Plasma radius time variation in a uniform magnetic field

Question

In this problem we consider a cylindrical plasma of radius $a_0$ in a uniform, time-dependent magnetic field $\overrightarrow{B(t)}=B_z(t)\cdot\overrightarrow{e_z}$.

From Faraday's law $\nabla\times \overrightarrow{E}=-\frac{\partial \overrightarrow{B_z}}{\partial t}$ , it follows that the temporally varying magnetic field induces an azimuthal electric field given by : $$\overrightarrow{E}= \frac{-r}{2}\cdot\frac{\partial \overrightarrow{B_z}}{\partial t}\cdot\overrightarrow{e_{\theta}}$$

Given that both $B > 0$ , and $\frac{\partial \overrightarrow{B_z}}{\partial t}$ $> 0$

We found the plasma drift velocity expression : $$V_{E\times B} = \frac{-r}{2B_z}\cdot\frac{\partial \overrightarrow{B_z}}{\partial t}\cdot\overrightarrow{e_r}$$

Question :

Using the expression of drift velocity $V_{E\times B}$ how can we determine the expression for time variation of plasma radius $a(t)$ if at $a(t=0)=a_0$ ?

I am open to any hints!

Answer 1

Taking r = a and $V_{E \times B} = {d a \over d t}$ we have the differential equation \begin{equation} {d a \over d t} = - {a \over 2 B_z} {d B_z \over d t} \implies {d \log a \over d t} = - {1 \over 2} {d \log B_z \over d t} \implies \log a(t) = - {\log B_z(t) \over 2} + c \end{equation} To fix the constant $c$ we use $a(0) = a_0$ to find \begin{equation} c = \log a_0 + {\log B_z(0) \over 2} \end{equation} so we get \begin{equation} a(t) = a_0 \sqrt{B_z(0) \over B_z(t)}. \end{equation}