- #1
azupol
- 17
- 0
Homework Statement
In class we were given the classic image problem of a point charge on the z-axis (at z=d) above an infinite grounded conducting plane, the xy plane in this case. We found the potential by getting rid of the plane and placing an image charge at a distance z= -d. Now the question is, how do we show that this potential obeys Poisson's equation in the region of interest, in this case z>0
Homework Equations
Poisson's Equation:
And the potential was found to be: V(x,y,z)=1/4piε [q/(x2 + y2 + (z-d)2 )1/2 ] -q/(x2 + y2 + (z+d)2 )1/2 ]Induced surface charge density (i.e. the induced charge in the conducting plane) is -qd/2pi(x2+y2+d2)3/2
The Attempt at a Solution
I attempted to twice differentiate the potential, and tried to make it equal to the induced surface charge density, to check that after simplification they were both equal, but got stuck at the second partial derivative of V with respect to z. There must be a better way to solve this.
We are using Griffith's Introduction to Electrodynamics, and our prof suggested we reference the section on the Dirac delta function. I'm failing to see how the Dirac Delta would help in this case