Determine the work required to bring an electric charge

Determine the work required to bring an electric charge $e_2=1$ (CGSE) from infinity to a distance $R$ from a fixed charge $e_1$.

Step 1: Potential energy at the two endpoints

\[U_\infty=\frac{e_1e_2}{\infty}=0,\qquad U_R=\frac{e_1e_2}{R}=\frac{e_1}{R}\quad(\text{since }e_2=1).\]

Step 2: Work by the external agent

\[W=U_R-U_\infty=\frac{e_1}{R}.\]

\[\boxed{W=\frac{e_1}{R}}\]

Step 3: Confirm via direct integration

The force on $e_2=1$ due to $e_1$ at distance $r$ is $F(r)=e_1/r^2$. Moving against this force from $\infty$ to $R$:

\[W=-\int_\infty^R\frac{e_1}{r^2}\,dr=-e_1\left[-\frac1r\right]_\infty^R=-e_1\left(-\frac1R-0\right)=\frac{e_1}{R},\]

matching Step 2.

Step 4: Interpretation

This is precisely the electric potential $V(R)=e_1/R$ due to $e_1$, multiplied by the unit charge $e_2=1$ — the standard definition of potential as "work per unit charge to bring it in from infinity." The sign of $W$ follows the sign of $e_1$: positive (work must be supplied against repulsion) if $e_1>0$, negative (the attractive force does the work) if $e_1<0$.