Magnetic Force on Currents and Charges

A magnetic field is invisible, and mostly it seems to do nothing at all — a fridge magnet just sits there. But set an electric charge moving through that field and something remarkable happens: the field reaches out and shoves the charge sideways, at right angles to the way it is going. Nudge a whole stream of charges — a current in a wire — and the wire itself jumps. This one sideways push runs the world's electric motors, steers beams of particles round kilometre-wide accelerators, sorts atoms by their mass, and paints the polar sky with the aurora.

You have already met one face of it in the motor effect, the force on a current-carrying wire. This page pushes deeper: down to the force on a single moving charge, and the beautiful consequence that a charge fired across a magnetic field does not fly straight or curve to a stop — it wheels round in a perfect circle.

The force on a current-carrying wire: F = BIL\sin\theta

Lay a straight wire of length L carrying current I across a magnetic field of flux density B, and it feels a force. When the wire sits at right angles to the field the force is at its greatest, F = BIL. Swing the wire round to make an angle \theta with the field and only the part of the field crossing the wire perpendicularly does any pushing, so the general result is

F = BIL\sin\theta.

Read off the two extremes. When \theta = 90^\circ, \sin\theta = 1 and the force is largest: F = BIL. When the wire lies along the field, \theta = 0^\circ, \sin\theta = 0 — and the force vanishes completely. A wire pointing the same way as the field feels no push at all.

The quantity B is the magnetic flux density — the true measure of a field's strength — and it is measured in tesla (T). One tesla is a very strong field: the Earth's own field is only about 5 \times 10^{-5}\ \text{T}, a fridge magnet a few hundredths of a tesla, and a hospital MRI scanner around 1.5\ \text{T}.

The force on a single moving charge: F = BQv\sin\theta

A current is nothing but charge on the move, so the wire's force must really be the sum of tiny forces on each charge carrier drifting along inside it. Zoom in on one charge Q travelling at speed v through the field, and the same rule applies:

F = BQv\sin\theta,

where \theta is the angle between the charge's velocity and the field. As before, the force is greatest when the charge cuts across the field at 90^\circ, giving F = BQv, and it is zero when the charge moves along the field lines.

Two features of this deserve their own spotlight, because they trip people up constantly. First, only a moving charge feels the force: put v = 0 and F = BQv = 0, so a stationary charge sitting in a magnetic field feels nothing. Second, only the component of velocity across the field counts — that is the job of the \sin\theta.

Which way? Fleming's left-hand rule

The force, the field and the current are mutually perpendicular — three directions each at right angles to the other two. To untangle them, hold up your left hand with thumb, first finger and second finger stuck out like the corner of a box:

Line the first finger up with the field and the second finger along the current, and the thumb points the way the force pushes. For a single positive charge, "current" is simply the direction the charge moves, so point the second finger along its velocity.

Here is the sharp edge that catches almost everyone. Conventional current runs the way positive charge flows. A negative charge (an electron, say) moving to the right is a conventional current to the left. So to find the force on a negative charge with the left-hand rule, point your second finger opposite to the charge's actual motion — and the force comes out reversed compared with a positive charge going the same way.

Why a charge in a field goes in a circle

Now the payoff. Because the magnetic force is always perpendicular to the velocity, it can never speed the charge up or slow it down — it can only bend its direction. A force that is forever at right angles to the motion, constant in size, doing nothing but turning the velocity: that is exactly the recipe for circular motion. A charge fired straight across a uniform field curves round into a perfect circle.

We can pin down the radius. The magnetic force supplies the whole of the centripetal force, so set the two equal:

BQv = \dfrac{mv^2}{r}.

Cancel one factor of v from each side and rearrange for the radius:

r = \dfrac{mv}{BQ}.

The radius grows with the particle's momentum mv — a heavier or faster particle carves a wider arc — and shrinks with a stronger field B or a larger charge Q. This single formula is the engine behind mass spectrometers and particle accelerators alike.

