Dynamics | Part IA, 2002

An electron of mass mm moving with velocity x˙\dot{\mathbf{x}} in the vicinity of the North Pole experiences a force

F=ax˙×xx3,\mathbf{F}=a \dot{\mathbf{x}} \times \frac{\mathbf{x}}{|\mathbf{x}|^{3}},

where aa is a constant and the position vector x\mathbf{x} of the particle is with respect to an origin located at the North Pole. Write down the equation of motion of the electron, neglecting gravity. By taking the dot product of the equation with x˙\dot{x} show that the speed of the electron is constant. By taking the cross product of the equation with x\mathbf{x} show that

mx×x˙axx=Lm \mathbf{x} \times \dot{\mathbf{x}}-a \frac{\mathbf{x}}{|\mathbf{x}|}=\mathbf{L}

where L\mathbf{L} is a constant vector. By taking the dot product of this equation with x\mathbf{x}, show that the electron moves on a cone centred on the North Pole.

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