Magnetic field and the Lorentz force

A needle that moved

In 1820 the Danish physicist Hans Christian Oersted held a compass near a wire. When a current started in the wire, the needle swung. A current had made a magnetic field.

Now think about the third law of Newton. If a current pushes a magnet, the magnet must push the current back. It does. A magnetic field pushes on moving charges. This lesson is about that push.

What is a magnetic field?

At every point near a magnet or a current there is a magnetic field. It is a vector, $\vec{B}$. Its direction is the direction in which the north pole of a small compass needle points. The size of $B$ is defined by the force it puts on a moving charge, and we will see how in a moment.

The unit of $B$ is the tesla (T). An older, smaller unit is the gauss: $1\ \text{G} = 10^{-4}\ \text{T}$. Here are some fields to give you a feel for the size.

• The Earth's field at its surface: between $3 \times 10^{-5}$ T and $6 \times 10^{-5}$ T.
• A fridge magnet, close to its face: a few $10^{-3}$ T.
• A strong electromagnet in a laboratory: 1 T to 2 T.
• A hospital MRI scanner: 1.5 T to 3 T.

The force on a moving charge

Imagine a beam of charged particles passing through a field. We watch what the field does to them. Three things happen.

1. A charge at rest feels no force. The field may be strong, but a magnetic force needs motion.
2. A charge moving along the field feels no force. The same is true if it moves against the field.
3. A charge moving across the field feels the largest force. The force is at right angles to the velocity and at right angles to the field.

All three are in one formula. Let a charge $q$ move with speed $v$ at an angle $\theta$ to a field $B$. Then

Only the part of the velocity across the field matters. That part is $v_\perp = v\sin\theta$, so $F = qv_\perp B$.

In vector form, $\vec{F} = q\,\vec{v} \times \vec{B}$. The cross product gives us the direction. Use the right-hand rule. Point the fingers of your right hand along $\vec{v}$. Turn them towards $\vec{B}$, through the smaller angle. Your thumb now points along $\vec{v} \times \vec{B}$. That is the direction of the force on a positive charge.

For a negative charge, the force is the other way. An electron and a proton that move the same way in the same field are pushed in opposite directions.

If you work with unit vectors along $x$, $y$ and $z$, remember $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$ and $\hat{k} \times \hat{i} = \hat{j}$. Reverse the order and the sign changes. In components,

Where does the tesla come from?

The formula gives the definition of $B$. For a charge at right angles to the field, $B = \dfrac{F}{qv}$. So

One tesla is the field in which a charge of $1$ C, moving at $1$ m/s across the field, feels a force of $1$ N. The second form, newton for each ampere metre, will be useful when we study the force on a wire.

A worked example

An electron moves at $3.0 \times 10^6$ m/s at right angles to a field of $0.40$ T. We find the force, its acceleration and how it compares with the weight of the electron. The mass of an electron is $9.1 \times 10^{-31}$ kg.

1. The force. The angle is $90^\circ$, so $\sin\theta = 1$. $F = evB = 1.6 \times 10^{-19}\ \text{C} \times 3.0 \times 10^{6}\ \text{m/s} \times 0.40\ \text{T} = 1.92 \times 10^{-13}\ \text{N}$.
2. The acceleration. We want $a$, so $a = \dfrac{F}{m} = \dfrac{1.92 \times 10^{-13}\ \text{N}}{9.1 \times 10^{-31}\ \text{kg}} = 2.1 \times 10^{17}\ \text{m/s}^2$.
3. The weight. $mg = 9.1 \times 10^{-31}\ \text{kg} \times 9.8\ \text{m/s}^2 = 8.9 \times 10^{-30}\ \text{N}$. The magnetic force is about $2 \times 10^{16}$ times larger. That is why we ignore gravity for charged particles.

A worked example with directions

A proton moves north in a field that points up. Which way is the force?

Let east be $x$, north be $y$ and up be $z$. The velocity is along $+y$ and the field is along $+z$. Then $\hat{j} \times \hat{k} = \hat{i}$, which is east. The proton is positive, so the force is towards the east. An electron moving north in the same field would be pushed west.

Why the magnetic force does no work

The force $q\,\vec{v} \times \vec{B}$ is at right angles to $\vec{v}$. The power of a force is $\vec{F} \cdot \vec{v}$, and the dot product of two vectors at right angles is zero. So the magnetic force does no work.

This is a very useful fact. The kinetic energy of the charge does not change, so its speed does not change. The field only changes the direction of the velocity. Compare this with an electric field. The force $q\vec{E}$ can be along the motion, and then it speeds the charge up or slows it down.

The magnetic force steers. The electric force accelerates.

The Lorentz force

A charge can be in a magnetic field and an electric field at the same time. The total force is the Lorentz force:

The first part, $q\vec{E}$, acts even on a charge at rest. The second part acts only when the charge moves across the field. In the next lesson we use both parts together.

Put the angle at $0^\circ$ and then at $180^\circ$. The force is zero for both, since the charge moves along the field or against it. Then find the angle with the largest force. What do you read in the last readout?

Practise with new numbers

Try it in Python

The program finds the force on a proton and on an electron with the cross product. It does it for four directions of the velocity. It also finds $\vec{F} \cdot \vec{v}$, which is the power of the force.

Common mistakes

• Using the wrong angle. In $F = qvB\sin\theta$, the angle $\theta$ is between $\vec{v}$ and $\vec{B}$. At $0^\circ$ the force is zero.
• Pointing the force along the velocity or along the field. The force is at right angles to both.
• Forgetting that a negative charge reverses the force. Use the right-hand rule, then turn the answer round for an electron.
• Thinking the field can speed a charge up. A magnetic force does no work. The speed does not change.
• Writing a force on a charge at rest. With no velocity there is no magnetic force. There may still be an electric one.

Remember these

• $F = qvB\sin\theta$. In vector form, $\vec{F} = q\,\vec{v} \times \vec{B}$.
• The force is at right angles to $\vec{v}$ and $\vec{B}$. Use the right-hand rule, and reverse it for a negative charge.
• No force when $v = 0$, or when $\vec{v}$ is along $\vec{B}$.
• $1\ \text{T} = 1\ \text{N}/(\text{A m})$. A gauss is $10^{-4}$ T.
• The magnetic force does no work. It changes the direction of the velocity, not the speed.
• Lorentz force: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A proton moves along a field of $0.80$ T at $5.0 \times 10^6$ m/s. What is the force on it?

2. An electron moves east in a field that points north. In which direction is the force on it?

3. A proton moves at $4.0 \times 10^6$ m/s at $30^\circ$ to a field of $0.50$ T. What is the force?

4. A student says that a magnetic field is used to speed up the particles in a machine. What is wrong with this?