The oscillator of a 96.1-MHz FM station has an inductance of 1.8μH. What value must the capacitance be?
1.5e-12 is wrong?
Xc = Xl = 6.28*96.1*10^6*1.8*10^-6 = 1086.3 Ohms.
C = 1/6.28*F*C.
C = 1/(6.28*96.1*10^6*1086.3) = 1.53*10^-12 Farads = 1.53 uuF = 1.53 picofarads.
Well, maybe 1.5e-12 is just a bit too shy to handle the frequency! It's like trying to fit a dinosaur in a bathtub. You need a capacitive tank that can handle the wild vibrations of a 96.1-MHz FM station.
So, let's do some calculations. The formula for calculating the capacitance required for an oscillator is:
f = 1 / (2π√(LC))
Where f is the frequency, L is the inductance, and C is the capacitance.
Plugging in the values given:
96.1e6 = 1 / (2π√(1.8e-6 × C))
Solving for C:
C = (1 / (2π × 96.1e6)²) / (1.8e-6)
C ≈ 1.7506e-12
So, it looks like the value of 1.5e-12 isn't quite right. You need a capacitance of approximately 1.7506e-12 to keep that FM station swinging!
To find the value of capacitance needed, we can use the formula for the resonant frequency of an LC circuit:
f = 1 / [2 * π * √(L * C)]
Where:
f = frequency in Hertz
π = Pi (approximately 3.14159)
L = inductance in Henrys (H)
C = capacitance in Farads (F)
Given:
Frequency (f) = 96.1 MHz = 96.1 * 10^6 Hz
Inductance (L) = 1.8 μH = 1.8 * 10^(-6) H
Substituting these values into the equation, we can solve for capacitance (C):
96.1 * 10^6 = 1 / [2 * 3.14159 * √(1.8 * 10^(-6) * C)]
Let's solve this equation to find the value of C:
96.1 * 10^6 = 1 / [2 * 3.14159 * √(1.8 * 10^(-6) * C)]
Multiplying both sides of the equation by 2 * 3.14159:
192.2 * 3.14159 * √(1.8 * 10^(-6) * C) = 1
Simplifying further:
602.8236 * √(1.8 * 10^(-6) * C) = 1
Taking the square root on both sides:
√(1.8 * 10^(-6) * C) = 1 / 602.8236
Squaring both sides:
1.8 * 10^(-6) * C = (1 / 602.8236)^2
Simplifying:
C = (1 / (1.8 * 10^(-6))) * ((1 / 602.8236)^2)
Calculating this expression gives us:
C ≈ 4.49 * 10^(-12) F
Therefore, the value of capacitance (C) needed is approximately 4.49 picofarads (pF), not 1.5 * 10^(-12) F.
To find the value of the capacitance needed for the oscillator, we can use the formula for the resonant frequency of an LC circuit:
f = 1 / (2 * π * √(LC))
Where:
f = Frequency (96.1 MHz = 96.1 * 10^6 Hz)
L = Inductance (1.8 μH = 1.8 * 10^-6 H)
C = Capacitance (unknown)
Rearranging the formula to solve for C:
C = 1 / (4 * π^2 * L * f^2)
Now, let's calculate the capacitance using the given values:
C = 1 / (4 * 3.14159265359^2 * 1.8 * 10^-6 * (96.1 * 10^6)^2)
C ≈ 1.5184 * 10^-12 F
So, the value of the capacitance needed for the oscillator is approximately 1.5184 picofarads (pF), not 1.5 * 10^-12 F.