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

Question

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

Answer 1

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).