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I have a stationary solenoid of radius $a$ and length $L$ with $n$ windings per unit length. There is a time varying current in the wire $I = kt$, with $k$ a constant. A conducting wire with radius $r$, mass $m$ and resistance $R$ is placed around the wire centred around the solenoid and is free to move.

I am at first interested in the induced electric field. However I am a bit confused.

I know that $$\oint_C \mathbf{E} \cdot d \mathbf{l} = - \frac{d \phi}{dt}$$

For my integral, we are used to choosing circular path, but I am unsure about what radius to choose for this? Shall I choose a radius $r$ or radius $a$? Similarly, what radius do I choose to use for the area that comes into play with the $\phi$?

And what is the difference between the cases $r \geq a$ and $r<a$?

Thanks :)

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Inside the solenoid:

The field is $$B(t)=\mu_0ni(t)$$

So for $r<a$ the flux is $$\phi_B=\mu_0ni(t)*\pi r^2$$

$$\oint E\,ds=\frac{d\phi_B}{dt}$$ $$E2\pi r=\mu_0nk\pi r^2$$ $$E=\frac{\mu_0nkr}{2}$$

Outside the solenoid:

$$B=0$$ So the magnetic flux is only due to the field inside the solenoid: $$\phi_B=\mu_0ni(t)*\pi a^2$$

$$\oint E\,ds=\frac{d\phi_B}{dt}$$ $$E2\pi r=\mu_0nk\pi a^2$$ $$E=\frac{\mu_0nk a^2}{2r}$$

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