Answer :
The value of the capacitance [tex]\( C_2 \)[/tex] that should be added to the circuit so that the circuit will have a resonant frequency that matches the generator frequency is [tex]\( 1.346 \, \mu\text{F} \).[/tex]
To find the value of the capacitance [tex]\( C_2 \)[/tex] that should be added to the circuit so that the circuit will have a resonant frequency that matches the generator frequency, we can use the formula for resonant frequency in an RLC circuit:
[tex]\[ f_0 = \frac{1}{2\pi \sqrt{LC}} \][/tex]
Capacitance of the original circuit [tex]\( C_1 = 3.7 \, \mu\text{F} \)[/tex]
Resonant frequency of the original circuit [tex]\( f_{01} = 5.3 \, \text{kHz} \)[/tex]
Generator frequency[tex]\( f_g = 3.5 \, \text{kHz} \)[/tex]
We can rearrange the resonant frequency formula to solve for the inductance L:
[tex]\[ L = \frac{1}{(2\pi f_0)^2 C} \][/tex]
We'll first calculate the inductance of the original circuit using the given values.
[tex]\[ L_1 = \frac{1}{(2\pi \times 5.3 \times 10^3)^2 \times 3.7 \times 10^{-6}} \][/tex]
[tex]\[ L_1 = 8.537 \times 10^{-4} \, \text{H} \][/tex]
Now, we'll use the inductance [tex]\( L_1 \)[/tex] and the generator frequency [tex]\( f_g \)[/tex] to calculate the capacitance [tex]\( C_2 \)[/tex] needed for the new resonant frequency:
[tex]\[ C_2 = \frac{1}{(2\pi f_g)^2 L_1} \][/tex]
[tex]\[ C_2 = \frac{1}{(2\pi \times 3.5 \times 10^3)^2 \times 8.537 \times 10^{-4}} \][/tex]
[tex]\[ C_2 = 1.346 \times 10^{-6} \, \text{F} \][/tex]
To find the value of the capacitance C2 to match the generator frequency in the RCL circuit, calculate the resonant frequency first and then determine the new capacitor value by incorporating the given parameters.
The capacitance in a series RCL circuit is C1 = 3.7 μF, and the resonant frequency isf01 = 5.3 kHz. The generator frequency is 3.5 kHz.
- Calculate the resonant frequency using the formula f0 = 1 / (2π√(LC)).
- Determine the new capacitor value C2 by using the formula f0 = 1 / (2π√((C1 + C2)L)).
- Substitute the given values to find the capacitance C2.
The resonant frequency of a series RLC circuit is given by:
f_0 = 1 / (2π√(LC) )
Where:
f_0 = the resonant frequency,
L = the inductance, and
C = the total capacitance.
In the given circuit, the capacitance is C_1 = 3.7 μF, and the resonant frequency is f_(01) = 5.3 kHz. rearrange the formula above to solve for L:
L = 1 / (4π²C1f01²)
Substituting the values into the formula:
L = 1 / (4π² × 3.7 ×10^-6 × (5.3 ×10³)²)
L ≈ 6.34 H (rounded to two decimal places)
Now, to calculate the new value of capacitance C_2, we rearrange the formula for the resonant frequency:
f_0 = 1 / (2π√((LC) ))
the new resonant frequency f0 to be 3.5 kHz. Rearranging the formula and substituting the known values:
C2 = 1 / (4π² × L × f02²)
C2 = 1 / (4π² × 6.34 × (3.5 ×10^3)²)
C2 ≈ 3.46 μF (rounded to two decimal places)
Therefore, the value of capacitance C_2 that should be added to the circuit is approximately 3.46 μF.
