Magnetic Force and Fields

Hold a compass anywhere on Earth and its needle swings north, tugged by a field that streams silently out of the planet's molten core. Point a beam of electrons at a bar magnet and the beam bends. Stand under the northern sky on a good night and watch the aurora ripple green and violet — charged particles from the Sun, funnelled by Earth's magnetism and slammed into the upper atmosphere. All three are the same story: a magnetic field pushing on moving electric charge.

That last phrase is the whole secret of the page, and it is worth saying slowly: a magnetic force acts only on charge that is moving. Park an electron at rest in the strongest magnet on Earth and it feels nothing at all. Let it drift, even slowly, and a sideways push appears from nowhere. This is what makes magnetism so different from the familiar electric force — it cares not just about how much charge you have, but about how fast and which way it is going. Below we pin down exactly how big that push is, which way it points, why it can never speed a particle up, and how it curls beams into circles and rivers of particles into the aurora.

The magnetic force: \vec F = q\,\vec v \times \vec B

A charge q moving with velocity \vec v through a magnetic field \vec B feels a force given by a cross product:

\vec F = q\,\vec v \times \vec B.

The cross product packs in both the size and the direction of the push. Its magnitude is

F = q\,v\,B\,\sin\theta,

where \theta is the angle between \vec v and \vec B. Read off the two extremes at once: a charge moving straight along the field (\theta = 0) feels no force — \sin 0 = 0 — while a charge moving square across the field (\theta = 90^\circ) feels the maximum push, F = qvB.

The direction comes from the right-hand rule for \vec v \times \vec B: point the fingers of your right hand along \vec v, curl them towards \vec B, and your thumb points along \vec v \times \vec B. For a positive charge that thumb is the force; for a negative charge the force points the opposite way. The magnetic force is always perpendicular to both \vec v and \vec B — it never points along the motion.

The field \vec B is measured in tesla (T). One tesla is a hefty field: the Earth's is about 5\times 10^{-5}\ \text{T}, a fridge magnet a few hundredths of a tesla, an MRI scanner around 1.5\text{–}3\ \text{T}. From F = qvB you can read the unit off directly: 1\ \text{T} = 1\ \text{N}/(\text{A}\cdot\text{m}) = 1\ \text{N}\,\text{s}/(\text{C}\cdot\text{m}).

Magnetism rarely acts alone. When both an electric field \vec E and a magnetic field \vec B are present, the total force on the charge is the Lorentz force:

The magnetic force does no work

Here is the single most important consequence of \vec F = q\,\vec v \times \vec B, and it follows purely from the geometry of the cross product. The force is always perpendicular to the velocity. Work is force along the direction of motion — \text{d}W = \vec F \cdot \vec v\,\text{d}t — and a force at right angles to \vec v contributes nothing to that dot product:

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

So a purely magnetic force does zero work. It can never change a particle's kinetic energy — it cannot speed the particle up or slow it down. All it can do is change the direction of the velocity, bending the path while the speed stays fixed. A magnet is a superb steering wheel and a hopeless engine. Keep this in your pocket: whenever a problem asks whether a magnetic field changes a particle's speed, the answer is always no.

Motion in a uniform field: circles and helices

Now let a charge fly across a uniform field, with \vec v perpendicular to \vec B. The force qvB always points at right angles to the motion, steering the particle sideways. A constant-magnitude force that is forever perpendicular to the velocity is exactly the recipe for uniform circular motion: the particle loops around at constant speed, the magnetic force playing the role of the centripetal force. Reveal the figure to see the field, the circling charge, its tangent velocity, and the force forever aimed at the centre.

Set the magnetic force equal to the centripetal force mv^2/r to find the radius:

q\,v\,B = \frac{m\,v^2}{r} \quad\Longrightarrow\quad r = \frac{m\,v}{q\,B}.

Faster particles and heavier particles swing round wider circles; a stronger field or a bigger charge winds them tighter. Now find how quickly the particle goes round. The angular frequency is \omega = v/r, and substituting r = mv/(qB) the speed cancels clean out:

That last fact is startling: every particle of a given charge-to-mass ratio takes the same time to complete a lap, no matter how fast it is going. A fast particle simply traces a bigger circle at higher speed and arrives back on schedule. And if the velocity has a component along \vec B as well as across it? That parallel part feels no force (\sin 0 = 0), so it sails on undisturbed while the perpendicular part keeps circling — the two together trace a helix, a corkscrew winding along the field lines. That is precisely how solar particles spiral down Earth's field lines toward the poles to light the aurora.

Try it: how the radius grows with speed

The relation r = mv/(qB) is a straight line through the origin: double the speed and you double the radius. Drag the slider to change the field strength B and watch the line tilt — a stronger field means a steeper price in force, so the same speed is bent into a tighter circle (a smaller radius). The curve is drawn for a proton (m = 1.67\times 10^{-27}\ \text{kg}, q = 1.6\times 10^{-19}\ \text{C}).

