Electric Potential Energy of a System of Point Charges — Question 6

Question 6

Two freely released charges divide the available energy according to their masses

Two positive charges q and 2q have masses m and 2m. They start at rest a distance a apart in isolation and are released. Ignore radiation and relativistic effects.

1. Find the total kinetic energy at very large separation.

2. Use momentum conservation to find the two final kinetic energies and speeds.

Question 6 - Solution

Two freely released charges divide the available energy according to their masses

(a) Method and calculation

$$U_i=\frac{2kq^2}{a},\quad U_f\to0\Rightarrow K_1+K_2=\frac{2kq^2}{a}.$$$$mv_1=2mv_2\Rightarrow v_1=2v_2,\quad K_1=2K_2.$$

(b) Interpretation and further calculation

$$K_1=\frac{4kq^2}{3a},\quad K_2=\frac{2kq^2}{3a}.$$$$v_1=\sqrt{\frac{8kq^2}{3ma}},\quad v_2=\sqrt{\frac{2kq^2}{3ma}}.$$

Physical check

The forces have equal magnitudes, but the lighter particle gains more kinetic energy because both particles have equal momentum magnitudes.