# Inductance of loop to a square wire? What is the induced current and the induced $\bar E$?

I am practising for the elementary-physics -course Tfy-0.1064, this puzzle is a simplified version of the problem 3A here (sorry not in English). My basic idea here is to understand a scenario with one loop and then understand the solenoid case.

Suppose a circle around the square of 20-cm-side and $\frac{dB}{dt}=35\frac{mT}{s}$ in the circle, please, see the picture below. What is the induced current and the induced $\bar E$ to the square by the circle?

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Calculations

We are going to use Biot-Savart

$$\Delta \bar B = \frac{\mu_0}{4\pi} \frac{I \Delta\bar l\times \hat r}{r^2}$$.

I have used books such as "Matter and Interactions 3rd Edition" by Chabay (particularly pages around 721) and "Introduction to Electrodynamics" by Griffiths (particularly pages around 78 and around 322).

3A

What is the induced EMF?

$\frac{dB}{dt}=35 mT/s$

Assume $\frac{dB}{dt}:=\frac{\partial B}{dt}$. Now by Faraday's law $\nabla\times \bar E = -\frac{\partial\bar B}{\partial t}$ and now by Stokes -theorem:

$$emf=\oint\bar E\cdot d\bar l = -\int\int_A \frac{\partial\bar B}{dt}\cdot d\bar A \\=-\frac{\partial\bar B}{dt} 20^2$$

so $$emf =-\frac{\partial\bar B}{dt} 20^2 (cm)^2=-(20cm)^2 35 mT/s$$

Griffiths about the page 322 covers this issue, reading... history with something about the magnetic field and Biot-Savart but not apparently needed for this question.

TODO 1: how can I get the electric field?

TODO 2: how can I get the current?

Other questions in the attachment

3B: potential between corners assuming static-field -approximation

$$V(\bar r)=\oint\bar E\cdot d\bar l$$ where $\theta\in[0,\pi/2]$ and, since potential does not depend on the path, you can choose any path like along the circle the easy one. Use the solution of $\bar E$ from the 3A.

3C: "Dynamic field cannot precisely be approximated with static potential, why?"

Because changing magnetic field have different characteristics to static fields such as the inductance. (Not sure, perhaps some better explanation exist).

Conclusion

Todo.

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...perhaps I could use $\bar S=\bar E\times\bar B$, somehow?! – hhh Sep 5 '12 at 1:20
Griffitts "Introduction to electrodynamics" p.307 has an equation $\oint\bar E\cdot d\bar{l}=\frac{-d\phi}{dt}=-\pi a^2\frac{dB}{dt}$ in some situation, I need to understand the flux better here...reading. Faraday -law that $emf=\frac{d\phi}{dt}$?! – hhh Sep 5 '12 at 1:30
...Faraday law is according to my teacher easier way of doing this...thinking.. – hhh Sep 5 '12 at 9:36