Notice the line always passes through the origin: a particle at rest has zero radius because a stationary charge feels no magnetic force at all. Notice too that a bigger B pushes the whole line down — the field grips the particle harder and reins it into a smaller loop.

Force on a current-carrying wire

A current is just charge on the move, so a wire carrying a current through a magnetic field feels a force too — it is the sum of the tiny q\vec v \times \vec B pushes on all its drifting charges. For a straight segment of length L carrying current I, the force is

\vec F = I\,\vec L \times \vec B, \qquad F = B\,I\,L\,\sin\theta,

where \vec L points along the wire in the direction of the current and \theta is the angle between the wire and the field. Same right-hand rule, same \sin\theta: a wire lying along the field feels nothing, a wire across it feels the full BIL. This is the force that spins every electric motor — a current loop in a magnetic field, pushed round and round.

The field of a long straight wire

We have been treating \vec B as given, but currents are also the source of magnetic fields. The simplest case is a long straight wire carrying a steady current I. The field it produces circles around the wire, with magnitude

(The full derivation — from the Biot–Savart law or Ampère's law — lives on its own page; here we just state the result and use it.) Now put two long parallel wires side by side, a distance d apart, each carrying a current. Wire 1 makes a field at wire 2, and wire 2's current feels a I\vec L \times \vec B force in it. Working the two right-hand rules through gives a beautifully simple result: the force per unit length is

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}.

And the sign of the effect is easy to remember: parallel currents attract, anti-parallel currents repel. Two wires carrying current the same way are pulled together; carry them in opposite directions and they push apart. (This force was once the very definition of the ampere.)

Worked examples

Example 1 — force on a moving charge. A proton (q = 1.6\times 10^{-19}\ \text{C}) moves at v = 2.0\times 10^{6}\ \text{m/s} through a field B = 0.50\ \text{T}, at an angle \theta = 30^\circ to the field. The magnetic force is

F = qvB\sin\theta = (1.6\times 10^{-19})(2.0\times 10^{6})(0.50)\sin 30^\circ. F = (1.6\times 10^{-19})(2.0\times 10^{6})(0.50)(0.5) \approx 8.0\times 10^{-14}\ \text{N}.

Tiny in newtons, but on a proton of mass 1.67\times 10^{-27}\ \text{kg} that is a colossal acceleration.

Example 2 — radius and period of circular motion. An electron (m = 9.11\times 10^{-31}\ \text{kg}, q = 1.6\times 10^{-19}\ \text{C}) moves at v = 3.0\times 10^{6}\ \text{m/s} perpendicular to a field B = 0.010\ \text{T}. Its circular radius is

r = \frac{mv}{qB} = \frac{(9.11\times 10^{-31})(3.0\times 10^{6})}{(1.6\times 10^{-19})(0.010)} \approx 1.7\times 10^{-3}\ \text{m} \approx 1.7\ \text{mm}.

And the time for one full loop, independent of the speed, is

T = \frac{2\pi m}{qB} = \frac{2\pi (9.11\times 10^{-31})}{(1.6\times 10^{-19})(0.010)} \approx 3.6\times 10^{-9}\ \text{s}.

Example 3 — force between two parallel wires. Two long wires d = 0.10\ \text{m} apart each carry I = 10\ \text{A} in the same direction. The force per metre is

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} = \frac{(4\pi\times 10^{-7})(10)(10)}{2\pi (0.10)} = 2.0\times 10^{-4}\ \text{N/m},

an attraction, because the currents run the same way. Reverse one current and the same-size force becomes a repulsion.

No — never, and this catches people constantly. Because \vec F = q\,\vec v \times \vec B is always perpendicular to \vec v, the magnetic force does zero work and cannot change kinetic energy. A magnetic field can bend a beam into a circle, steer it, focus it, turn it around — but the speed coming out equals the speed going in. If a charged particle actually gains energy (as in a cyclotron or synchrotron), the energy is supplied by an electric field; the magnetic field only does the steering. Two more traps in the same family: a stationary charge (v = 0) feels no magnetic force at all, and a charge moving parallel to \vec B (\theta = 0) also feels none. And don't forget the sign flip for negative charges: the right-hand rule gives the direction of \vec v \times \vec B, but the force on an electron points the opposite way.

The magic ingredient is that the cyclotron period T = 2\pi m/(qB) is independent of speed. In a cyclotron, protons spiral inside two hollow D-shaped electrodes bathed in a uniform magnetic field, and every time a proton crosses the gap between the Ds an alternating electric field gives it a kick, speeding it up. As it speeds up its circle grows — but because the period never changes, the same steady electrical rhythm stays perfectly in step with the particle, lap after lap, all the way out to the rim. Ernest Lawrence's first cyclotron in 1930 was just 13\ \text{cm} across and fit in the palm of a hand, yet the trick scaled up to machines that split atoms and, eventually, to the giant rings of modern particle physics. (At very high speeds relativity slowly stretches the period, which is why the biggest accelerators must ramp their field or frequency to keep pace — but that is a story for another page.)