The antenna broadcasts radio signals at the resonant frequency of the tank circuit. Suppose that a certain tank circuit in a shortwave radio transmitter has a fixed capacitance of 1.8 x 10-11 F and a variable inductance. If the antenna is intended to broadcast radio signals ranging in frequency from 3.1 MHz to 9.2 MHz, find the (a) minimum and (b) maximum inductance of the tank circuit.
To find the minimum and maximum inductance of the tank circuit, we need to use the formula for the resonant frequency of a tank circuit:
f = 1 / (2π√(LC))
Where:
f = resonant frequency in Hertz (Hz)
L = inductance in Henrys (H)
C = capacitance in Farads (F)
π = pi, approximately 3.14159
We can rearrange the formula to solve for L:
L = (1 / (4π²f²C))
Let's calculate the minimum and maximum inductance:
(a) Minimum Inductance:
Given frequency (f) = 3.1 MHz = 3.1 × 10⁶ Hz
Capacitance (C) = 1.8 × 10⁻¹¹ F
Substituting these values into the formula:
L = (1 / (4π²(3.1 × 10⁶)²(1.8 × 10⁻¹¹)))
Simplifying the expression:
L ≈ 2.072 × 10⁻⁷ H
Therefore, the minimum inductance of the tank circuit is approximately 2.072 × 10⁻⁷ Henrys.
(b) Maximum Inductance:
Given frequency (f) = 9.2 MHz = 9.2 × 10⁶ Hz
Capacitance (C) = 1.8 × 10⁻¹¹ F
Substituting these values into the formula:
L = (1 / (4π²(9.2 × 10⁶)²(1.8 × 10⁻¹¹)))
Simplifying the expression:
L ≈ 1.802 × 10⁻⁶ H
Therefore, the maximum inductance of the tank circuit is approximately 1.802 × 10⁻⁶ Henrys.
So, the minimum inductance is approximately 2.072 × 10⁻⁷ H and the maximum inductance is approximately 1.802 × 10⁻⁶ H.