A charged ellipsoid

Determining how the charges distribute on the surface of a conductor is, in general, a very difficult problem. We know that if we charge a conductor the charges go to the surface and redistribute so that the electric field in the conductor vanishes. One of the few shapes for which this distribution can be determined analytically is the ellipsoid. x2a2+y2b2+z2c2=1. {\frac{x^{2}}{a^{2}} +\frac{y^{2}}{b^{2}}+ \frac{z^{2}}{c^{2}}}=1. Here, a, b and c are the ellipsoid's semi-axes. One can prove that for an ellipsoidal conductor the surface charge density is given by

σ(x,y,z)=Q4πabc1x2a4+y2b4+z2c4 \sigma (x,y,z)=\frac{Q}{4\pi a b c}\frac{1}{\sqrt{ \frac{x^{2}}{a^{4}} +\frac{y^{2}}{b^{4}}+ \frac{z^{2}}{c^{4}}}} where Q is the the net charge of the conductor. Note that if we set a=b=c=Ra=b=c=R we obtain the uniform charge distribution Q4πR2\frac{Q}{4 \pi R^{2}}, corresponding to a spherical conductor. Suppose that we measure the electric field near the surface of a charged ellipsoid with Q=1nCQ=1nC, a=2cm,b=5cma=2 cm, b=5 cm and c=3cm c=3cm. What is the maximum value in in volts per meter of the electric field?

Details and assumptions

k=14πϵ0=9×109m/F k=\frac{1}{4\pi \epsilon_{0}}= 9\times 10^{9} m/F

×

Problem Loading...

Note Loading...

Set Loading...