Revise Motion in electromagnetic fields for IB Physics (D.3)

A wire carrying a current is a stream of moving charges, so when it lies in a magnetic field it feels a force. This is the motor effect. The size of the force depends on the field strength $B$, the current $I$, the length $L$ of wire that lies in the field, and the angle $\theta$ between the current and the field. The force is a maximum when the wire is perpendicular to the field and drops to zero when the wire runs along the field lines.

$F = BIL\sin\theta$

Here $B$ is in tesla (T), $I$ in amperes (A), $L$ in metres (m) and the force $F$ in newtons (N). The direction of the force is perpendicular to BOTH the current and the field, and you find it with Fleming's left-hand rule.

The force on a wire is really the combined force on all the individual charges drifting through it. Zoom in on a single charge $q$ moving with speed $v$ through a field $B$ and the force on it is $F = qvB\sin\theta$, where $\theta$ is the angle between the velocity and the field. Just like the wire, a stationary charge feels no magnetic force, and a charge moving along the field lines feels none either — only the component of velocity perpendicular to the field matters.

$F = qvB\sin\theta$

Fire a charge into a uniform magnetic field so that its velocity is perpendicular to the field, and something elegant happens. The magnetic force has a constant magnitude $qvB$ (the speed cannot change, because the force does no work) and it always points at right angles to the velocity. A constant-magnitude force that is always perpendicular to the motion is exactly the definition of a centripetal force, so the charge is pushed round a circle at constant speed. Equating the magnetic force to the centripetal force gives the radius of that circle.

$qvB = \frac{mv^2}{r} \quad\Longrightarrow\quad r = \frac{mv}{qB}$

The radius grows with the particle's momentum ($mv$) and shrinks as the charge or field increases. This is why a mass spectrometer can separate ions: with the same speed, charge and field, heavier ions swing round on wider arcs. Notice the speed $v$ cancels from neither side by accident — it stays in the formula, so faster particles do trace larger circles even though the field does no work on them.

Because every current-carrying wire in a field feels a force, a coil of wire in a field feels a turning effect. In a simple d.c. motor the current runs up one side of a coil and down the other, so by Fleming's left-hand rule the two sides are pushed in opposite directions. That pair of opposite forces on either side of the axle is a couple, and it spins the coil — the motor effect turned into rotation. A commutator flips the current direction every half turn so the coil keeps rotating the same way.

Two parallel wires interact through the same effect. Each wire produces a magnetic field in the space around it, and the other wire, carrying current through that field, feels a force $F = BIL$. Working through the directions with Fleming's left-hand rule gives a clean qualitative rule: currents in the same direction attract, and currents in opposite directions repel. This mutual magnetic force is how the ampere was historically defined.

When a charge moves through a region containing both an electric field and a magnetic field, it feels an electric force $F_E = qE$ (which acts on the charge whether or not it is moving) and a magnetic force $F_B = qvB$ (which acts only because it is moving). A velocity selector arranges the two fields at right angles so that these forces point in opposite directions. The electric force $qE$ is independent of speed, but the magnetic force $qvB$ grows with speed, so the two can balance for only one particular speed.

The magnetic force on a moving charge is the engine behind an enormous amount of physics and technology. The circular-motion result $r = mv/qB$ is the working principle of mass spectrometers and cyclotrons, and the same force that spins a motor, run in reverse, is what generates electricity in the induction topic that follows. Master the habit here — decide whether you have a current or a single charge, pick the matching force law, watch the angle, use the left hand and reverse for negatives, and set magnetic force equal to centripetal force for circular paths — and the field topics that follow become variations on a method you already own.