Moving Charges & Magnetism

A magnetic field exerts a force only on moving charges. The Lorentz force on a charge $q$ moving with velocity $v$ in a field $B$ has magnitude $F = qvB\sin\theta$, where $\theta$ is the angle between $v$ and $B$. The direction is given by the right-hand rule and is always perpendicular to both $v$ and $B$ — a defining feature with deep consequences.

Because the force is perpendicular to the velocity, it can change the direction of motion but never the speed. A magnetic force therefore does no work on a charged particle, so the kinetic energy and speed stay constant — one of the most-tested conceptual points in this chapter. The force is zero when the charge moves parallel to the field ($\theta = 0$) and maximum when it moves perpendicular ($\theta = 90^{\circ}$).

When a charge enters a uniform field perpendicular to it, the constant perpendicular force makes it move in a circle. The magnetic force supplies the centripetal force, $qvB = \dfrac{mv^{2}}{r}$, giving the radius $r = \dfrac{mv}{qB}$. The time period $T = \dfrac{2\pi m}{qB}$ is striking because it is independent of speed — faster particles simply trace larger circles in the same time, the principle behind the cyclotron.

If the velocity has a component along the field as well, the path becomes a helix: the parallel component carries the particle steadily forward while the perpendicular component drives the circular motion. When both electric and magnetic fields are present, the combined Lorentz force is $F = qE + qvB$, and a velocity selector uses balanced electric and magnetic forces to let through only particles of a chosen speed $v = E/B$.

Since a current is a stream of moving charges, a current-carrying wire in a magnetic field experiences a force. For a straight wire of length $L$ carrying current $I$ in a field $B$, the force is $F = BIL\sin\theta$, where $\theta$ is the angle between the wire and the field. The direction follows Fleming's left-hand rule — a reliable shortcut for NEET, with thumb, forefinger and middle finger giving force, field and current.

This force is the working principle of every electric motor and loudspeaker. It is maximum when the wire is perpendicular to the field and zero when the wire lies along the field. A useful exam habit is to resolve the wire or the field into perpendicular and parallel parts, since only the perpendicular part contributes.

A current loop in a uniform field feels no net force (the forces on opposite sides cancel) but does feel a torque that tries to rotate it. For a coil of $N$ turns, area $A$, carrying current $I$, the torque is $\tau = NIAB\sin\theta$, where $\theta$ is measured from the field. This is the basis of the moving-coil galvanometer and the DC motor.

The product $NIA$ behaves like a magnetic dipole moment $m = NIA$, so a current loop acts exactly like a tiny bar magnet, with the torque written compactly as $\tau = mB\sin\theta$. This dipole analogy — a current loop equals a magnet — is a recurring NEET theme that links this chapter to magnetism of materials.

Currents not only feel magnetic forces, they also produce magnetic fields. The Biot–Savart law gives the field of a small current element: $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^{2}}$, where $\mu_0 = 4\pi\times10^{-7}\ \text{T m A}^{-1}$ is the permeability of free space. The field is perpendicular to the plane containing the element and the point, falling off as $1/r^{2}$.

Integrating this gives standard results worth memorising. The field at distance $r$ from a long straight wire is $B = \dfrac{\mu_0 I}{2\pi r}$, circling the wire (right-hand grip rule) and falling as $1/r$. At the centre of a circular loop of radius $R$ it is $B = \dfrac{\mu_0 I}{2R}$, and for $N$ turns it is $N$ times this — a very common NEET numerical.

A solenoid, a tightly wound coil, produces a strong, nearly uniform field inside and a weak field outside. The interior field is $B = \mu_0 n I$, where $n$ is the number of turns per unit length — notably independent of the solenoid's radius. A solenoid behaves like a bar magnet, with one end acting as a north pole and the other as a south pole, linking neatly to the dipole idea.

A toroid is a solenoid bent into a doughnut; its field is confined entirely within the ring, $B = \dfrac{\mu_0 N I}{2\pi r}$, with essentially zero field outside. These geometry-specific formulas — straight wire, loop, solenoid, toroid — are the toolkit NEET expects you to apply quickly without re-deriving.

Ampere's circuital law is the magnetic counterpart of Gauss's law: the line integral of the magnetic field around any closed loop equals $\mu_0$ times the total current threading the loop, $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}$. For problems with enough symmetry it gives the field far faster than the Biot–Savart law, and it cleanly reproduces the straight-wire and solenoid results.

Because each current produces a field and each current feels a force in a field, two parallel wires exert forces on each other. The force per unit length is $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$. The crucial NEET rule of thumb is the direction: currents in the same direction attract, currents in opposite directions repel — the reverse of how electric charges behave. This mutual force is, in fact, the definition of the ampere.

The torque on a current loop is harnessed in the moving-coil galvanometer. A coil pivoted in a radial magnetic field deflects until the magnetic torque balances the restoring torque of a spring, so the deflection is directly proportional to the current: $I \propto \phi$. A radial field is used precisely so that the scale is linear, a detail NEET likes to ask about.

Two figures of merit describe a galvanometer. Its current sensitivity is the deflection per unit current, and its voltage sensitivity the deflection per unit voltage. As covered in current electricity, adding a small shunt in parallel converts it to an ammeter, while a large series resistance converts it to a voltmeter — closing the loop between magnetism and circuit measurement.