Q. The displacement of a charge $\mathcal{Q}$in the electric field $\mathbf{E} = e_{1}\widehat{\mathbf{i}} + e_{2}\widehat{\mathbf{j}} + e_{3}\widehat{\mathbf{k}}\ is\ \mathbf{r} = a\widehat{\mathbf{i}} + b\widehat{\mathbf{j}}.$ The work done is
A) $\mathcal{Q}\left( ae_{1} + be_{2} \right)$
B) $\mathcal{Q}\sqrt{\left( ae_{1} \right)^{2} + \left( be_{2} \right)^{2}}$
C) $\mathcal{Q}\left( e_{1} + e_{2} \right)\sqrt{a^{2} + b^{2}}$
D) $\mathcal{Q}\left( \sqrt{e_{1}^{2} - e_{2}^{2}} \right)(a + b)$
Explanation
(a) By using $W = \mathcal{Q}\left( \mathbf{E}.\mathrm{\Delta}\mathbf{r} \right)$
$\Longrightarrow W = \mathcal{Q}\left\lbrack e_{1}\widehat{\mathbf{i}} + e_{2}\widehat{\mathbf{j}} + e_{3}\widehat{\mathbf{k}} \right).(a\widehat{\mathbf{i}} + b\widehat{\mathbf{j}})\rbrack$
$\mathcal{= Q}\left( e_{1}a + e_{2}b \right)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.