GATE 2026 EE Q.15: solution and answer

Official answer: CSource: Official answer key, GATE 2026 EE (IIT Guwahati)

1. Put the ring in the $xy$-plane, centred at the origin. The charge is spread uniformly along the circumference $2\pi R$, so the line charge density is ${\rho }_L=\frac{Q}{2\pi R}$.

2. Take a charge element $dQ={\rho }_{L}Rd\varphi$ at angle $\varphi$, i.e. at position $R{\hat{a}}_{\rho }$ with ${\hat{a}}_{\rho }=\cos \varphi {\hat{a}}_x+\sin \varphi {\hat{a}}_y$. By Coulomb's law its field at the centre has magnitude $dE=\frac{dQ}{4\pi ϵR^2}$ and points from the element towards the centre, i.e. along $-{\hat{a}}_{\rho }$:
   $$d\mathbf{E}=-\frac{{\rho }_{L}Rd\varphi }{4\pi ϵR^2}{\hat{a}}_{\rho }$$

3. Superpose over the whole ring. Because ${\hat{a}}_{\rho }$ changes direction with $\varphi$, write it in the fixed Cartesian unit vectors before integrating:
   $$\mathbf{E}(0)=-\frac{{\rho }_L}{4\pi ϵR}{\int }_0^{2\pi }\left(\cos \varphi {\hat{a}}_x+\sin \varphi {\hat{a}}_y\right)d\varphi =0$$
   since ${\int }_0^{2\pi }\cos \varphi d\varphi ={\int }_0^{2\pi }\sin \varphi d\varphi =0$.

4. Symmetry check: every element $dQ$ at angle $\varphi$ has an equal partner at $\varphi +\pi$, diametrically opposite. Their fields at the centre are equal in magnitude and opposite in direction, so they cancel in pairs. Cross-check with the standard on-axis result for a ring, $E_z=\frac{Qz}{4\pi ϵ{\left(R^2+z^2\right)}^{3/2}}$, which is $0$ at $z=0$.

5. Force on the point charge: $\mathbf{F}=q\mathbf{E}(0)=0$, so $|\mathbf{F}|=0$ for any $Q$, $q$, $R$ and $ϵ$.

6. Checking the options: (A) $\frac{Qq}{4\pi ϵR}$ has the units of energy. It is the potential energy $qV$ of the point charge, where $V=\frac{Q}{4\pi ϵR}$ is the (non-zero) potential at the centre. It is not a force. (B) is what you get by adding the magnitudes of all the elemental forces and ignoring their directions. (D) does have the units of force ($\frac{q}{4\pi ϵR}$ is in $\text{V}$, $\frac{Q}{2\pi R}={\rho }_L$ is in $\text{C/m}$, and $\text{V}\cdot \text{C/m}=\text{J/m}=\text{N}$), but it is non-zero, while the vector sum is exactly zero. Only (C) is correct.