Answer Check: Displacement Currents and Capacitors

[TFM]

Homework Statement

A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap in the wire, of width w << a, forms a parallel-plate capacitor, as shown in the figure above. Find the magnetic field in the gap, at a distance s < a from the axis.

{Figure given below}

Homework Equations

$$displacement current, J_d = \epsilon_0 \frac{\partial E}{\partial t}$$ --- (1)

$$\frac{\partial E}{\partial t} = \frac{1}{\epsilon_0 A}I$$ --- (2)

$$B(r) = \frac{\mu_0 I}{2 \pi r}$$ --- (3)

The Attempt at a Solution

Okay so firstly, I have put together (1) and (2) to get:


$$J_d = \epsilon_0 \frac{1}{\epsilon_0 A}I$$

I got this to cancel down into:

$$J_d = \frac{I}{A}$$

I then made a very bold assumption that the current in B(r) = the displacement current J_d

So I Inserted the values and got:

$$B(r) = \frac{\mu_0 \left( \frac{I}{A}\right)}{2 \pi r}$$

This seems very quick and straight forwards, though...

Does this look correct?

TFM

 

[TFM]

Does this look right?

 

[thomate1]

Your solution looks mostly correct, but there are a few things to consider. First, in your equation (2), you have used the constant current I instead of the time derivative of the current, which is what is needed for the displacement current term. So the correct equation should be:

\frac{\partial E}{\partial t} = \frac{1}{\epsilon_0 A}\frac{\partial I}{\partial t}

Next, when you cancel out the \epsilon_0 terms in (1), you should also take into account the fact that the electric field E is not constant throughout the gap, but rather varies with distance from the axis. So the correct equation for the displacement current would be:

J_d = \epsilon_0 \frac{\partial E}{\partial t} = \frac{1}{A}\frac{\partial I}{\partial t}

Then, as you correctly noted, the displacement current is equal to the current in the magnetic field equation (3), so we can write:

B(r) = \frac{\mu_0}{2 \pi r}\frac{\partial I}{\partial t}

This means that the magnetic field in the gap will vary with time, as the current is changing in time. To find the magnetic field at a specific distance s from the axis, you can use the equation:

B(s) = \frac{\mu_0}{2 \pi s}\frac{\partial I}{\partial t}

Overall, your approach is correct, but make sure to include all the necessary variables and consider the time dependence of the electric field and current.