A radio can tune over the frequency range of a portion of MW broadcast band: (800 kHz to 1200 kHz). If its LC circuit has an effective inductance of 200μH, what must be the range of its variable capacitor? [Hint: For tuning, the natural frequency i.e., the frequency of free oscillations of the LC circuit should be equal to the frequency of the radio wave.]

The range of frequency (ν) of a radio is 800 kHz to 1200 kHz.

Lower tuning frequency, ν1 = 800 kHz = 800 x 10³ Hz

Upper tuning frequency, ν2 = 1200 kHz = 1200 x 10³ Hz

Effective inductance of circuit L = 200μH = 200 x 10⁻⁶ H

Capacitance of variable capacitor for ν1 is given as_:

C₁ = 1/(ω₁)²_L

Where, ω₁ = Angular frequency for capacitor C₁= 2πν1= 2π x  800 x 10³ rad/s

Therefore,

C₁ = 1/(2π x 800 x 10³ )² x 200 x 10⁻⁶  = 1.9809 x 10⁻¹⁰ F = 198 pF

Capacitance of variable capacitor for ν2 is given as ;

C2 = 1/(ω₂)²_L

Where, ω₂ = Angular frequency for capacitor C2 = 2πν2 = 2π x 1200 x 10³ rad/s

Therefore ,

C2 = 1/(2π x 1200 x 10³ )² x 200 x 10⁻⁶  = 0.8804  x 10⁻¹⁰ F = 88 pF

Hence, the range of the variable capacitor is from 88.04 pF to 198.1 pF.