All Problems Constant Electric Field in a Vacuum
Problem 3.37
The potential of a certain electrostatic field has the form \(\varphi=a\left(x^{2}+y^{2}\right)+b z^{2},\) where \(a\) and \(b\) are constants. Find the magnitude and direction of the electric field strength vector. What shape have the equipotential surfaces in the following cases: (a) \(a>0, b>0\); (b) \(a>0, b<0 ?\)
\[ \text { 3.37. } \mathbf{E}=-2(a x \mathbf{i}+a y \mathbf{j}+b z \mathbf{k}), \quad E=2 \sqrt{a^{2}\left(x^{2}+y^{2}\right)+b^{2} z^{2}} \] (a) An ellipsoid of revolution with semiaxes \(\sqrt{\varphi / a}\) and \(\sqrt{\varphi / b}\). In the case of \(\varphi>0,\) a single-cavity hyperboloid of revolution; when \(\varphi=0,\) a right round cone; when \(\varphi<0,\) a two-cavity hyperboloid of revolution.