Electromagnetism - RLC Circuit

A series RLC circuit is used in a radio to tune an FM station broadcasting at 103.7 MHz. The resistance in the circuit is 10 ohms and the inductance is 2.0 microhenries. What is the best estimate of the capacitance that should be used?

A. 200 pF
B. 50 pF
C. 1 pF
D. 0.2 pF
E. 0.02 pF

(GR9677 #26)

Solution:

X_L = X_C
ωL = 1 / ωC
C = 1 / ω²L 

Given:
f  = 103.7 × 10⁶ Hz
ω = 2πf  = 2π × 103.7 × 10⁶ ≈ 2π × 10² × 10⁶ = 2π × 10⁸
ω² = (2π × 10⁸)² ≈ 4π² × 10¹⁶ 
L  = 2.0 microhenries = 2 × 10⁻⁶ H

C = 1 / (4π² × 10¹⁶ × 2 × 10⁻⁶) 

= 1 / (8π² × 10¹⁰)

Take π² ≈ 10

C  ≈ 1/8 × 10⁻¹¹ 
= 1.25 × 10⁻¹² 
= 1.25 pF

Answer: C