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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!

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!

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 : $V_{E\times B}$ = $\frac{-r}{2B_z}\cdot\frac{\partial \overrightarrow{B_z}}{\partial t}\cdot\overrightarrow{e_r}$

Now the question is as follows: 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!

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!

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 : $V_{E\times B}$ = $\frac{-r}{2B_z}\cdot\frac{\partial \overrightarrow{B_z}}{\partial t}\cdot\overrightarrow{e_r}$

Now the question is as follows: 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 $a(t=0)=a_0$

Am open to any hints!

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 : $V_{E\times B}$ = $\frac{-r}{2B_z}\cdot\frac{\partial \overrightarrow{B_z}}{\partial t}\cdot\overrightarrow{e_r}$

Now the question is as follows: 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!