Question

An electron enters a magnetic field at an angle theta and it leaves at an angle A degrees . Find the relation between A and theta if electron enters through origin in magnetic field inside the plane of paper and magnetic field is starting from positive xy plane

Answer 1

tan A = (2 tan θ)/(1 – tan² θ)

Answer 2

We will show that by drawing the circle for the electron’s path the answer can be expressed in a “double‐angle” form.

1. When a charged particle (here an electron of charge –e) enters a uniform magnetic field the Lorentz force makes it move in a circle of radius

   R = (mv)/(eB).

2. Suppose the electron enters at the origin with velocity making an angle θ with the x‑axis. For “in‐plane” motion the magnetic force always acts at right angles to the velocity so that the electron follows a circle. (Its centre is located a distance R away from the entry point in a direction perpendicular to the velocity; the sign of the charge (–e) tells you which side.)

3. Let the electron leave the magnetic field at some point on the circular arc so that the tangent at the exit makes an angle A with the x‑axis. A little geometric analysis of the circle shows that the angle subtended at the centre by the arc is 2θ. (That is, the chord joining the entry and exit points subtends a angle 2θ at the centre.)

4. Since for a circle the tangent and the radius are perpendicular, one finds that the relation between the two tangents (at entry and exit) is exactly that of a “double‐angle.” In other words one deduces that

   tan A = (2 tan θ)/(1 – tan² θ).

It is also clear that if the angles are small then tan A ≈ 2 tan θ so that A ≈ 2θ.