Motion in Electromagnetic Fields

Between two oppositely charged parallel plates the electric field is uniform: the field lines are evenly spaced, parallel, and run from the positive plate to the negative plate. A charge $q$ in this region experiences a constant force, directed along the field for a positive charge and opposite to it for a negative charge, given by:

$$F = qE$$

A constant force means a constant acceleration $a = \frac{qE}{m}$, so the motion is exactly analogous to projectile motion under gravity.

• A charge released from rest accelerates in a straight line along the field.
• A charge entering perpendicular to the field keeps a constant velocity component across the plates while accelerating along the field, tracing a parabolic path.

For example, the electron beam in an oscilloscope is steered up and down by a pair of deflection plates in exactly this way.

You should be able to split the motion into independent components, apply the kinematic (suvat) equations along the field direction, and calculate the work done $W = qEd$ as the charge moves a distance $d$ along the field, which equals its gain in kinetic energy.

A magnetic field pushes on a moving charge with force $F = qvB\sin\theta$, where $\theta$ is the angle between the velocity and the field. The force is perpendicular to both the velocity and the field, with a direction given by hand rules (reversed for a negative charge).

When a charged particle enters a uniform magnetic field perpendicularly, the magnetic force acts as a centripetal force: because it is always perpendicular to the velocity, its magnitude stays constant at $F = qvB$ while its direction changes continuously, which is exactly the condition for uniform circular motion:

$$qvB = \frac{mv^2}{r}$$

From this we derive the orbital radius $r = \frac{mv}{qB}$ and the period of rotation $T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$. Crucially, the period is completely independent of the particle's speed and radius, which is the fundamental physics principle behind cyclotron accelerators.

Since the force never has a component along the motion, the field does no work: the speed and kinetic energy are unchanged and only the direction turns. A proton and an electron entering the same field curve in opposite senses, and the much lighter electron follows a far tighter circle.

You must be able to find the force direction, calculate $r$ and $T$, and sketch circular arcs of constant radius.

When a region contains both an electric field and a magnetic field, a moving charge feels two forces at once: the electric force $qE$, which does not depend on the motion, and the magnetic force $qvB$ (for velocity perpendicular to the field), which does.

Arranging the two fields at right angles to each other, so-called crossed fields, makes these forces point in opposite directions for a particle moving perpendicular to both. The forces balance when $qE = qvB$, which happens for one single speed:

$$v = \frac{E}{B}$$

Notice that this speed is independent of the particle's charge and mass. This arrangement is a velocity selector: only particles with exactly this speed travel straight through, while faster particles are deflected by the now-dominant magnetic force and slower ones by the electric force.

A velocity selector is used to prepare a beam of known speed, for example before the particles enter a separate field region where their paths are analysed. You should be able to set up the force balance, derive $v = \frac{E}{B}$, and predict which way an off-speed particle deflects.

A current is a flow of charge, so a current-carrying wire placed in a magnetic field feels the combined magnetic force on all of its moving charges. For a straight conductor of length $L$ carrying current $I$ at angle $\theta$ to a field $B$, the force is given by:

$$F = BIL\sin\theta$$

The force is largest when the wire is perpendicular to the field and zero when the wire lies along the field lines. The direction is perpendicular to both the wire and the field and comes from Fleming's left-hand rule:

• first finger along the field
• second finger along the current
• thumb giving the force

This motor effect is what makes a coil turn in a DC motor: the two opposite sides of the coil carry current in opposite directions, so they feel forces in opposite directions, producing a turning effect.

For example, a $0.50\ \text{m}$ wire carrying $3.0\ \text{A}$ perpendicular to a $0.20\ \text{T}$ field experiences $F = 0.20 \times 3.0 \times 0.50 = 0.30\ \text{N}$.

Be ready to identify $\theta$ correctly and to connect this equation to the definition of magnetic flux density and the tesla.

Every current creates its own magnetic field, so two current-carrying wires inevitably act on each other. The force between two long parallel wires is an elegant demonstration of Newton's third law and of field interaction.

Wire 1 produces a concentric magnetic field of strength $B_1 = \frac{\mu_0 I_1}{2\pi r}$ at the position of Wire 2, a distance $r$ away. Wire 2, carrying current $I_2$, therefore experiences a force $F = B_1 I_2 L$ along a length $L$. Substituting for $B_1$ gives the force per unit length:

$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r}$$

By symmetry, Wire 2 exerts an equal and opposite force on Wire 1.

• Currents in the same direction attract: the field in the gap between the wires is weakened while the field outside is strengthened, pushing the wires together.
• Currents in opposite directions repel.

You can see this in practice when closely spaced cables carrying large currents jump apart at switch-on.

You should be able to calculate $\frac{F}{L}$ using $\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}$, identify the force direction from the two current directions, and explain the attraction or repulsion using the resultant field pattern.