Apply the magnetic force on a moving charge and on a current-carrying conductor, including the motor effect

What this dot point is asking

WACE wants you to predict the size and direction of the force a magnetic field exerts, both on a lone moving charge and on a wire carrying a current, and then to use that force to explain circular motion in a field and the operation of a simple DC motor. The key idea is that a magnetic force is always perpendicular to the velocity, so it changes direction but never speed.

Force on a moving charge

A charge $q$ moving with speed $v$ at an angle $\theta$ to a magnetic field $B$ experiences a force

The force is greatest when the velocity is perpendicular to the field ($\theta=90^\circ$) and zero when the charge moves along the field ($\theta=0$). Because the force is always at right angles to $v$, it does no work: it cannot change the kinetic energy, only the direction.

Direction comes from a hand rule. Using the right hand for conventional (positive) current, point the fingers in the direction of $v$, curl them toward $B$, and the thumb gives $F$; for a negative charge such as an electron the force is reversed. Many WACE students prefer the right-hand slap rule (fingers along $B$, thumb along $v$, palm pushes in the direction of $F$ for a positive charge).

Charges moving in circles

When a charge enters a uniform field at right angles, the constant perpendicular force supplies a centripetal force, so the charge follows a circular path. Setting the magnetic force equal to the centripetal force,

Faster or heavier particles curve in larger circles; stronger fields or larger charges tighten the radius. This is the principle behind mass spectrometers and the bending magnets in particle accelerators.

Force on a current-carrying conductor

A current is simply moving charge, so a straight wire of length $L$ carrying current $I$ in a field $B$ feels

where $\theta$ is the angle between the current and the field. The direction follows the same right-hand rule, now with the fingers (or thumb) pointing along the conventional current. Two parallel wires therefore exert forces on each other: currents in the same direction attract, opposite currents repel.

The motor effect

Place a rectangular current loop in a magnetic field and the two sides carrying current across the field feel opposite forces, producing a turning effect (a torque) that rotates the coil. This is the motor effect. A split-ring commutator reverses the current every half turn so the torque keeps driving the coil the same way, giving continuous rotation. The torque is largest when the coil plane is parallel to the field and zero when the plane is perpendicular, which is why a real motor uses many turns and several coils to smooth the output.

Reading the geometry

Exam questions live or die on the angle $\theta$ and the direction. Always identify whether the velocity or current is perpendicular to the field, draw the field, velocity and force on three mutually perpendicular axes, and state clearly which hand rule you used and whether the carrier is positive or negative.