A tank circuit in a radio transmitter is a series RCL circuit connected to an antenna. 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 consider the resonant frequency of the circuit. At resonance, the frequency of the radio signals matches the natural frequency of the tank circuit.

The resonant frequency of a series RCL circuit can be calculated using the formula:

f = 1 / (2π√LC)

where:
f is the frequency in Hertz (Hz)
L is the inductance in Henrys (H)
C is the capacitance in Farads (F)
π is a mathematical constant approximately equal to 3.1416

Given the fixed capacitance of 1.8 x 10^(-11) F, we can substitute it into the formula.

To calculate the minimum inductance, we'll use the maximum frequency of 9.2 MHz.

a) Minimum Inductance:

f = 9.2 MHz = 9.2 x 10^6 Hz
C = 1.8 x 10^(-11) F

Substituting the values into the formula and solving for L:

9.2 x 10^6 Hz = 1 / (2π√L(1.8 x 10^(-11) F))

Simplifying the equation:

2π√L = 1 / (9.2 x 10^6 Hz x 1.8 x 10^(-11) F)
2π√L = 5.43 x 10^6

Squaring both sides of the equation:

4π^2L = (5.43 x 10^6)^2
L = ((5.43 x 10^6)^2) / (4π^2)

Now we can calculate the value of L using a calculator:

L ≈ 1.998 x 10^(-6) H (or 1.998 μH)

So, the minimum inductance of the tank circuit is approximately 1.998 μH.

b) Maximum Inductance:

Similarly, we'll use the minimum frequency of 3.1 MHz to calculate the maximum inductance.

f = 3.1 MHz = 3.1 x 10^6 Hz

Substituting the values into the formula and solving for L:

3.1 x 10^6 Hz = 1 / (2π√L(1.8 x 10^(-11) F))

2π√L = 1 / (3.1 x 10^6 Hz x 1.8 x 10^(-11) F)
2π√L = 9.61 x 10^6

Squaring both sides of the equation:

4π^2L = (9.61 x 10^6)^2
L = ((9.61 x 10^6)^2) / (4π^2)

Calculating the value of L using a calculator:

L ≈ 1.127 x 10^(-6) H (or 1.127 μH)

Therefore, the maximum inductance of the tank circuit is approximately 1.127 μH.