Electrodynamics - Finding Surface Charge

[metgt4]

Homework Statement

Consider a very thin flat plate positioned in the x-z plane. The plate is semi-infinite with an edge running along the z axis. The plate is held at potential V₀. Using techniques from teh theory of complex variables a solution is determined to be:

V(x,y) = A/(2)^1/2[(x² + y²)^1/2 - x]^1/2 + V₀

Where A is a constant.

Evaluate an expression for the y component of the electric field. Then carefully take limits for x > 0 and y approaching zero to show that the surface charge on the plate is given by:

s(x) = -A(epsilon₀)/(x^1/2)

Homework Equations

The Attempt at a Solution

I've worked out a few other parts of this problem, but this part has me stumped. I take the derivative of V(x,y) with respect to y and that gives you the electric field for varying y, right? Then you would use Gauss' law, right? I can't seem to work it out though. My work is scanned and attached to this message.

Thanks!
Andrew

 

[gabbagabbahey]

I take the derivative of V(x,y) with respect to y and that gives you the electric field for varying y, right?

No, the electric field is given by the negative gradient of the potential;

\textbf{E}=-\mathbf{\nabla}V=-\frac{\partial V}{\partial x}\hat{\mathbf{i}}-\frac{\partial V}{\partial y}\hat{\mathbf{j}}-\frac{\partial V}{\partial z}\hat{\mathbf{k}}

As for determining the surface charge, your text should tell you the boundary conditions on the electric field crossing a surface charge, use those.