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Q.
The potential difference $V_{A B}$ between $A(0,0,0)$ and $B(1,1,0)$ in an electric field $\vec{E} = x \hat{i} + z \hat{k}$, is
Electrostatic Potential and Capacitance
Solution:
$d V =-\vec{E} \cdot d \vec{r}$
$ \int\limits_{B}^{A} d V=-\int\limits_{(1,1,0)}^{(0,0,0)}(x \hat{i}+z \hat{k}) \cdot(d x \hat{i}+d y \hat{j}+d z \hat{k}) $ or
$ V_{A}-V_{B}=-\int\limits_{(1,1,0)}^{(0,0,0)}(x d x+z d z) $ or
$ V_{A}-V_{B}=-\left[\frac{x^{2}}{2}+\frac{z^{2}}{2}\right]_{(1,1,0)}^{(0,0,0)}$
$=-\left[\frac{0}{2}+\frac{0}{2}-\frac{1}{2}-\frac{0}{2}\right]=\frac{1}{2} V$