Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 37A / c

Solution

The four-velocity and particle energy are

u^{μ} = γ (c, v), E = γm c^{2}, p^{0} = m u^{0} = \frac{E}{c} .

With the conventions for the given metric,

F^{0 i} = E_{i} / c

. The temporal component of the covariant Lorentz force is therefore

\frac{d ( E / c )}{d τ} = q F^{0 i} u_{i} = \frac{q}{c} γ E \cdot v .

Since

d t / d τ = γ

, cancellation of

γ / c

\frac{d E}{d t} = q E \cdot v .

The absence of

B

reflects that magnetic force is perpendicular to the velocity and does no work.

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