Faraday's law and a uniform magnetic field

[ehrenfest]

[SOLVED] faraday's law

Homework Statement

A uniform magnetic field $\mathbf{B}(t)$ in the z-direction, fills a circular region in the x-y plane. If B is changing with time, what is the direction of $\mathbf{E}$/

Homework Equations

The Attempt at a Solution

My book says it is circumferential, just like the magnetic field inside a long straight wire carrying a uniform current density.

Apparently they are using the analogy between Faraday's Law and Ampere's Law. But I do not see the logic at all.

This is Griffiths Example 7.7.

 

[Ben Niehoff]

They are mathematically very similar. Faraday's law:

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

And Ampere's Law (for electrostatics):

$$\nabla \times \vec B = \mu_0 \vec J$$

 

[ehrenfest]

I know, but why does that imply that the E-field is circumferential?

 

[Ben Niehoff]

Because $\partial \vec B / \partial t$ is vertical.