# Electric field due to changing uniform magnetic field

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?