Dynamics | Part IA, 2006

A particle of mass mm and charge q>0q>0 moves in a time-dependent magnetic field B=(0,0,Bz(t))\mathbf{B}=\left(0,0, B_{z}(t)\right).

Write down the equations of motion governing the particle's x,yx, y and zz coordinates.

Show that the speed of the particle in the (x,y)(x, y) plane, V=x˙2+y˙2V=\sqrt{\dot{x}^{2}+\dot{y}^{2}}, is a constant.

Show that the general solution of the equations of motion is

x(t)=x0+V0tdtcos(0tdtqBz(t)m+ϕ)y(t)=y0+V0tdtsin(0tdtqBz(t)m+ϕ)z(t)=z0+vzt\begin{aligned} &x(t)=x_{0}+V \int_{0}^{t} d t^{\prime} \cos \left(-\int_{0}^{t^{\prime}} d t^{\prime \prime} q \frac{B_{z}\left(t^{\prime \prime}\right)}{m}+\phi\right) \\ &y(t)=y_{0}+V \int_{0}^{t} d t^{\prime} \sin \left(-\int_{0}^{t^{\prime}} d t^{\prime \prime} q \frac{B_{z}\left(t^{\prime \prime}\right)}{m}+\phi\right) \\ &z(t)=z_{0}+v_{z} t \end{aligned}

and interpret each of the six constants of integration, x0,y0,z0,vz,Vx_{0}, y_{0}, z_{0}, v_{z}, V and ϕ\phi. [Hint: Solve the equations for the particle's velocity in cylindrical polars.]

Let Bz(t)=βtB_{z}(t)=\beta t, where β\beta is a positive constant. Assuming that x0=y0=z0=x_{0}=y_{0}=z_{0}= vz=ϕ=0v_{z}=\phi=0 and V=1V=1, calculate the position of the particle in the limit tt \rightarrow \infty (you may assume this limit exists). [Hint: You may use the results 0dxcos(x2)=0dxsin(x2)=\int_{0}^{\infty} d x \cos \left(x^{2}\right)=\int_{0}^{\infty} d x \sin \left(x^{2}\right)= π/8.]\sqrt{\pi / 8} .]

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