Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2009

(a) A particle of charge qq moves with velocity vv in a constant magnetic field B. Give an expression for the Lorentz force F\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 QQ, are fixed at the four corners of a square with sides of length 2a2 a. Another point particle, of positive charge qq, 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 qq 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 QQ and qq separated by distance rr is Qq/4πϵ0r.]\left.Q q / 4 \pi \epsilon_{0} r .\right]

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