Moving Charges and Magnetism

A charge $q$ moving with velocity $\vec{v}$ through a magnetic field $\vec{B}$ experiences the magnetic (Lorentz) force:

Because this is a cross product, the force is always perpendicular to both $\vec{v}$ and $\vec{B}$. Its magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$: maximum when velocity is perpendicular to the field ($\theta=90^\circ$), and zero when the charge moves along the field ($\theta=0^\circ$), in which case it feels no magnetic force at all. The direction follows the right-hand rule: point the fingers along $\vec{v}$, curl them toward $\vec{B}$, and the thumb gives the direction of $\vec{v}\times\vec{B}$ (for a positive charge; reverse it for a negative one).

A force perpendicular to velocity does zero work on the charge, no matter how strong the field: magnetic forces can redirect a charged particle, but they can never speed it up or slow it down.

When a charge enters a uniform field with its velocity exactly perpendicular to $\vec{B}$, the magnetic force is always perpendicular to the velocity, which is exactly the condition for uniform circular motion: the force supplies the centripetal force, with a fixed magnitude and continuously changing direction.

This is the operating principle of the cyclotron, a particle accelerator that repeatedly kicks a charged particle to higher speeds using an alternating voltage held at one fixed frequency. Because the particle’s own revolution frequency never changes as it speeds up, that fixed external frequency stays in step with it, lap after lap, letting a single alternating voltage keep accelerating the particle indefinitely (this synchrony eventually fails at speeds approaching light speed, outside CBSE scope).

A current is just a stream of moving charges, so a current-carrying wire in a magnetic field feels a force too, the sum of the Lorentz forces on every charge drifting through it. For a straight wire of length $L$ carrying current $I$ in a uniform field $B$:

where $\theta$ is the angle between the wire (current direction) and the field. The force is maximum when the wire is perpendicular to $B$, and zero when the current flows along the field.

Section 4.3 took a magnetic field as given and asked what force it exerts. This section asks the reverse question: where does the field itself come from? Every magnetic field traced back far enough comes from a moving charge or a current, and the Biot–Savart law tells us exactly how much field a small piece of current produces.

Take a tiny element of wire, length $d\vec{l}$, carrying current $I$. At a point P, a displacement $\vec{r}$ away from the element (with $\hat{r}$ the unit vector along it), this element contributes a field:

where $\mu_0 = 4\pi\times10^{-7}\text{ T m/A}$ is the permeability of free space, the magnetic counterpart of $\varepsilon_0$ in Coulomb’s law. In magnitude, writing $\theta$ for the angle between $d\vec{l}$ and $\vec{r}$:

The direction of $d\vec{B}$ is perpendicular to the plane containing $d\vec{l}$ and $\vec{r}$ (the right-hand screw rule: curl the fingers from $d\vec{l}$ toward $\vec{r}$, and the thumb gives the direction of $d\vec{l}\times\hat{r}$). To get the actual field at P due to a real, finite wire, every element’s contribution has to be summed (in practice, integrated) over the whole length of the wire, using the same principle of superposition that governs the electric field of a continuous charge distribution.

As a first real application of the Biot–Savart law, consider a circular loop of radius $R$, carrying current $I$, and ask for the field at a point P on its axis, a distance $x$ from the centre.

Every element $d\vec{l}$ of the loop sits a distance $r=\sqrt{x^2+R^2}$ from P, and is perpendicular to that displacement, so each contributes $dB=\frac{\mu_0}{4\pi}\frac{I\,dl}{x^2+R^2}$. That field has a component along the axis and a component perpendicular to it; but for every element, there is a diametrically opposite element whose perpendicular component points exactly the other way, so all the perpendicular components cancel in pairs, leaving only the axial component to survive the sum. That axial component is $dB_x = dB\cos\phi$, where $\cos\phi = R/\sqrt{x^2+R^2}$ (the angle between $d\vec{B}$ and the axis), so:

Summing $dl$ all the way around the loop just gives its circumference, $2\pi R$:

Setting $x=0$ gives the field at the centre of the loop itself, a clean special case worth remembering on its own:

The Biot–Savart law is always true, but integrating it by brute force can be hard work. For situations with enough symmetry, Ampere’s circuital law gives the same physics in a far more convenient package, exactly the way Gauss’s law is Coulomb’s law repackaged for symmetric charge distributions.

The integral runs around any closed loop (an “Amperian loop”), and $I_{\text{enclosed}}$ is the total current passing through the surface that loop bounds. In the common case where a loop can be chosen on which $B$ is tangential and has the same magnitude all the way around (or is zero, or purely perpendicular, over the rest of the loop), this collapses to simple algebra:

with $L$ the length of the loop over which $B$ is tangential and $I_e$ the current it encloses.

A solenoid is a tightly wound helical coil. Deep inside a long solenoid, far from its ends, the magnetic field is uniform and directed along its axis, essentially zero just outside.

Unlike the field near a single wire, this field doesn’t fall off inside the solenoid, it stays essentially constant across the entire interior cross-section, independent of position, right up to the ends.

Each current-carrying wire creates its own magnetic field (now derived properly in §4.6.1), which exerts a force on any other current sitting in it. Two long, straight, parallel wires carrying currents $I_1$ and $I_2$, separated by a distance $d$, exert a force on each other per unit length of:

Currents in the same direction attract; currents in opposite directions repel, the reverse of what happens between like and unlike electric charges.

This result is historically how the ampere itself was defined: one ampere is the constant current which, flowing in each of two infinitely long parallel wires held 1 m apart in vacuum, produces a force of exactly $2\times10^{-7}\text{ N}$ on each metre of either wire.

A current loop of $N$ turns, each of area $A$, carrying current $I$, placed in a field $B$ making angle $\theta$ with the loop’s own magnetic moment, experiences a torque that tries to rotate it into alignment with the field:

This has exactly the same mathematical form as the torque on any magnetic dipole, $\vec{\tau} = \vec{m}\times\vec{B}$, since a current loop simply is a magnetic dipole, with dipole moment $m = NIA$.

The torque vanishes only when $\vec{m}$ is parallel or antiparallel to $\vec{B}$: these are the loop’s two equilibrium orientations. Parallel ($\theta=0$) is stable, any small rotation away from it is pushed straight back; antiparallel ($\theta=180^\circ$) is unstable, the restoring torque there actually grows the rotation instead of opposing it. This is the same stable/unstable pair, for exactly the same reason, as a bar magnet’s two alignments with an external field.

A moving-coil galvanometer puts §4.9’s torque to work as a measuring instrument. A coil of $N$ turns and area $A$ sits in a radial magnetic field, radial by design so that $B$ is always perpendicular to the coil’s plane, whatever angle the coil has rotated to, keeping $\sin\theta=1$ throughout its swing. The magnetic torque $NIAB$ is balanced against a spring’s restoring torque, $k\phi$, where $k$ is the spring’s torsional constant and $\phi$ the steady deflection angle:

the deflection ends up directly proportional to the current, which is exactly what lets the instrument measure current in the first place. The quantity in brackets is called the current sensitivity, $\phi/I$, a fixed property of a given galvanometer.

Two of the chapter’s own exercise questions, with original worked solutions, both exercising the field-generation formulas derived in §4.5 and §4.6.1 above.