What is the electric field outside a cylindrical solenoid?

Question

What is the electric field outside a cylindrical solenoid when inside is turned on a magnetic field? The question is related to the question aharonov-bohm-effect-electricity-generation

Answer 1

The magnetic field inside of a cylindrical solenoid of radius $R$ is given by

$$ \textbf{B} = \mu_0 n I \hat{z}, $$

where n is the turn density in turns/m and I is the current. The field everywhere outside the solenoid is zero. Let's assume that the current $I(t)$ is linearly increasing, so

$$ I(t) = at . $$

For a cylinder of radius $r \ge R$, the flux through its center is

$$ \Phi(t) = \int_{S} \textbf{B} \cdot d\textbf{A} = (\mu_0 n at)(\pi R^2). $$

We can apply Faraday's law to find the electric field,

$$ \int_{\partial S} \textbf{E} \cdot d\textbf{r} = -\frac{d}{dt} \int_S \textbf{B} \cdot d\textbf{A}, $$

and use the cylindrical symmetry to assume that $\textbf{E}$ is constant along the circular boundary. Thus,

$$ \textbf{E}(r, t) = - \hat{\phi} \frac{1}{2 \pi r} \frac{d}{dt} (\mu_0 \pi R^2 n a t) = - \frac{\mu_0 na R^2}{2r} \hat{\phi}. $$

In this problem, all distances are measured from the center of the solenoid ($r = 0$). Unlike the magnetic field, the electric field is everywhere nonzero. For the more general problem, where $I(t)$ is an arbitrary function, the solution is

$$ \textbf{E}(r, t) = - \frac{\mu_0 n R^2}{2r} \frac{dI}{dt} \hat{\phi}. $$

Answer 2

For what it's worth, it is stated in http://arxiv.org/abs/1407.4826 and references therein in the context of the Aharonov-Bohm effect that even a constant-current solenoid has outside electric fields: "always there is an electric field outside stationary resistive conductor carrying constant current. In such ohmic conductor there are quasistatic surface charges that generate not only the electric field inside the wire driving the current, but also a static electric field outside it...These fields are well-known in electrical engineering." Sorry, I have not checked that, but it sounds plausible.

EDIT (07/25/2014) Seems there is a confirmation here: http://www.astrophysik.uni-kiel.de/~hhaertel/PUB/voltage_IRL.pdf , see, especially, Fig.4 therein.