GATE 2022 IN Q.59: solution and answer

GATE 2022 · INQ.59 · 2 marks · NAT

A capacitor is constructed using two concentric spheres and air as the dielectric medium (permittivity of air = $8.854 \times 1 0^{- 12}$ F/m). The radii of the inner and outer spheres are $a = 10$ cm and $b = 15$ cm, respectively. The capacitance (in picofarads) is ________ (round off to 2 decimal places)

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Answer and solution

ANSWER

The capacitance of the air-filled spherical capacitor is $C = \frac{4 π ε _{0} ab}{b - a} = 4 π (8.854 \times 1 0^{- 12}) (0.3) \approx 33.38 pF$ .

Official answer: 32.8 to 33.9Source: Official answer key, GATE 2022 IN (IIT Kharagpur)

Assume a charge. Put $+ Q$ on the inner sphere (radius $a$ ) and $- Q$ on the outer sphere (radius $b$ ).
Gauss's law over a concentric sphere of radius $r$ ( $a < r < b$ ): the enclosed charge is $Q$ and $E$ is radial with constant magnitude on that sphere, so $4 π r^{2} E = Q / ε_{0}$ , giving $E = \frac{Q}{4 π ε _{0} r ^{2}} \overset{a}{^}_{r}$
Potential difference between the inner and outer spheres, integrating from the outer sphere to the inner one: $V = V_{a} - V_{b} = - \int_{b}^{a} \frac{Q}{4 π ε _{0} r ^{2}} d r = \frac{Q}{4 π ε _{0}} (\frac{1}{a} - \frac{1}{b})$
Capacitance $C = Q / V$ : $C = \frac{4 π ε _{0}}{\frac{1}{a} - \frac{1}{b}} = \frac{4 π ε _{0} ab}{b - a}$
Geometry factor in SI units: $a = 0.10 m$ and $b = 0.15 m$ , so $\frac{ab}{b - a} = \frac{0.10 \times 0.15}{0.05} = 0.3 m$ .
Substitute $ε_{0} = 8.854 \times 1 0^{- 12} F/m$ : $C = 4 π \times 8.854 \times 1 0^{- 12} \times 0.3 = 111.2626 \times 1 0^{- 12} \times 0.3 = 33.379 \times 1 0^{- 12} F$
Result: $C \approx 33.38 pF$ .

KEY IDEA

Between two concentric spheres the field is that of a point charge, $E = Q / (4 π ε_{0} r^{2})$ . Integrating it from $b$ to $a$ gives $V = \frac{Q}{4 π ε _{0}} (\frac{1}{a} - \frac{1}{b})$ , so $C = \frac{4 π ε _{0} ab}{b - a}$ . Only the geometry factor $ab / (b - a)$ , measured in metres, depends on the radii.

WATCH OUT FOR

Substituting the radii in centimetres. That gives $ab / (b - a) = 30$ instead of $0.3 m$ and makes the answer 100 times too large (about 3338 pF). The other common slip is using the isolated-sphere formula $4 π ε_{0} a$ (about 11.13 pF), which ignores the outer sphere.

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GATE 2022 IN Q.59: solution and answer

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