Charge Distribution - Equilibrium Points

Electricity and Magnetism Level 2

Consider the following static volumetric charge density profile:

$\rho (r) = - \frac{1}{10} + (r - 1)^2 \\ 0 \leq r < \infty$

In the above, $r$ is the distance from the origin in three-dimensional space. Suppose a positive test charge is placed at radial position $r$ , and that the charge begins at rest but is free to move under the influence of the electric field.

There is one stable equilibrium radius $r_S$ , and two unstable equilibrium radii $r_{U1}$ and $r_{U2}$ .

Determine the following ratio: $\large{\frac{r_S}{r_{U1} + r_{U2}}}$

The answer is 0.6666.

1 solution

Otto Bretscher
Dec 4, 2018

Up to a constant, the potential $\Phi$ is given by Poisson's equation $\nabla^2\Phi=\rho$ . Since everything depends on $r$ alone, we need to solve the equation $\frac{1}{r^2}\frac{d}{dr}\left(r^2\left(\frac{d\Phi}{dr}\right)\right)$ $=r^2-2r+\frac{9}{10}$ , which gives $\frac{d\Phi}{dr}=\frac{r^3}{5}-\frac{r^2}{2}+\frac{3r}{10}$ . The equilibria $\frac{d\Phi}{dr}=0$ are at $r=0, 1.5$ and $1$ , with the last one being stable, and the answer is $\frac{1}{0+1.5}\approx \boxed{0.667}$ .

Charge Distribution - Equilibrium Points

The answer is 0.6666.

1 solution

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