Dynamics | Part IA, 2005

A particle of mass mm and charge qq moving in a vacuum through a magnetic field B\mathbf{B} and subject to no other forces obeys

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

where r(t)\mathbf{r}(t) is the location of the particle.

For B=(0,0,B)\mathbf{B}=(0,0, B) with constant BB, and using cylindrical polar coordinates r=\mathbf{r}= (r,θ,z)(r, \theta, z), or otherwise, determine the motion of the particle in the z=0z=0 plane if its initial speed is u0u_{0} with z˙=0\dot{z}=0. [Hint: Choose the origin so that r˙=0\dot{r}=0 and r¨=0\ddot{r}=0 at t=0t=0.]

Due to a leak, a small amount of gas enters the system, causing the particle to experience a drag force D=μr˙\mathbf{D}=-\mu \dot{\mathbf{r}}, where μqB\mu \ll q B. Write down the new governing equations and show that the speed of the particle decays exponentially. Sketch the path followed by the particle. [Hint: Consider the equations for the velocity in Cartesian coordinates; you need not apply any initial conditions.]

Typos? Please submit corrections to this page on GitHub.