The capacitance in a series RCL circuit is C1 = 1.2 μF, and the corresponding resonant frequency is f01 = 7.2 kHz. The generator frequency is 1.2 kHz. What is the value of the capacitance C2 that should be added to the circuit so that the circuit will have a resonant frequency that matches the generator frequency?
To find the value of the capacitance C2 that should be added to the circuit, we need to determine the resonant frequency for the new circuit configuration.
In a series RCL circuit, the resonant frequency can be calculated using the formula:
f0 = 1 / (2π √(L C))
where f0 is the resonant frequency, L is the inductance, and C is the capacitance.
Given that the initial capacitance C1 is 1.2 μF and the corresponding resonant frequency f01 is 7.2 kHz, we can rearrange the formula to solve for the inductance:
f01 = 1 / (2π √(L C1))
Simplifying the equation:
(2π √(L C1)) = 1 / f01
Squaring both sides:
4π²LC1 = 1 / (f01)²
Now, let's substitute the given values:
4π²L(1.2 μF) = 1 / (7.2 kHz)²
Simplifying further, converting units to get consistent values:
4π²L(1.2 × 10⁻⁶ F) = 1 / (7.2 × 10³ Hz)²
Simplifying the expression:
L = 1 / (4π² × 1.2 × 10⁻⁶ × (7.2 × 10³)²)
Calculating this expression will give us the inductance L.
Once we have the inductance value for the initial circuit, we can use the new desired resonant frequency and the same formula to find the value of the capacitance C2:
f02 = 1 / (2π √(L C2))
Now, substitute the values:
1.2 kHz = 1 / (2π √(L C2))
Simplifying:
(2π √(L C2)) = 1 / (1.2 kHz)
Squaring both sides:
4π²L C2 = 1 / (1.2 kHz)²
Now, rearrange the equation to solve for C2:
C2 = 1 / (4π²L (1.2 kHz)²)
Calculate this expression using the inductance value obtained earlier to find the value of the capacitance C2 that should be added to the circuit.