Chapter 3.4, Problem 3.38P

Here’s an alternative derivation of Eq. 3.10 (the surface charge density induced on a grounded conducted plane by a point charge q a distance d abovethe plane). This approach¹⁹ (which generalizes to many other problems) does notrely on the method of images. The total field is due in pan to q, and in pan to theinduced surface charge. Write down the z components of these fields—in terms ofq and the as-yet-unknown $\sigma \left(x,y\right)$ − just below the surface. The sum must be zero, of course, because this is inside a conductor. Use that to determine $\sigma$ .