Electric Potential Energy of a System of Point Charges — Question 7

Question 7

Three equal repelling charges expand self-similarly from an equilateral triangle

Three particles each of charge q>0 and mass m start at rest at the vertices of an equilateral triangle of side a and are released in isolation.

1. Use symmetry and energy conservation to find each speed when the side length has grown to b>a.

2. Find each limiting speed and explain why the particles remain at the vertices of an expanding equilateral triangle.

Question 7 - Solution

Three equal repelling charges expand self-similarly from an equilateral triangle

(a) Method and calculation

$$U(a)=3kq^2/a,\quad U(b)=3kq^2/b.$$$$3\left(\tfrac12mv^2\right)=3kq^2(1/a-1/b)\Rightarrow v=\sqrt{\frac{2kq^2}{m}\left(\frac1a-\frac1b\right)}.$$

(b) Interpretation and further calculation

$$v_\infty=\sqrt{2kq^2/(ma)}.$$$$\mathbf F_i\parallel(\mathbf r_i-\mathbf r_{\mathrm{center}}),\quad |\mathbf F_1|=|\mathbf F_2|=|\mathbf F_3|.$$Identical radial evolution preserves all threefold symmetries.

Physical check

The displayed v is each particle speed relative to the fixed center of mass; the side-length rate is $\dot b=\sqrt3v$.