Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2019

(a) A particle of mass mm and positive charge qq moves with velocity x˙\dot{\mathbf{x}} in a region in which the magnetic field B=(0,0,B)\mathbf{B}=(0,0, B) is constant and no other forces act, where B>0B>0. Initially, the particle is at position x=(1,0,0)\mathbf{x}=(1,0,0) and x˙=(0,v,v)\dot{\mathbf{x}}=(0, v, v). Write the equation of motion of the particle and then solve it to find x\mathbf{x} as a function of time tt. Sketch its path in (x,y,z)(x, y, z).

(b) For B=0B=0, three point particles, each of charge qq, are fixed at (0,a/3,0)(0, a / \sqrt{3}, 0), (a/2,a/(23),0)(a / 2,-a /(2 \sqrt{3}), 0) and (a/2,a/(23),0)(-a / 2,-a /(2 \sqrt{3}), 0), respectively. Another point particle of mass mm and charge qq is constrained to move in the z=0z=0 plane and suffers Coulomb repulsion from each fixed charge. Neglecting any magnetic fields,

(i) Find the position of an equilibrium point.

(ii) By finding the form of the electric potential near this point, deduce that the equilibrium is stable.

(iii) Consider small displacements of the point particle from the equilibrium point. By resolving forces in the directions (1,0,0)(1,0,0) and (0,1,0)(0,1,0), show that the frequency of oscillation is

ω=Aqmϵ0a3,\omega=A \frac{|q|}{\sqrt{m \epsilon_{0} a^{3}}},

where AA is a constant which you should find.

[You may assume that if two identical charges qq are separated by a distance dd then the repulsive Coulomb force experienced by each of the charges is q2/(4πϵ0d2)q^{2} /\left(4 \pi \epsilon_{0} d^{2}\right), where ϵ0\epsilon_{0} is a constant.]

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