Paper 4, Section II, A

(a) A particle of charge $q$ moves with velocity $v$ in a constant magnetic field B. Give an expression for the Lorentz force $\mathbf{F}$ experienced by the particle. If no other forces act on the particle, show that its kinetic energy is independent of time.

(b) Four point particles, each of positive charge $Q$, are fixed at the four corners of a square with sides of length $2 a$. Another point particle, of positive charge $q$, is constrained to move in the plane of the square but is otherwise free.

By considering the form of the electrostatic potential near the centre of the square, show that the state in which the particle of charge $q$ is stationary at the centre of the square is a stable equilibrium. Obtain the frequency of small oscillations about this equilibrium.

[The Coulomb potential for two point particles of charges $Q$ and $q$ separated by distance $r$ is $\left.Q q / 4 \pi \epsilon_{0} r .\right]$

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