Calculating the Field from the Potential — Question 6

Question 6

A fitted local potential reveals the charge density required to support it

A measured three-dimensional potential is $V=V_0+ax+by+c(x^2+y^2+z^2)$ over a region. Use $\nabla\cdot\mathbf E=\rho/\epsilon_0$.

1. Find E and the implied volume charge density.

2. If c>0, find the equilibrium position for a positive test charge q of mass m and classify its local stability.

Question 6 - Solution

A fitted local potential reveals the charge density required to support it

(a) Method and calculation

$$\mathbf E=-(a+2cx)\hat{\mathbf x}-(b+2cy)\hat{\mathbf y}-2cz\hat{\mathbf z}.$$$$\rho=\epsilon_0\nabla\cdot\mathbf E=-6\epsilon_0c.$$

(b) Interpretation and further calculation

$$\mathbf r_{\mathrm{eq}}=\left(-\frac a{2c},-\frac b{2c},0\right).$$$$U=qV=q c|\mathbf r-\mathbf r_{\mathrm{eq}}|^2+\text{constant},\quad\omega=\sqrt{2qc/m}.$$For q>0 and c>0 the local equilibrium is stable if the fitted region includes it.

Physical check

This stable electrostatic minimum requires nonzero negative source charge density, so it is not a charge-free trap.