What is the direction of the induced E field from a changing uniform magnetic field? If a stationary charge is placed in a changing uniform magnetic field to measure the induced E field at some point, what would be the direction of E?
I think E must equal zero normal to B to maintain the symmetry of the problem, leaving E entirely along B.
Is this correct?
 A: Faraday's law of induction - one of Maxwell's equations - says
$$\nabla \times E = - \frac{\partial B}{\partial t}.$$
So the time derivative of $B$ induces an electric field. However, what you incorrectly assumed was that the time derivative of $B$ will directly be proportional to the induced electric field. In reality, it is the "curl" of the electric field that is equal to the time derivative of $B$.
In other words, if $B$ and its time derivative is uniform in space, the field $E$ will not be uniform. For example, if $-\partial B / \partial t = (0,0,\beta)$ in the $z$-direction, the most natural $E$ will depend linearly on spatial coordinates, and may be e.g.
$$ E = (-y\beta/2, x\beta/2, 0) $$
This electric field is rotating around the $z$-axis - around the direction of the magnetic field (its change in time).
Note that it is constructed so that its curl is equal to $(0,0,\beta)$. However, this configuration is also nicely rotational symmetric because up to the extra factor of $\beta$, the vector $E$ is nothing else than the vector $r$ projected to the $xy$-plane and rotated by 90 degrees around the $z$-axis - and this definition doesn't depend on the choice of axes in the $xy$-plane, so the configuration is rotationally symmetric.
It is not translationally symmetric in the $x$ and $y$ directions, however. There exists a specific axis - in my case, the $z$-axis - where the induced $E$ vanishes. The solution to Maxwell's equations is not unique. I may add any other uniform $E$ field (or, more generally, any electromagnetic wave solution of Maxwell's equations with no right hand side) without spoiling the validity of the Maxwell's equations - this is a kind of trivial artifact of the linearity of these equations.
