GATE 2024 PH – Question 46
Two charges +q are 2d apart. A grounded solid conducting sphere of radius a is midway between them. Assume $a^2\ll d^2$. Which statement(s) is/are true?
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Correct answer: (A) For $a>d/8$ the net forces point toward each other; (B) The sphere surface potential is zero; (C) Total induced charge is $-2aq/d$
Explanation
Place the grounded conducting sphere (radius $a$) at the midpoint between the two charges $+q$, each at a distance $d$ from the centre.
**Image charge.** A charge $q$ at distance $d$ from the centre of a grounded sphere has an image charge $q^\prime=-\dfrac{aq}{d}$ at a distance $\dfrac{a^2}{d}$ from the centre.
- **B. The sphere surface potential is zero.** True: the sphere is grounded. ✓
- **C. The total induced charge is $-2aq/d$.** True: each external charge induces $-aq/d$ on the sphere, and there are two. ✓
- **D. The potential at the centre is nonzero.** False: the conductor is an equipotential, so the potential at the centre is also zero.
- **A. For $a>d/8$ the net forces point toward each other.** Each charge $q$ is repelled by the other charge ($2d$ away) with force $\dfrac{q^2}{4\pi\epsilon_0(2d)^2}=\dfrac{q^2}{16\pi\epsilon_0d^2}$ and is attracted by the two image charges (for $a^2\ll d^2$, each image is near the centre, a distance about $d$ away): the combined attraction is about $\dfrac{2aq^2}{4\pi\epsilon_0d^3}$. The attraction wins when
$$\frac{2a}{d^3}>\frac1{4d^2}\;\Rightarrow\;a>\frac d8 .$$
✓
Answer **A, B and C**.