# Is the induced electric field due to time varying magnetic flux always circular?

According to Faraday's law, changing magnetic flux induces an electric field. Is that electric field always circular?

-
Circular in the sense of a circle -> no. Circular in the sense that the flux lines are closed -> yes. –  Fabian Dec 18 '12 at 22:56

Your question is like asking do charge particles always produce a spherically symmetric electric field? The answer is yes and no. A single electric charge, like an electron, will produce a spherically symmetric electric field, but if I have a collection of charged particles, then the shape of the resultant electric field will depend upon the geometry of the distribution of the charged particles.

Let us examine Faraday's law:

$$\nabla \times E = - \frac{\partial B}{\partial t}.$$

Using the Stoke's theorem we arrive at:

$$\int E . ds = - \frac{\partial \int\int B. dA}{\partial t}. = - \frac{\partial magnetic flux}{\partial t}$$

Let us suppose that we have cylindrical symmetric magnetic field heading from North to South and varying in intensity over time, thus giving a non-zero value for $\frac{\partial magnetic flux}{\partial t}$. Now if we choose the line integral to be a circle around this magnetic field, due to symmetry, $E$ must be the same around all points of this line, and thus we could say that this magnetic field produced a circular electric field.

However, this example is a non-realistic case of a cylindrically symmetric magnetic field. In reality, the distribution of the magnetic field won't be so symmetric, thus causing non circular electric fields (although still closed loops of electric fields).

-