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A region contains a uniform electric field $\mathbf{E}=(10 \hat{\mathbf{i}}+30 \hat{\mathbf{j}}) \mathrm{Vm}^{-1} . A$ and $B$ are two points in the field at $(1,2,0) \mathrm{m}$ and $(2,1,3) \mathrm{m}$, respectively. The work done when a charge of $0.8 \mathrm{C}$ moves from $A$ to $B$ in a parabolic path is
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Verified Answer
The correct answer is:
$16\ J$
Using $\mathrm{E}=\frac{d V}{d r}$ or, $\mathrm{dV}=\mathrm{E} \mathrm{dr}$
Displacement, $\mathrm{dr}=\mathrm{r}_{\mathrm{B}}-\mathrm{r}_{\mathrm{A}}$
$\begin{aligned}
& =(2 \hat{i}+\hat{j}+3 \hat{k})-(\hat{i}+2 \hat{j}) \\
& =(10 \hat{i}+30 \hat{j})(\hat{i}-\hat{j}+3 \hat{k}) \\
& =10 \hat{i} \cdot \hat{i}-30 \hat{i} \cdot \hat{j}+0 \\
& =10-30=-20
\end{aligned}$
Work done, $\mathrm{W}=\mathrm{q} \times|\mathrm{dV}|=0.8(20)=16 \mathrm{~J}$
Displacement, $\mathrm{dr}=\mathrm{r}_{\mathrm{B}}-\mathrm{r}_{\mathrm{A}}$
$\begin{aligned}
& =(2 \hat{i}+\hat{j}+3 \hat{k})-(\hat{i}+2 \hat{j}) \\
& =(10 \hat{i}+30 \hat{j})(\hat{i}-\hat{j}+3 \hat{k}) \\
& =10 \hat{i} \cdot \hat{i}-30 \hat{i} \cdot \hat{j}+0 \\
& =10-30=-20
\end{aligned}$
Work done, $\mathrm{W}=\mathrm{q} \times|\mathrm{dV}|=0.8(20)=16 \mathrm{~J}$
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