 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ia/paper-4/10a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

m \ddot{x} = q \dot{x} \times B .

Ω = qB / m

, the component equations are

\overset{x}{¨} = Ω \overset{y}{˙}, \overset{y}{¨} = - Ω \overset{x}{˙}, \overset{z}{¨} = 0.

The initial velocity therefore gives

\overset{x}{˙} = v sin (Ω t), \overset{y}{˙} = v cos (Ω t), \overset{z}{˙} = v,

and integration with the initial position gives

x (t) = (1 + \frac{v}{Ω} [1 - cos (Ω t)], \frac{v}{Ω} sin (Ω t), v t) .

v /Ω

around the line

x = 1 + v /Ω

y = 0

, with its axis parallel to the magnetic field.

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