Find the potential $V$ of an electrostatic field $\vec E = a\left( {y\hat i + x\hat j} \right)$, where $a$ is a constant.
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$\mathrm{E}=a(y i+x j)$
$\mathrm{V}_{2}-\mathrm{V}_{1}=-\int(\mathrm{ayd} \mathrm{x}+\mathrm{axd} \mathrm{y})$
Take $\mathrm{V}_{1}=\mathrm{C}$ and $\mathrm{V}_{2}=\mathrm{V} .$ Then
${\mathrm{V}=-\mathrm{a} \int(\mathrm{ydx}+\mathrm{xdy})+\mathrm{C}} $
${=-\mathrm{a} \int \mathrm{d}(\mathrm{yx})+\mathrm{C}=-\mathrm{axy}+\mathrm{C}}$
$\mathrm{O} \mathrm{R}$
Verify with $ \overrightarrow{\mathrm{E}}=-\nabla \mathrm{V}=-\left(\hat{i} \frac{\partial}{\partial \mathrm{x}}+\hat{j} \frac{\partial}{\partial \mathrm{y}}\right)(-\mathrm{axy}+\mathrm{C})$
$=a y \hat{i}+a x \hat{j}+0=a(y \hat{i}+x \hat{j})$
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