mass spectrometer. In this subject work is denoted by $W$ , because the letter $L$ denotes inductance.

Proof

The power of the magnetic force is

P = F \cdot v = q (v \times B) \cdot v = 0,

because the cross product $v \times B$ is perpendicular to $v$ . Hence the work in a time interval $[t_{1}, t_{2}]$ is $W = \int_{t_{1}}^{t_{2}} P d t = 0$ . If the magnetic force is the only force, by the work-energy theorem we have $K_{2} - K_{1} = W = 0$ , therefore $\frac{1}{2} m v^{2}$ is constant, and with it the speed $v$ .

The theory continues with 6 more sections (motion of a charge in a uniform magnetic field, velocity selector and mass spectrometer, hall effect…), plus all worked examples, the solutions and the PDF.

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