Dynamics | Part IA, 2004

A particle of mass mm and charge qq moves through a magnetic field B\mathbf{B}. There is no electric field or external force so that the particle obeys

mr¨=qr˙×B,m \ddot{\mathbf{r}}=q \dot{\mathbf{r}} \times \mathbf{B},

where r\mathbf{r} is the location of the particle. Prove that the kinetic energy of the particle is preserved.

Consider an axisymmetric magnetic field described by B=(0,0,B(r))\mathbf{B}=(0,0, B(r)) in cylindrical polar coordinates r=(r,θ,z)\mathbf{r}=(r, \theta, z). Determine the angular velocity of a circular orbit centred on r=0\mathbf{r}=\mathbf{0}.

For a general orbit when B(r)=B0/rB(r)=B_{0} / r, show that the angular momentum about the zz-axis varies as L=L0qB0(rr0)L=L_{0}-q B_{0}\left(r-r_{0}\right), where L0L_{0} is the angular momentum at radius r0r_{0}. Determine and sketch the relationship between r˙2\dot{r}^{2} and rr. [Hint: Use conservation of energy.] What is the escape velocity for the particle?

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