Electric field due to changing uniform magnetic field

Question

Suppose in an infinite region of space, there is a uniform magnetic field which changes at a rate $\frac{d\phi}{dt}$. I want to find the induced electric field at a certain point so, I draw an imaginary circular loop passing through that point. Since the loop and surroundings are similar from all directions the field must be similar at all points of loop too, then from Faraday's law $$\int{\vec{E}\cdot d\vec{\ell}} = -\frac{d\phi}{dt}.$$ I will then take out $\vec{E}$ because its same for all ${\rm d}\vec{\ell}$ and integrate ${\rm d}\vec{\ell}$ which will be $2\pi r$. Now the problem is, I could've taken any loop, therefore there could've been different radii, and electric field at a point cannot assume multiple values simultaneously, so what's wrong.

I think what I'm taking out common, $|\vec{E}\,|$ is actually a component of field along tangent to the loop , but still I don't know how the produced electric fields will look like. Can someone please help with these two?

Answer 1

You are assuming there is single unique electric field to be found, determined by the assumption of uniform magnetic field changing in time.

This is not true. As you can see from the equation relating magnetic field and curl of electric field, there can be infinity of electric fields obeying the same equation. Two possible solutions differ by a vector field that has zero curl.

The electric field could be determined if some other conditions were imposed on the EM field in addition to the knowledge of magnetic field. In practice, boundary conditions are sometimes apparent. For example, one could study induced electric field near two cylindrical poles of an electromagnet opposing each other. In that case the boundary condition would be that electric and magnetic field at infinity is zero and field on the poles would have values that copy symmetry of the poles. The induced electric field would have rotational symmetry (rotation about the axis of the system), so it would have circular lines of force. With this assumption, you could then use the Faraday law to estimate strength of the electric field on any such line of force.

In simpler words, the electric field is determined by the electric charge and current distribution in the body producing the field. In the above case, it is the wires and ferrite core who create the induced electric field, so the field has kind of similar shape.