Electrostatic potential energy

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Question

An electric field is given by $\vec{E} = (y \hat{i} + x \hat{j}) N / C$ . Find the work done (in J) by electric field in moving a 1C charge from $\vec{r_{A}} = (\sqrt{2} \hat{i} + 2 \hat{j}) m$ to $\vec{r_{B}} = (4 \hat{i} + \hat{j}) m$

Moderate

Solution

$W = q \int_{r_{A}}^{r_{B}} \vec{E} . d \vec{r} = 1 \int_{(\sqrt{2}, 2)}^{(4, 1)} E_{x} d x + E_{y} d y = 1 \int_{(\sqrt{2}, 2)}^{(4, 1)} (y d x + x d y) = \int_{(\sqrt{2}, 2)}^{(4, 1)} d (x y) = {[x y]}_{(\sqrt{2}, 2)}^{(4, 1)} = (4 \times 1) - (\sqrt{2} \times 2) = 4 - 2 \sqrt{2} = 1.17$

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