GATE 2024 IN Q.20: solution and answer

GATE 2024 · INQ.20 · 1 mark · MCQ

The capacitance formed between two concentric spherical metal shells having radii $x$ and $y$ with $y > x$ is

Note: $ϵ$ is the permittivity of the medium between the shells.

(A)
$4 π ϵ (\frac{x y}{y - x})$
(B)
$4 π ϵ (\frac{x ^{2}}{y - x})$
(C)
$4 π ϵ (\frac{y ^{2}}{y - x})$
(D)
$4 π ϵ (\frac{y ^{2} - x y}{x})$

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Answer and solution

ANSWER

The capacitance of the spherical capacitor is $C = 4 π ϵ (\frac{x y}{y - x})$ , which is option (A).

Official answer: ASource: Official answer key, GATE 2024 IN (IISc Bengaluru)

Put charge $+ Q$ on the inner shell (radius $x$ ) and $- Q$ on the outer shell (radius $y$ ). The charges sit on the facing surfaces, so the field exists only in the gap $x < r < y$ . Inside the inner shell no charge is enclosed, and outside the outer shell the net enclosed charge is zero.
Apply Gauss's law over a concentric sphere of radius $r$ in the gap: $ϵ E_{r} (4 π r^{2}) = Q$ , so $E = \frac{Q}{4 π ϵ r ^{2}} \overset{a}{^}_{r}, x < r < y .$
Find the potential of the inner shell relative to the outer one, $V_{x y} = - \int_{y}^{x} E_{r} d r$ : $V_{x y} = - \frac{Q}{4 π ϵ} \int_{y}^{x} \frac{d r}{r ^{2}} = \frac{Q}{4 π ϵ} (\frac{1}{x} - \frac{1}{y}) = \frac{Q}{4 π ϵ} \cdot \frac{y - x}{x y} .$ This is positive because $y > x$ .
Capacitance is $C = Q / V_{x y}$ : $C = 4 π ϵ (\frac{x y}{y - x}) .$
Check the limit. As $y \to \infty$ , $\frac{x y}{y - x} \to x$ , so $C \to 4 π ϵ x$ , which is the capacitance of an isolated sphere of radius $x$ . Option (B) goes to $0$ in this limit, and options (C) and (D) grow without bound, so none of them reduces to the isolated sphere. Answer: (A).

KEY IDEA

Assume charges $\pm Q$ on the two shells. Gauss's law gives $E = Q / (4 π ϵ r^{2})$ in the gap. Integrate from $x$ to $y$ to get $V = \frac{Q}{4 π ϵ} (\frac{1}{x} - \frac{1}{y})$ , then $C = Q / V = 4 π ϵ \frac{x y}{y - x}$ . To check the answer quickly, let the outer radius go to infinity: the result must become the isolated-sphere value $4 π ϵ x$ .

WATCH OUT FOR

Students often slip when combining $\frac{1}{x} - \frac{1}{y}$ into one fraction. Some write $\frac{x - y}{x y}$ , which is negative, or $\frac{y - x}{y ^{2}}$ , and then pick (C) or (B). Others use the coaxial-cable result $2 π ϵ L / ln (y / x)$ , which applies to cylinders, not spheres. Putting $y \to \infty$ exposes all of these errors at once.

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GATE 2024 IN Q.20: solution and answer

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