Chapter 23 Problem 72

Problem

The cube with eight charges at the corners

At each corner of a cube of side $l$ there is a point charge $Q$,

(a) What is the potential at the center of the cube ($V=0$ at $r=\infty$)?

(b) What is the potential at each corner due to the other seven charges?

(c) What is the total potential energy of this system?

Solution

The distances

(a)

The center is equidistant from all the corners, and the distance is $d={\sqrt {{\frac {l^{2}}{4}}+{\frac {2l^{2}}{4}}}}={\frac {\sqrt {3}}{2}}l$

The potential at the center of the cube is just the scalar sum of potential due to all charges:

$V_{\textrm {center}}=8\cdot {\frac {1}{4\pi \epsilon _{0}}}{\frac {Q}{{\frac {\sqrt {3}}{2}}l}}={\frac {16}{\sqrt {3}}}{\frac {1}{4\pi \epsilon _{0}}}{\frac {Q}{l}}$

(b)

Each corner charge sees 3 edge pairs ($d=l$), 3 face diagonals ($d={\sqrt {2}}l$), and one body diagonal charge (${\frac {\sqrt {3}}{2}}l$). The potential is scalar sum of all these pair potential terms.

$V_{\textrm {corner}}=\left(3+{\frac {3}{\sqrt {2}}}+{\frac {1}{\sqrt {3}}}\right){\frac {1}{4\pi \epsilon _{0}}}{\frac {Q}{l}}=5.7{\frac {1}{4\pi \epsilon _{0}}}{\frac {Q}{l}}$

(c)

Be careful here! (Here are lions!)

The total energy is due to 28 pair potential terms (12 edge pairs, 12 face diagonal pairs, and 4 body diagonal pairs). If you simply multiply $V_{\textrm {corner}}$ by 8, since it contained 3 edge pairs, 3 face diagonals and one body diagonals, you end up with double of what you should count!

so,

$U={\frac {1}{2}}8\left(QV_{\textrm {corner}}\right)\approx 22.8{\frac {1}{4\pi \epsilon _{0}}}{\frac {Q^{2}}{l}}$