First slide

Electrostatic potential energy

Question

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

Moderate

Solution

$\begin{array}{l} A = (2, 2) and B = (4, 1); \\ W_{A \to B} = q \int_{A}^{B} d V = - \int_{A}^{B} q \bar{E} \cdot d \vec{r} \\ = q \int_{A}^{B} (y \hat{i} + \hat{x j}) \cdot (d x \hat{i} + d y \hat{j}) = q \int_{A}^{B} (y d x + x d y) \\ = - q \int_{(2, 2)}^{(4, 1)} d (x y) = - q [x y]_{(2.2)}^{(4, 1)} = - q [4 - 4] = 0 \end{array}$

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