An electric field is given by E→$$=(yi^+xj^)NC$$. Find the work done (in$$ $$J) in moving a $$1$$ C charge from r→$$A=(2i^+2j^)m$$ to r→$$B=(4i^+j^)m$$ .

$$A=(2$$,$$2)$$ and $$B=(4$$,$$1)$$;WA→$$B=q$$∫AB $$dV=$$−∫AB qE¯⋅dr→$$=q$$∫AB (yi^+xj^)⋅(dxi^+dyj^)=q$$∫AB $$(ydx+xdy)=$$−q∫$$(2$$,$$2)(4$$,$$1) $$d(xy)=$$−q[xy]$$(2.2)(4$$,$$1)=$$−q[$$4$$−$$4$$]$$=0$$

Electrostatic potential energy

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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 ${\vec{r}}_{A} = (2 \hat{i} + 2 \hat{j}) m to {\vec{r}}_{B} = (4 \hat{i} + \hat{j}) m .$

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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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Electrostatic potential energy

Remember concepts with our Masterclasses.

An electric field is given by E→=(yi^+xj^)NC. Find the work done (in J) in moving a 1 C charge from r→A=(2i^+2j^)m to r→B=(4i^+j^)m .

A=(2,2) and B=(4,1);WA→B=q∫AB dV=−∫AB qE¯⋅dr→=q∫AB (yi^+xj^)⋅(dxi^+dyj^)=q∫AB (ydx+xdy)=−q∫(2,2)(4,1) d(xy)=−q[xy](2.2)(4,1)=−q[4−4]=0

Ready to Test Your Skills?

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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 ${\vec{r}}_{A} = (2 \hat{i} + 2 \hat{j}) m to {\vec{r}}_{B} = (4 \hat{i} + \hat{j}) m .$