# why isn't the net force considered while calculating potential due to a system of charges?

The textbook I'm reading defines potential at a point as

work done per unit charge by an external agent to move the test charge from the reference point to the point under consideration (without changing it's kinetic energy)

During calculation of potential at a point due to a system of charges, why isn't the work done against the net force due to the system considered instead of simply adding up the work done against separate forces caused by individual charges?

PS: Wherever the explanation requires math, kindly also provide it's physical implications

...why isn't the work done against the net force due to the system considered instead of simply adding up the work done against separate forces caused by individual charges?

They're both equivalent, due to the principle of superposition.

Basically, the net force is what you get when you add up the separate forces from the individual charges acting on the test charge, so when you calculate the work done against the net force, it's the same as adding up the work done against the separate forces.

General two particle system:

Imagine you have two charged particles in space, $Q_1$ and $Q_2$, and your test charge. When you move your test charge, the work done against the electrostatic force, $\mathbf{F_1}$, of $Q_1$ is $W_1=-\int{\mathbf{F_1}\cdot\mathrm{d}\mathbf{r}}$. Similarly, the work done against charge 2 is $W_2=-\int{\mathbf{F_2}\cdot\mathrm{d}\mathbf{r}}$.

What is the total work done? $$W_T=W_1+W_2=-\int{\mathbf{F_1}\cdot\mathrm{d}\mathbf{r}}-\int{\mathbf{F_2}\cdot\mathrm{d}\mathbf{r}}$$ $$=-\int{(\mathbf{F_1}+\mathbf{F_2})\cdot\mathrm{d}\mathbf{r}}$$ But wait, what is the resultant force on the test charge? It's $\mathbf{F_T}=\mathbf{F_1}+\mathbf{F_2}$. Therefore $$W_T=-\int{\mathbf{F_T}\cdot\mathrm{d}\mathbf{r}}$$

• @Apoorv Are you familiar with line integrals, i.e., that $W=\int \mathbf{F}\cdot \mathrm{d} \mathbf{r}$? – binaryfunt May 17 '15 at 12:17
• yes, I'm pretty comfortable with those – Apoorv May 17 '15 at 12:21