Question

Work done in assembling two identical charges each having a charge 'q', separated by a distance $r$ is:

• A. $\frac{1}{4\pi\varepsilon_0}\frac{q^2}{r}$
• B. $\frac{1}{4\pi\varepsilon_0}\frac{2q^2}{r}$
• C. $\frac{1}{4\pi\varepsilon_0}\frac{q^2}{2r}$
• D. $\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}$

Hint:

The work done in assembling point charges is always equal to the electrostatic potential energy of the final configuration.
For two charges, it is simply Coulomb's potential energy formula: $U = \frac{k q_1 q_2}{r}$.

Answer 1

The Correct Option is A

Step 1: Build the work done from the force itself, not the energy formula.
Bring the first charge $q$ in for free. To bring the second charge $q$ in from infinity to a distance $r$, we push it against the repulsive Coulomb force $F(r') = \frac{1}{4\pi\varepsilon_0}\frac{q^2}{r'^2}$.
Step 2: Integrate the force over the path. \[ W = \int_{\infty}^{r} F(r')\, dr' = \frac{q^2}{4\pi\varepsilon_0}\int_{\infty}^{r} \frac{dr'}{r'^2} \]
Step 3: Evaluate the integral. \[ \int_{\infty}^{r} \frac{dr'}{r'^2} = \left[-\frac{1}{r'}\right]_{\infty}^{r} = \frac{1}{r} \]
Step 4: Combine to get the work done. \[ \boxed{W = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q^2}{r}} \]