The potential difference across the wings of a moving airplane due to the magnetic field of the Earth Let's assume that the airplane is nothing but its wings and that the two wings form a long thin solid metal box.
This would be the top view:

Assume that the velocity of the airplane does not change.
Assume the magnetic field of the Earth is constant and pointing to the ground as depicted in the picture.
For each electron of the wing we have:
$$\mathbf{F}_B
= (-e) \ \mathbf{v}\times \mathbf{B}
= evB\mathbf{\hat{x}}$$
This force will alter the charge distribution on the wing.
In the stationary state, this equation should hold for each electron in the wing:
$$\mathbf{F}_E
= -\mathbf{F}_B$$
Which implies: 
$$\mathbf{E}
= vB\mathbf{\hat{x}} \qquad  (\text{inside})$$
The magnetic field is assumed to be constant so $\nabla\times \mathbf{E}=0$. Thus we can write $\mathbf{E}=-\nabla V$.
But we know that every point of the wing must be at the same potential since it is a conductor.
This implies that the electric field inside the conductor must be zero (grad(constant)=0).
But this contradicts with what I just concluded: $\mathbf{E}
= vB\mathbf{\hat{x}} \qquad  (\text{inside})$
What is my mistake?
 A: 
But we know that every point of the wing must be at the same potential since it is a conductor.

That is often true if the conductor is in static state, but not always.
"Potential being the same everywhere in conductor" is the same state as "macroscopic electrostatic field being zero in conductor".
Electric field is zero throughout the body if the only macroscopic force acting on the charge carriers inside is due to electric field -- if the charged particles are to stay at rest, the force must be zero, hence electric field must be zero.
If there are other forces acting on the charged particles, like magnetic forces, then the condition of static state implies only that sum of electric and magnetic forces is zero. So it is possible that both electric and magnetic field inside are not zero and this is what happens inside the wing flying through magnetic field.
A: Electric field produced due to magnetic fields is different from electric fields in a purely electrostatic environment. The fact that electric field inside a conducting metallic conductor is zero corresponds to the latter case. Hence a conductor can have electric field inside it provided it is produced by magnetic fields.
