Question
Electric field in a region is given by $\overrightarrow{E} = Ax\widehat{i} + By\widehat{j}$, where $A = 10\text{ }V/m^{2}$ and $B = 5\text{ }V/m^{2}$. If the electric potential at a point $(10,20)$ is 500 V , then the electric potential at origin is $\_\_\_\_$ V.
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Solution
$\overrightarrow{E} = 10x\widehat{i} + 5y\widehat{j}$
$${V_{\text{at~}(10,20)} = 500\text{ }V }{\Delta V - \int\overrightarrow{E} \cdot d\overrightarrow{r} }{500 - V_{0} = - \int_{(0,0)}^{(10,20)}\mspace{2mu}(10x\widehat{i} + 5y\widehat{j}) \cdot (dx\widehat{i} + dy\widehat{j}) }{500 - V_{0} = - \left\lbrack 5x^{2} + \frac{5y^{2}}{2} \right\rbrack_{(0,0)}^{(10,20)} }{V_{0} - 500 = \left( 500 + 5 \times \frac{400}{2} \right) - (0 - 0) }{V_{0} - 500 = 500 + 1000 }{V_{0} = 2000\text{ }V}$$
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