Paper 4, Section II, A

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

(b) For $B=0$, three point particles, each of charge $q$, are fixed at $(0, a / \sqrt{3}, 0)$, $(a / 2,-a /(2 \sqrt{3}), 0)$ and $(-a / 2,-a /(2 \sqrt{3}), 0)$, respectively. Another point particle of mass $m$ and charge $q$ is constrained to move in the $z=0$ 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 $(1,0,0)$ and $(0,1,0)$, show that the frequency of oscillation is

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

where $A$ is a constant which you should find.

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

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