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I was wondering about the graph of the emf around a changing magnetic field for all radii of a closed loop centered at the changing magnetic field.

Here is what I think.

This is Faraday's law:

$$\textrm{emf} = -~\frac{\mathrm d\phi_B}{\mathrm dt}$$

And if we assume that the magnetic field that is changing has a radius of R, then it would become:

$$\textrm{emf} = -~A \frac{\mathrm dB}{\mathrm dt} = -~\pi R^2\frac{\mathrm dB}{\mathrm dt}$$

Now, if we were inside the magnetic field with a radius r < R, then the equation would be:

$$\textrm{emf} = -~ \pi r^2 \frac{\mathrm dB}{\mathrm dt}$$ which means that $\textrm{emf} \propto r^2$

But when we are outside the magnetic field with a radius r > R, then the equation would be:

$\textrm{emf} = -~ \pi R^2 \frac{\mathrm dB}{\mathrm dt}$ which means that Emf is constant for all r > R. Is this true? If I have a closed loop with a super super long radius, will there still be a potential on that loop? The equation $V = Ed$ certainly agrees, but that doesn't make sense to me? Are there any limits to Faraday's Law? Any help is much appreciated!

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In theory there are no limits although how you contain the magnetic field is another matter.
As you make the loop bigger the magnetic field lines emanating from your device producing the magnetic field will be looping back to the other end so the overall change in magnetic flux through your loop will decrease.

In practice you will have lots of other varying magnetic and electric field "in the air" (background noise) which will make it more difficult to observe the effect of your changing magnetic field.

Just connect a loop of wire to the input of a CRO, turn up the gain and observe what you get. A 50 or 60 Hz signal will be the result of mains supply, a Mhz signal will probably radio signals, spikes will be from switches being turned on or off or motors running, etc.

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