What capacitance in needed in series with an 785 µH inductor to form a circuit that radiates a wavelength of 199 m?
L = V/F = 199 m.
F = V/199 = 300*10^6/199=1.51*10^6 cy/s=
1.51*10^6 Hz.
WL = 1/WC
C = 1/W^2*L
C=1/(6.28*1.51*10^6)^2*785*10^-6=0.0142
nanofarads.
To determine the capacitance needed in series with an inductor to form a circuit that radiates a specific wavelength, you can use the formula for the resonance frequency of an LC circuit:
f = 1 / (2π√(LC))
Here, f represents the frequency corresponding to the desired wavelength and L is the inductance of the inductor. In this case, the wavelength is 199 m.
First, let's convert the wavelength from meters to frequency in Hz. The speed of light (c) is approximately 3 x 10^8 meters per second. The frequency (f) is equal to the speed of light divided by the wavelength (λ):
f = c / λ
f = (3 x 10^8 m/s) / (199 m)
f ≈ 1.51 x 10^6 Hz
Now, we have the desired frequency (f), and we know the inductance (L = 785 µH = 785 x 10^(-6) H).
Rearranging the resonance frequency formula, we can solve for the capacitance (C):
C = 1 / (4π^2f^2L)
Substituting the values:
C = 1 / (4π^2 x (1.51 x 10^6 Hz)^2 x 785 x 10^(-6) H)
Calculating this expression will give you the required capacitance value in farads (F).