Question
Derive the formula for electrostatic potential due to a point charge.
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Answer
→As shown in fig., charge $Q(Q>0)$ is placed on the origin of the cartesian co-ordinates system. We want to find electric potential at some point P , having its position vector $\vec{r}$ from origin O. For this we should calculate work done by external force in bringing unit positive charge (test charge) from infinity to point $P$.
→Suppose, there is a point $P ^{\prime}$ at distance $r^{\prime}$ in between the path from infinity to P .
→Force on unit positive charge kept at point $P ^{\prime}$ is
$\overrightarrow{ F }^{\prime}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{ Q }{r^{\prime^2}} \cdot \hat{r}^{\prime}$
Where $\hat{r}^{\prime}$ is the unit vector in the direction of $\overrightarrow{ OP ^{\prime}}$.
→The work done against the (field) force in giving $\overrightarrow{\Delta r^{\prime}}$ displacement to the unit positive charge,
$\Delta W =-\frac{1}{4 \pi \varepsilon_0} \cdot \frac{ Q }{r^{\prime 2}} \cdot \Delta r^{\prime}$
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