Electric field caused by magnetic field

Question

Does electric field caused by time varying magnetic field form closed loops(electric field starts from a positive charge and ends at a negative charge)? and are they conservative or non-conservative fields?

Answer 1

The field lines certainly can form closed loops, for example in a transformer. But they don't have to either, an example being a plane EM wave.

For a specific example, consider the following long solenoid of radius $a$ and turn density $n$ with a time-varying current running through it: $I(t)$.

The solenoid is oriented along the x-axis, and current runs in the direction of $-x$ to $+x$. This current induces a magnetic field inside the solenoid:

$$\mathbf{B}=\mu_0 nI(t) \mathbf{i}$$

Now, by Faraday's Law:

$$\int \mathbf{E} \cdot \mathrm{d} \mathbf{s}=-\frac{\partial}{\partial t} \int \mathbf{B} \cdot \mathrm{d} \mathbf{A}$$

Which gives us a relationship the time derivative of magnetic flux through a loop and the electric field along that loop. Using a loop that is concentric with the solenoid gives:

$$2\pi r E=-\frac{\partial}{\partial t} \int \mu_0 n I(t) \mathrm{d}A=-\mu_0 n I'(t) \pi a^2$$

$$\Rightarrow E=-\frac{1}{2}\mu_0 n a^2~ \frac{I'(t)}{r}$$

where $I'(t)=\partial I / \partial t$, and we know $E$ is oriented along a closed loop.