A particle of mass m m m and charge q > 0 q>0 q > 0 moves in a time-dependent magnetic field B = ( 0 , 0 , B z ( t ) ) \mathbf{B}=\left(0,0, B_{z}(t)\right) B = ( 0 , 0 , B z ( t ) ) .
Write down the equations of motion governing the particle's x , y x, y x , y and z z z coordinates.
Show that the speed of the particle in the ( x , y ) (x, y) ( x , y ) plane, V = x ˙ 2 + y ˙ 2 V=\sqrt{\dot{x}^{2}+\dot{y}^{2}} V = x ˙ 2 + y ˙ 2 , is a constant.
Show that the general solution of the equations of motion is
x ( t ) = x 0 + V ∫ 0 t d t ′ cos ( − ∫ 0 t ′ d t ′ ′ q B z ( t ′ ′ ) m + ϕ ) y ( t ) = y 0 + V ∫ 0 t d t ′ sin ( − ∫ 0 t ′ d t ′ ′ q B z ( t ′ ′ ) m + ϕ ) z ( t ) = z 0 + v z t \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} x ( t ) = x 0 + V ∫ 0 t d t ′ cos ( − ∫ 0 t ′ d t ′ ′ q m B z ( t ′ ′ ) + ϕ ) y ( t ) = y 0 + V ∫ 0 t d t ′ sin ( − ∫ 0 t ′ d t ′ ′ q m B z ( t ′ ′ ) + ϕ ) z ( t ) = z 0 + v z t
and interpret each of the six constants of integration, x 0 , y 0 , z 0 , v z , V x_{0}, y_{0}, z_{0}, v_{z}, V x 0 , y 0 , z 0 , v z , V and ϕ \phi ϕ . [Hint: Solve the equations for the particle's velocity in cylindrical polars.]
Let B z ( t ) = β t B_{z}(t)=\beta t B z ( t ) = β t , where β \beta β is a positive constant. Assuming that x 0 = y 0 = z 0 = x_{0}=y_{0}=z_{0}= x 0 = y 0 = z 0 = v z = ϕ = 0 v_{z}=\phi=0 v z = ϕ = 0 and V = 1 V=1 V = 1 , calculate the position of the particle in the limit t → ∞ t \rightarrow \infty t → ∞ (you may assume this limit exists). [Hint: You may use the results ∫ 0 ∞ d x cos ( x 2 ) = ∫ 0 ∞ d x sin ( x 2 ) = \int_{0}^{\infty} d x \cos \left(x^{2}\right)=\int_{0}^{\infty} d x \sin \left(x^{2}\right)= ∫ 0 ∞ d x cos ( x 2 ) = ∫ 0 ∞ d x sin ( x 2 ) = π / 8 . ] \sqrt{\pi / 8} .] π / 8 . ]