Physics - Electrostatics Question with Solution | TestHub

The work done by an external agent in slowly moving a test charge $+q$ from point A to point B is given by $W_{ext}=q(V_B-V_A)$, where $V_A$ and $V_B$ are the electrostatic potentials at points A and B respectively. The potential due to a point charge $+Q$ at a distance $r$ is $V=\frac{Q}{4\pi {ϵ}_{0}r}$. At point $A(a,0)$, the distance from the origin is $r_A=a$. So, $V_A=\frac{Q}{4\pi {ϵ}_{0}a}$. At point $B(a,a)$, the distance from the origin is $r_B=\sqrt{a^2+a^2}=a\sqrt{2}$. So, $V_B=\frac{Q}{4\pi {ϵ}_{0}a\sqrt{2}}$. Therefore, $W_{ext}=q(V_B-V_A)=q\left(\frac{Q}{4\pi {ϵ}_{0}a\sqrt{2}}-\frac{Q}{4\pi {ϵ}_{0}a}\right)=\frac{Qq}{4\pi {ϵ}_{0}a}\left(\frac{1}{\sqrt{2}}-1\right)$.