Cube with charges at each vertex

[archaic]

Homework Statement

Determine an expression for the magnitude of the electric field at the center of any face of the cube.

Relevant Equations

$$E=k\frac{q}{r^2}$$

For convenience, I take the center of the upper face.
The charges at the top cancel each other's effects, and those at the bottom cancel each other's horizontal effects, so I get$$E=4k\frac{q}{r^2}\sin\theta$$I have found that $\theta=\arcsin\left(\frac{s}{r}\right)=\arcsin{\sqrt\frac{2}{3}}$, with $r=\sqrt{\frac{3}{2}}s$:$$E=4\frac{2}{3}\sqrt\frac23k\frac{q}{s^2}=\frac{8\sqrt6}{9}k\frac{q}{s^2}$$
Correct, right?

 

[etotheipi]

The method looks right, but I think you have dropped a factor of $\sqrt{2}$ in your very final answer.

 

[archaic]

The method looks right, but I think you have dropped a factor of $\sqrt{2}$ in your very final answer.

should have been $\sqrt6$ instead of $\sqrt3$. thanks, corrected!