Coulomb's Law — Question 10

Question 10

Balancing one corner of a square does not balance the whole configuration

A square of side $a$ has positive charge $q$ at $(0,0)$, $(a,0)$ and $(0,a)$. An adjustable charge $Q$ is fixed at $(a,a)$. Use Coulomb law with constant $k$ and ignore gravity.

1. Find $Q$ so the electrostatic force on the origin charge is zero.

2. For that value of $Q$, find the force components on the charge at $(a,0)$ in units $F_0=kq^2/a^2$. Determine whether all four charges can remain at rest without supports.

Question 10 - Solution

Balancing one corner of a square does not balance the whole configuration

(a) Method and calculation

$$\mathbf F_{\mathrm{adjacent,on\ origin}}=-F_0(\hat{\mathbf i}+\hat{\mathbf j}).$$$$\mathbf F_{Q,\mathrm{on\ origin}}=-\frac{kqQ}{2\sqrt2a^2}(\hat{\mathbf i}+\hat{\mathbf j}).$$$$q^2+qQ/(2\sqrt2)=0\Rightarrow Q=-2\sqrt2q.$$

(b) Interpretation and further calculation

$$\frac{F_x(a,0)}{F_0}=1+\frac1{2\sqrt2},\quad\frac{F_y(a,0)}{F_0}=2\sqrt2-\frac1{2\sqrt2}.$$$$\mathbf F(a,0)\ne\mathbf0.$$The origin charge is balanced, but at least one other charge is not. Fixed source positions therefore require external support forces.

Physical check

The diagonal separation is $\sqrt2a$ and its force must also be projected by $1/\sqrt2$ onto each axis.