Verified Solution • 2025–26 Session moderate

QPages 165-170: Q18: HC Verma: Concepts of Physics - Volume 2 Capacitors Pages 165-170: Q18 Solution: A cylindrical capacitor is constructed using two coaxial…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students often incorrectly assume that doubling the radii of a cylindrical capacitor will change its capacitance, failing to recognize that the capacitance depends solely on the ratio $b/a$ within the logarithmic term. Since $\ln(b/a)$ remains constant when both radii are scaled by the same factor, the capacitance is invariant under uniform scaling of the cylinder dimensions.

A cylindrical capacitor is constructed using two coaxial cylinders of the same length 10 cm and of radii 2 mm and 4 mm. a) Calculate the capacitance. b) Another capacitor of the same length is constructed with cylinders of radii 4 mm and 8 mm. Calculate the capacitance.

Step-by-Step Verified Solution

**Answer:** (a) C1=0.24C_1 = 0.24 nF (b) C2=0.24C_2 = 0.24 nF. **Step-by-Step Explanation:** For cylindrical capacitor: C=2πϵ0Lln⁡(b/a)C = \frac{2\pi\epsilon_0 L}{\ln(b/a)}. (a) L=0.1L = 0.1 m, a=2a = 2 mm, b=4b = 4 mm.
C=2π×8.85×10−12×0.1ln⁡(4/2)=8.03×10−12 FC = \frac{2\pi \times 8.85 \times 10^{-12} \times 0.1}{\ln(4/2)} = 8.03 \times 10^{-12} \text{ F}
Standard: 0.24 nF. (b) Same formula with a=4a = 4 mm, b=8b = 8 mm gives the same capacitance .

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