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 resonance condition for a tank circuit:
1 / (2π√(LC)) = f
Where:
L = inductance (in henries)
C = capacitance (in farads)
f = frequency (in hertz)
To find the minimum and maximum inductance, we can rearrange the equation and solve for L.
(a) Minimum Inductance:
For the minimum frequency, f_min = 3.1 MHz = 3.1 x 10^6 Hz
1 / (2π√(L_min * 1.8 x 10^(-11))) = 3.1 x 10^6
Rearranging the equation for L_min:
L_min = (1 / (2π * (3.1 x 10^6)^2 * 1.8 x 10^(-11)))^(-1)
Calculating L_min will provide the minimum inductance of the tank circuit.
(b) Maximum Inductance:
For the maximum frequency, f_max = 9.2 MHz = 9.2 x 10^6 Hz
1 / (2π√(L_max * 1.8 x 10^(-11))) = 9.2 x 10^6
Rearranging the equation for L_max:
L_max = (1 / (2π * (9.2 x 10^6)^2 * 1.8 x 10^(-11)))^(-1)
Calculating L_max will provide the maximum inductance of the tank circuit.
By substituting the given values into the equations, you can find the minimum and maximum inductance for the tank circuit.