Answer :
Final answer:
The frequency of the generator in the series RC circuit is approximately 1.03 Hz.
Explanation:
In a series RC circuit, the resonant frequency is the frequency at which the inductive reactance (XL) and the capacitive reactance (XC) cancel each other out, resulting in a purely resistive circuit. The resonant frequency can be calculated using the formula:
fr = 1 / (2π√(LC))
where fr is the resonant frequency, L is the inductance, and C is the capacitance.
In this case, the resonant frequency is given as 230 H. We can use this information to find the frequency of the generator.
First, we need to determine the values of XL and XC. Since the circuit is not at the resonant frequency, the ratio of XL to XC is observed to be 5.12. This means that XL is 5.12 times greater than XC.
Using the formula for XL and XC:
XL = 2πfL
XC = 1 / (2πfC)
where f is the frequency, we can set up the following equation:
5.12 = XL / XC = (2πfL) / (1 / (2πfC))
Simplifying the equation:
5.12 = 4π²f²LC
Since we know the resonant frequency (230 H), we can substitute the values into the equation:
5.12 = 4π²(230)²LC
Solving for LC:
LC = 5.12 / (4π²(230)²)
Now, we can substitute the value of LC into the formula for the resonant frequency:
230 = 1 / (2π√(LC))
Solving for f:
f = 1 / (2π√(LC))
Substituting the value of LC:
f = 1 / (2π√(5.12 / (4π²(230)²)))
Simplifying the expression:
f ≈ 1.03 Hz
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Final answer:
The frequency of the generator in the series RC circuit is approximately 1.03 Hz.
Explanation:
In a series RC circuit, the resonant frequency is the frequency at which the inductive reactance (XL) and the capacitive reactance (XC) cancel each other out, resulting in a purely resistive circuit. The resonant frequency can be calculated using the formula:
fr = 1 / (2π√(LC))
where fr is the resonant frequency, L is the inductance, and C is the capacitance.
In this case, the resonant frequency is given as 230 H. We can use this information to find the frequency of the generator.
First, we need to determine the values of XL and XC. Since the circuit is not at the resonant frequency, the ratio of XL to XC is observed to be 5.12. This means that XL is 5.12 times greater than XC.
Using the formula for XL and XC:
XL = 2πfL
XC = 1 / (2πfC)
where f is the frequency, we can set up the following equation:
5.12 = XL / XC = (2πfL) / (1 / (2πfC))
Simplifying the equation:
5.12 = 4π²f²LC
Since we know the resonant frequency (230 H), we can substitute the values into the equation:
5.12 = 4π²(230)²LC
Solving for LC:
LC = 5.12 / (4π²(230)²)
Now, we can substitute the value of LC into the formula for the resonant frequency:
230 = 1 / (2π√(LC))
Solving for f:
f = 1 / (2π√(LC))
Substituting the value of LC:
f = 1 / (2π√(5.12 / (4π²(230)²)))
Simplifying the expression:
f ≈ 1.03 Hz
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