Electric field vector in a region is given by $\mathbf{E}=(3 \hat{\mathbf{i}}+4 y \hat{\mathbf{j}})…
Electric field vector in a region is given by $\mathbf{E}=(3 \hat{\mathbf{i}}+4 y \hat{\mathbf{j}}) \mathrm{V}-\mathrm{m}^{-1}$. The potential at the origin is zero. Then, the potential at a point $(2,1) \mathrm{m}$ is
7 V
8 V
-8 V
-7 V
Solution
$
\begin{gathered}
\text { As, } \quad E=\frac{-\partial V_x}{\partial x} \hat{\mathrm{i}}+\frac{-\partial V_y}{\partial y} \hat{\mathrm{j}} \\
\frac{-\partial V_x}{\partial x}=3 \text { and } \frac{-\partial V_y}{\partial y}=4 y \\
\Rightarrow \quad V_x=-3 x \text { and } V_y=-2 y^2
\end{gathered}
$
So, $V=$ potential function
$
=-\left(3 x+2 y^2\right)
$
Potential at point $(2,1)$ is
$
V=-\left(3 \times 2+2 \times 1^2\right)=-8 \mathrm{~V}
$