First slide

Electrostatic potential energy

Question

The displacement of a charge Q in the electric filed $\vec{E} = e_{1} \overset{\land}{i} + e_{2} \overset{\land}{j} + e_{3} \overset{\land}{k}$ is $\vec{r} = a \overset{\land}{i} + b \overset{\land}{j}$ . The work done is

Easy

A

$Q (a e_{1} + b e_{2})$
B

$Q \sqrt{{(a e_{1})}^{2} + {(b e_{2})}^{2}}$
C

$Q (e_{1} + e_{2}) \sqrt{a^{2} + b^{2}}$
D

$Q (\sqrt{e_{1}^{2} + e_{2}^{2}}) (a + b)$

Solution

By using
$W = Q (V) = Q (\vec{E} . \vec{r})$
$\Rightarrow W = Q [(e_{1} \overset{\land}{i} + e_{2} \overset{\land}{j} + e_{3} \overset{\land}{k}) . (a \overset{\land}{i} + b \overset{\land}{j})] = Q (e_{1} a + e_{2} b) (∵ \overset{\land}{i} . \overset{\land}{i} = \overset{\land}{j} . \overset{\land}{j} = \overset{\land}{k} . \overset{\land}{k} = 1 & \overset{\land}{i} . \overset{\land}{j} = \overset{\land}{j} . \overset{\land}{k} = \overset{\land}{k} . \overset{\land}{i} = 0)$

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