On the shape of magnetic and electric fields in an electromagnetic wave Electromagnetic waves are generally depicted like this:

Where the electric fields and magnetic fields exist in the planes perpendicular to the direction of propagation. I also realize that as the electric field changes while the wave is propagating, a magnetic field is induced and vice versa (by faraday's and maxwell's laws of induction). But, those laws predict that the fields will be circular. So, won't the electric and magnetic fields look different? Won't they be circles along arrows that are drawn in the figure? I haven't seen anything written about this anywhere.
 A: You write the integral formulation of Faraday's law, but there is also the equivalent differential formalism:
$$
\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}\tag{1}
$$
which can be proven in a straight-forward manner (and ought to be done in your standard E&M textbooks).
Using standard planar waves equations,
$$
E_y=E_0\sin\left(kx-\omega t\right) \\
B_z=B_0\sin\left(kx-\omega t\right), \\
$$
then Equation (1) gives us that
$$
kE_0\cos(kx-\omega t)=\omega B_0\cos(kx-\omega t)
$$
Which enforces the well-known relation that $E_0=cB_0$. So clearly plane waves do satisfy Faraday's law.
A similar situation exists with Ampere's law, usually written as
$$
\oint\mathbf B\cdot d\mathbf l=\frac{1}{c^2}\frac{d\phi_E}{dt}
$$
which leads to
$$
\nabla\times\mathbf B=\frac{1}{c^2}\frac{\partial\mathbf E}{\partial t}
$$
(I'm ignoring the current density here).
A: You shouldn't think of electromagnetic wave as the simple figure depicted as an arrow and two planes. That's too simplified illustration and surely misleading in your case. 
Electromagnetic waves fill certain 3D space. It's actually hard to accurately visualize it with a picture. In mathematical and a bit more rigorous language, at every point (x, y, z) in the space where electromagnetic waves exist, there's electric and magnetic field with value E(x, y, z) and H(x, y, z). Also they vary with time, so in the end E(x, y, z, t) and H(x, y, z, t) represent the wave. These two functions satisfy Maxwell's equations. 
As for the familiar picture of electromagnetic wave, it's trying to describe a plane wave in the whole space (let's say propagating in x direction), which means fields are varying in sinusoidal and cosinunoidal way with x, but invariant with y and z. In this sense, it allows to use the oversimplified arrow-plane illustration. 
