Physics - Electrostatics Question with Solution | TestHub

$\mathrm{A} = (2, 2)$and$\mathrm{B} = (4, 1)$

Now$W_{\text{A}⟶\text{B}}=q\left(V_{\text{B}}-V_{\text{A}}\right)$... (1)

${\int }_{\text{A}}^{\text{B}}\text{d}\mathit{\text{V}}=-{\int }_{\text{A}}^{\text{B}}\overrightarrow{\mathit{\text{E}}}·\overrightarrow{\text{d}}\mathit{\text{r}}$

or$V_{\text{B}}-V_{\text{A}}=-{\int }_{\left(2,2\right)}^{\left(4,1\right)}\left(y\hat{\text{i}}+x\hat{\text{j}}\right)·\left(\text{d}\mathit{\text{x}}\hat{\text{i}}+\text{d}\mathit{\text{y}}\hat{\text{j}}+\text{d}\mathit{\text{z}}\hat{\text{k}}\right)$

or$V_{\text{B}}-V_{\text{A}}=-{\int }_{\left(2,2\right)}^{\left(4,1\right)}\left(\mathit{\text{y}}\text{d}\mathit{\text{x}}+\mathit{\text{x}}\text{d}\mathit{\text{y}}\right)$

$=-{\int }_{\left(2,2\right)}^{\left(4,1\right)}\text{d}\left(\mathit{\text{xy}}\right)={\left[-\mathit{\text{xy}}\right]}_{\text{2,2}}^{\text{4,1}}=0$

$\therefore$${\mathit{\text{W}}}_{\text{A}\to \text{B}}=0$[from Eq. (1)]