Find induced E field inside a disk in an uniform magnetic field.

[yungman]

Homework Statement

Uniform time varying magnetic field $\vec B_{(t)}$ pointing at z direction, filling up a circular region on xy-plane. Find the induced E field.

I tried two different ways and get two different answers. Please tell me what did I do wrong.

Homework Equations

$$Emf =-\frac{d\Phi}{dt}$$

$$\nabla\times \vec E = -\frac{\partial \vec B}{\partial t}$$

The Attempt at a Solution

We know $$\vec E = \hat {\phi} E$$

1) Using

$$Emf =-\frac{d\Phi}{dt}$$

$$\Phi=\int_{S'} \vec B_{(t)} \cdot d S' = \pi s^2 B_{(t)} \;\hbox { where s is the radius of the circle and B is uniform. }$$

$$\frac {d \Phi}{dt} = \pi s^2 \frac{\partial \vec B}{\partial t} \;\rightarrow\; Emf = \int_C \vec E \cdot d\vec l = 2\pi s E = -\pi s^2 \frac{\partial \vec B}{\partial t} \; \Rightarrow \; \vec E = -\hat {\phi} \frac s 2 \frac{\partial \vec B}{\partial t}$$

The above is the same as in the book.




2) This one I use the fact of uniform B and $\vec E = \hat \phi E$

$$\nabla \times \vec E = \frac 1 r \left |\begin {array}{ccc} \hat r & r\hat {\phi} & \hat z \\ \frac {\partial }{\partial r} & \frac {\partial }{\partial \phi} & \frac {\partial }{\partial z}\\ 0 & rE_{\phi} & 0 \end {array}\right |_{r=s} = \hat z \frac{E_{\phi_{(t)}}}{s} =-\frac {\partial \vec B}{\partial t} \;\Rightarrow \; \vec E = -\hat {\phi} s \frac{\partial \vec B}{\partial t}$$




As you can see, the two methods differ by 1/2! What did I do wrong?

 

[yungman]

Anyone? I don't see how there is a difference of 1/2!

 

[qbert]

E depends on the r - coordinate.
So you calculated the curl wrong.

 

[yungman]

E depends on the r - coordinate.
So you calculated the curl wrong.

Can you show where I did wrong?

Thanks

 

[stevenb]

Can you show where I did wrong?

Thanks

Your second method does not directly give you the solution right away. Instead it gives you a differential equation to solve. I won't go through the process of solving the differential equation, but will just point out that the first solution you found clearly is a solution for your differential equation.

So, the first step is to calculate the differential equation from the curl equation. You get the following.

$${{1}\over{r}}{{d}\over{dr}}(rE_\phi)=-{{\partial B}\over{\partial t}}$$

This then leads to the following.

$${{dE_\phi}\over{dr}}+{{1}\over{r}}E_\phi=-{{\partial B}\over{\partial t}}$$

Now, you can solve this differential equation, but since we already know the answer, we need only verify that solution. Do it out and you will see that it does work out.

Uniform time varying magnetic field pointing at z direction, filling up a circular region on xy-plane.

Now, there is another aspect to the question. Strictly, the problem says that the region of constant flux change is restricted to a circular area. Hence, the solution requires that we center the origin in the middle of the circle and express the full solution as follows.

$$E_\phi={{-r}\over{2}}{{\partial B}\over{\partial t}}$$ if $$r<R$$

and

$$E_\phi={{-R^2}\over{2r}}{{\partial B}\over{\partial t}}$$ if $$r>R$$

where R is the radius of the region of flux.

Notice that even this second solution outside the region of flux (i.e. the region has dB/dt=0) also obeys your differential equation. $${{dE_\phi}\over{dr}}+{{1}\over{r}}E_\phi=0$$

 

[yungman]

Your second method does not directly give you the solution right away. Instead it gives you a differential equation to solve. I won't go through the process of solving the differential equation, but will just point out that the first solution you found clearly is a solution for your differential equation.

So, the first step is to calculate the differential equation from the curl equation. You get the following.

$${{1}\over{r}}{{d}\over{dr}}(rE_\phi)=-{{\partial B}\over{\partial t}}$$

This then leads to the following.

$${{dE_\phi}\over{dr}}+{{1}\over{r}}E_\phi=-{{\partial B}\over{\partial t}}$$

Now, you can solve this differential equation, but since we already know the answer, we need only verify that solution. Do it out and you will see that it does work out.



Now, there is another aspect to the question. Strictly, the problem says that the region of constant flux change is restricted to a circular area. Hence, the solution requires that we center the origin in the middle of the circle and express the full solution as follows.

$$E_\phi={{-r}\over{2}}{{\partial B}\over{\partial t}}$$ if $$r<R$$

and

$$E_\phi={{-R^2}\over{2r}}{{\partial B}\over{\partial t}}$$ if $$r>R$$

where R is the radius of the region of flux.

Notice that even this second solution outside the region of flux (i.e. the region has dB/dt=0) also obeys your differential equation. $${{dE_\phi}\over{dr}}+{{1}\over{r}}E_\phi=0$$

Thanks

Got it.