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## Homework Statement

Uniform time varying magnetic field [itex]\vec B_{(t)}[/itex] 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

[tex]Emf =-\frac{d\Phi}{dt}[/tex]

[tex]\nabla\times \vec E = -\frac{\partial \vec B}{\partial t}[/tex]

## The Attempt at a Solution

We know [tex]\vec E = \hat {\phi} E [/tex]

1) Using

[tex]Emf =-\frac{d\Phi}{dt}[/tex]

[tex]\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. }[/tex]

[tex] \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} [/tex]

The above is the same as in the book.

2) This one I use the fact of uniform

**B**and [itex] \vec E = \hat \phi E [/itex]

[tex] \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} [/tex]

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

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