Q. Charge $Q$ is given a displacement $\overrightarrow{r} = a\widehat{\dot{i}}$ +b$\widehat{\dot{j}}$ in an electric field $\overrightarrow{E} = E_{1}\widehat{\dot{i}}$ +$E_{2}b\widehat{\dot{j}}.$ The work done is
A) $Q(E_{1}a + E_{2}b)$
B) $Q{\sqrt{\left( E_{1}a \right)^{2} + \left( E_{2}b \right)^{2}}}^{\ }$
C) $Q(E_{1} + E_{2})\sqrt{a^{2} + b^{2}}$
D) $Q(\sqrt{{E_{1}^{2}}_{\ } + E_{2}^{2}})\sqrt{a^{2} + b^{2}}$
Explanation
(a) $\overrightarrow{E}\ $=${\ E}_{1}\widehat{\dot{i}}\ $+${\ E}_{2}\widehat{\dot{j}}$
As $W = \overrightarrow{F}.\ \overrightarrow{r} = Q\overrightarrow{E}.\ \overrightarrow{r}$
$= Q({\ E}_{1}\widehat{\dot{i}}\ $+${\ E}_{2}\widehat{\dot{j}}).(a\widehat{\dot{i}}$ +$\ b\widehat{\dot{j}}$)
$= {QE}_{1}a\ $+${\ QE}_{2}b{= Q(E}_{1}a\ $+${\ E}_{2}b$)
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.