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.

Question

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.

Answer 1

f=.159/sqrt(LC)

L=(.159/f)^2 /C

at 10 mhz,

L=.159^2/(1E7)^2 * 1/1.8E-11=1.40E-5 henrys.

Answer 2

To find the minimum and maximum inductance of the tank circuit, we can use the formula for the resonant frequency of a tank circuit:

Resonant frequency (f) = 1 / (2π√(LC))

where L is the inductance and C is the capacitance.

For the minimum inductance, we'll use the maximum frequency of 9.2 MHz:

9.2 MHz = 1 / (2π√(Lmin x 1.8 x 10^(-11) F))

Simplifying the equation, we get:

Lmin x 1.8 x 10^(-11) F = 1 / (2π x 9.2 MHz)

Lmin = (1 / (2π x 9.2 MHz)) / (1.8 x 10^(-11) F)

Now we can calculate the minimum inductance:

Lmin ≈ 7.0 μH

For the maximum inductance, we'll use the minimum frequency of 3.1 MHz:

3.1 MHz = 1 / (2π√(Lmax x 1.8 x 10^(-11) F))

Simplifying the equation, we get:

Lmax x 1.8 x 10^(-11) F = 1 / (2π x 3.1 MHz)

Lmax = (1 / (2π x 3.1 MHz)) / (1.8 x 10^(-11) F)

Now we can calculate the maximum inductance:

Lmax ≈ 33.6 μH

So, the minimum inductance of the tank circuit is approximately 7.0 μH, and the maximum inductance is approximately 33.6 μH.

Answer 3

To find the minimum and maximum inductance of the tank circuit in the shortwave radio transmitter, we need to use the resonant frequency formula for the tank circuit.

The resonant frequency of a tank circuit is given by the formula:

f = 1 / ( 2π √(L * C) )

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 3.14159)

Now, let's calculate the minimum and maximum inductance using the given values:

(a) Minimum Inductance:
To find the minimum inductance, we use the maximum frequency (9.2 MHz) in the resonant frequency formula.

Given:
f = 9.2 MHz = 9.2 x 10^6 Hz
C = 1.8 x 10^-11 F

Substituting these values into the formula, we get:
9.2 x 10^6 = 1 / (2π √(L * 1.8 x 10^-11))

Rearranging the equation to solve for L, we have:
L = (1 / (2π (9.2 x 10^6)^2 * 1.8 x 10^-11))^2

Calculating this expression will give us the minimum inductance.

(b) Maximum Inductance:
To find the maximum inductance, we use the minimum frequency (3.1 MHz) in the resonant frequency formula.

Given:
f = 3.1 MHz = 3.1 x 10^6 Hz
C = 1.8 x 10^-11 F

Substituting these values into the formula, we get:
3.1 x 10^6 = 1 / (2π √(L * 1.8 x 10^-11))

Rearranging the equation to solve for L, we have:
L = (1 / (2π (3.1 x 10^6)^2 * 1.8 x 10^-11))^2

Calculating this expression will give us the maximum inductance.