Question: Q. 1. (i) Write two properties by which electric potential is related to the electric field.
(ii) Two point charges $q_{1}$ and $q_{2}$, separated by a distance of $r_{12}$ are kept in an external electric field. Derive an expression for the potential energy of the system of two charges in the field.
A [Delhi Compt. I, II, III 2017]
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Solution:
Ans. (i) Properties
(ii) Derivation of expression of potential energy 2
(i) (a) Electric field is in the direction in which potential decreases at the maximum rate.
$1 / 2$
(b) Magnitude of electric field is given by change in the magnitude of potential per unit displacement normal to a charged conducting surface.
(ii) Work done in bringing the charge $q_{1}$ to a point against external electric field
$$ \begin{equation*} W_{1}=q_{1} V \overrightarrow{r_{1}} \tag{1} \end{equation*} $$
Work done in bringing the charge $q_{2}$ against the external electric field and electric field produced due to charge $q_{1}$
$W_{2}=q_{2} V\left(\overrightarrow{r_{2}}\right)+\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{1} q_{2}}{r_{12}}$
Therefore, Total work done $=$ Electrostatic potential energy :
$U=q_{1} V\left(\overrightarrow{r_{1}}\right)+q_{2} V\left(\overrightarrow{r_{2}}\right)+\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{1} q_{2}}{r_{12}}$
[CBSE Marking Scheme, 2017]