And notice what the force does not do: because it is perpendicular to the motion at every instant, the magnetic force does no work. The particle's speed — and so its kinetic energy — is unchanged all the way round. A magnetic field can steer a beam of particles, but it can never, on its own, make them go faster.

Worked examples

Example 1 — force on a wire (F = BIL). A straight wire of length L = 0.5\ \text{m} carries a current I = 3\ \text{A} at right angles to a field of B = 0.2\ \text{T}. Find the force.

F = BIL = 0.2 \times 3 \times 0.5 = 0.3\ \text{N}.

Example 2 — force on a moving charge (F = BQv). A proton of charge Q = 1.6 \times 10^{-19}\ \text{C} moves at v = 2.0 \times 10^{6}\ \text{m/s} at right angles to a B = 0.5\ \text{T} field. Find the magnetic force on it.

F = BQv = 0.5 \times (1.6 \times 10^{-19}) \times (2.0 \times 10^{6}) = 1.6 \times 10^{-13}\ \text{N}.

Tiny in absolute terms — but a proton is fantastically light, so this force whips it round a very tight circle.

Example 3 — the radius of the circle (r = \dfrac{mv}{BQ}). An electron (mass m = 9.1 \times 10^{-31}\ \text{kg}, charge Q = 1.6 \times 10^{-19}\ \text{C}) enters a B = 0.4\ \text{T} field at v = 3.0 \times 10^{6}\ \text{m/s}, perpendicular to the field. What is the radius of its circular path?

r = \dfrac{mv}{BQ} = \dfrac{(9.1 \times 10^{-31})(3.0 \times 10^{6})}{(0.4)(1.6 \times 10^{-19})} = \dfrac{2.73 \times 10^{-24}}{6.4 \times 10^{-20}} \approx 4.3 \times 10^{-5}\ \text{m}.

About 43\ \mu\text{m} — a circle thinner than a human hair. Double the speed and the circle doubles in size; double the field and it halves.

See it curl: the radius r = \dfrac{mv}{BQ}

Below, a positive charge +q moves through a field pointing into the page (the ⊗ marks). At every instant the magnetic force F = BQv (green) points straight in towards the centre — always at 90^\circ to the velocity v (blue tangent) — so the charge sweeps round the dashed circle.

Turn up the speed and watch the circle swell (more momentum, wider arc); turn up the field and watch it tighten. That is r = \dfrac{mv}{BQ} in action, shown here with the mass and charge set to one unit so r = v/B.

Four traps that catch people out with the magnetic force:

A mass spectrometer turns r = \dfrac{mv}{BQ} into a set of scales for atoms. First it strips electrons off a sample to make ions, then accelerates them all to the same speed and fires them into a uniform magnetic field. Each ion curves round a semicircle and lands on a detector — but heavier ions, carrying more momentum mv, swing round a wider arc before coming back. Because r \propto m (with v, B and Q fixed), the landing spot reveals the mass with astonishing precision. This is how chemists tell one isotope from another, how doping tests catch banned drugs, and how spacecraft sniff the make-up of an alien atmosphere.

The very same curving is what makes the aurora. Charged particles streaming from the Sun get caught by the Earth's magnetic field and spiral tightly along its field lines — because a charge moving at a slant to a field corkscrews rather than circles — funnelling down towards the magnetic poles, where they crash into the upper atmosphere and set it glowing green and red.

Where this force is working right now

The magnetic force on charges is not a laboratory curiosity — it is doing heavy lifting all around you. Every electric motor spins because of the force on its current-carrying coils. A cyclotron or synchrotron uses a field to bend charged particles onto a circular track, whipping them round again and again to enormous speeds while electric fields do the actual accelerating — the field bends, the voltage boosts. Mass spectrometers weigh atoms by their circular radius, and old cathode-ray televisions steered their electron beam with magnetic coils. From the fusion plasmas held in a magnetic bottle to the shimmering aurora overhead, it all comes back to one idea: a moving charge in a magnetic field gets pushed sideways.