FM radio stations use radio waves with frequencies from 88.0 to 108 MHz to broadcast their signals. Assuming that the inductance has a value of 4.60 x 10^-7 H, determine the range of capacitance values that are needed so the antenna can pick up all the radio waves broadcasted by FM stations.

Question

FM radio stations use radio waves with frequencies from 88.0 to 108 MHz to broadcast their signals. Assuming that the inductance has a value of 4.60 x 10^-7 H, determine the range of capacitance values that are needed so the antenna can pick up all the radio waves broadcasted by FM stations.

_____F (lowest acceptable capacitance)

____F (highest acceptable capacitance)

Answer 1

Find Xl (2PI f L)

then find C such that
Xc= 1/2PI*f*C where Xc= Xl

do it at each frequency end.

Answer 2

To determine the range of capacitance values needed for an antenna to pick up all FM radio waves, we can use the formula for the resonant frequency of an LC circuit:

f = 1 / (2π√(LC))

Where:
f = resonant frequency in Hz
L = inductance in Henrys
C = capacitance in Farads

In this case, the resonant frequency should be within the FM radio frequency range of 88.0 to 108 MHz, which is equivalent to 88.0e6 to 108e6 Hz.

First, let's convert the frequency range to the resonant frequency range by dividing by 2π:

88.0e6 Hz / (2π) = 13.98e6 Hz
108e6 Hz / (2π) = 17.16e6 Hz

Now, we can rearrange the formula to solve for capacitance:

C = 1 / (4π^2f^2L)

Plugging in the values:

For the lowest acceptable capacitance:
C = 1 / (4π^2(13.98e6 Hz)^2(4.60e-7 H))
C ≈ 9.75e-13 F

For the highest acceptable capacitance:
C = 1 / (4π^2(17.16e6 Hz)^2(4.60e-7 H))
C ≈ 5.59e-13 F

Therefore, the range of capacitance values needed for the antenna to pick up all FM radio waves is roughly 5.59e-13 F to 9.75e-13 F.

So,

Lowest acceptable capacitance = 5.59e-13 F
Highest acceptable capacitance = 9.75e-13 F