GATE 2026 EE – Question 63
A uniform spherical volume charge distribution of radius 2 m, centered at the origin, has a strength of $\dfrac3\pi\times10^{-5}$ C/m³. A point charge of strength $\pi\times8.854\times10^{-12}$ C is moved from $(-3,0,-4)$ to $(0,0,4)$ in Cartesian coordinate system. The relative permittivity of the medium is 1 and the coordinate values are in meters.
The work done during the process is ______ μJ.
(Round off to two decimal places)
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 3.98 to 4.02
Explanation
The sphere's total charge is $Q_s=\rho\cdot\tfrac43\pi a^3=\dfrac3\pi\times10^{-5}\times\dfrac43\pi\times8=3.2\times10^{-4}$ C. Both points are outside the sphere ($r_1=5$ m, $r_2=4$ m), so the field is that of a point charge. $W=q\,(V_2-V_1)=\dfrac{qQ_s}{4\pi\epsilon_0}\left(\dfrac1{4}-\dfrac1{5}\right)$. With $q=\pi\epsilon_0$: $\dfrac{q}{4\pi\epsilon_0}=\dfrac14$, so $W=\dfrac14\times3.2\times10^{-4}\times0.05=4\times10^{-6}$ J $=4.00\ \mu$J.