Paper 4, Section II, C

Write down the expression for the momentum of a particle of rest mass $m$, moving with velocity $\mathbf{v}$ where $v=|\mathbf{v}|$ is near the speed of light $c$. Write down the corresponding 4-momentum.

Such a particle experiences a force $\mathbf{F}$. Why is the following expression for the particle's acceleration,

$\mathbf{a}=\frac{\mathbf{F}}{m}$

not generally correct? Show that the force can be written as follows

$\mathbf{F}=m \gamma\left(\frac{\gamma^{2}}{c^{2}}(\mathbf{v} \cdot \mathbf{a}) \mathbf{v}+\mathbf{a}\right)$

Invert this expression to find the particle's acceleration as the sum of two vectors, one parallel to $\mathbf{F}$ and one parallel to $\mathbf{v}$.

A particle with rest mass $m$ and charge $q$ is in the presence of a constant electric field $\mathbf{E}$ which exerts a force $\mathbf{F}=q \mathbf{E}$ on the particle. If the particle is at rest at $t=0$, its motion will be in the direction of $\mathbf{E}$ for $t>0$. Determine the particle's speed for $t>0$. How does the velocity behave as $t \rightarrow \infty$ ?

[Hint: You may find that trigonometric substitution is helpful in evaluating an integral.]

*Typos? Please submit corrections to this page on GitHub.*