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The problem: A cylindrical thin shell of electric charge has length $l$ and radius $a$, where $l \gg a$. The surface charge density on the shell is $\sigma$. The shell rotates about its axis with angular velocity $\omega = kt$, where $k$ is a constant and $t \geq 0$. (The problem can be found here.)

I am then asked to find the magnetic field inside the cylinder. Of course we need to find the current distribution and use Ampére's law, and I have no trouble doing that and obtaining the same result as in the above link. However, since this is not a magnetostatic situation (the surface current density $K = \sigma akt$ is time-dependent), Ampére's law is not a priori valid without displacement current. How does one argue that, in fact, the electric field is static? Of course, the charge distribution is static, and from this it follows that the time derivative of the electric field is divergenceless, but that doesn't seem to be enough.

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The electric field inside the cylinder won't be exactly static, because the charges are accelerating in the direction perpendicular to their radius vectors and hence they will produce induced electric field whose lines of force will turn in circles inside the cylinder, the highest field being near the cylinder wall.

So the Ampere law is not really applicable, because there will be some displacement current which it neglects.

However, a good estimate of the magnetic field, even when there is some displacement current present, is usually possible to obtain by using the Biot-Savart law. For cylinder with surface current, it turns out that the magnetic field inside calculated that way is uniform and along the axis of symmetry.

Once you obtain that estimate of magnetic field value as a function of time, you can estimate the electric field inside the cylinder as well, from the Maxwell equation

$$ \oint \mathbf E\cdot d\mathbf s = -\frac{d\Phi_B}{dt}. $$

Then you can calculate flux of displacement current through any rectangle passing through the cylinder and apply the Maxwell-Ampere law to determine the correction to magnetic field at every distance from the axis.

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  • $\begingroup$ I have done this and obtained the same result as in the solution I linked to above. However, the electric field I find using the Maxwell-Faraday equation is static, so it seems like there would be no displacement current, and thus no correction when using the Maxwell-Ampère law. I'm not quite sure what to make of that. Aren't all of Maxwell's equations satisfied then? Furthermore, if I had instead found a dynamic electric field, would the correction from Faraday's law give me the exact magnetic field? It doesn't seem like it would in general, but I just want to be explicit. $\endgroup$
    – Danny
    Commented Jun 15, 2018 at 9:49
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    $\begingroup$ I am not able to read text on that page, the page rejects showing it to me. I think you are right, if the angular velocity is not too high, the shape of the cylinder won't change much, the tangential acceleration of the charges will be constant and thus the electric field inside will be constant and there is no correction to the Biot-Savart result. Eventually, however, the cylinder/charge distribution will expand to higher radius due to centrifugal effect and that will probably cause electric field inside to decrease. $\endgroup$
    – Ján Lalinský
    Commented Jun 15, 2018 at 18:30
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I have a naive explanation, isn't the potential equation $\phi(\vec{x},t)=k \int \frac{\rho(\vec{x},t)}{r}d^3x$ enough to conclude the potential thus the gradient of the potential (Electric field) is static for a static charge distribution (Here the Vector potential is a linear function of t so its time derivative is const)?

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