The exact question goes like this:
In a certain electronic tube, electrons are emitted from a hot plane metal surface, and collected by a plane metal plane parallel to the emitter, at a distance $d$ away. (The distance $d$ is small compared with the lateral dimensions of the plates)
The electric potential between the plates is given by $\varphi =kx^{4/3} $ where $x$ is the distance from the emitter.
a) What is the surface charge density $\sigma $ on the emitter? On the collector?
b) What is the volume charge density $\rho(x) $ for $0 < x < d$?
Now I tried to do just do a Laplacian on the electric potential so that it would give me a charge density. But I'm confused.
By taking Laplacian on the electric potential, it would give me :
$$\nabla^{2}\varphi = -\frac{\rho}{\varepsilon_{0}}$$
where $\rho$ is charge density. How do I know if it's surface charge density, or volume charge density?