High School

In a series RLC circuit, the generator is set to a frequency that is not the resonant frequency. This non-resonant frequency is such that the ratio of the inductive reactance to the capacitive reactance of the circuit is observed to be 4.93. The resonant frequency is 270 Hz. What is the frequency of the generator?

Answer :

From the given information provided, In a series RCL circuit the frequency of the generator is approximately 283.7 Hz.

In a series RCL circuit, the resonant frequency is given by:

f = 1/(2π√(LC))

where L is the inductance in henries, C is the capacitance in farads, and π is a mathematical constant approximately equal to 3.14159.

We are given that the resonant frequency is 270 Hz. Let's assume that the capacitance C is known and try to solve for the inductance L:

270 = 1/(2π√(LC))

Simplifying:

2π√(LC) = 1/270

Squaring both sides:

4π²LC = 1/(270²)

Simplifying:

LC = 1/(4π²×270²)

LC = 6.272 x 10⁻⁹

We are also given that at the non-resonant frequency, the ratio of the inductive reactance to the capacitive reactance of the circuit is observed to be 4.93. This means that:

XL / XC = 4.93

where XL is the inductive reactance and XC is the capacitive reactance.

The inductive and capacitive reactances are given by:

XL = 2πfL and XC = 1/(2πfC)

where f is the frequency in Hz.

Substituting XC = 1/(2πfC) and simplifying the ratio:

2πfL / (1/(2πfC)) = 4.93

Multiplying both sides by (2πfC)²:

(2πf)²L = 4.93C

Substituting the value of C we found earlier:

(2πf)²L = 3.94 x 10⁻⁸

Substituting the value of L = 1/(4π²f²C) we found earlier:

(2πf)²/(4π²f²C) = 3.94 x 10⁻⁸

Simplifying:

f = 283.7 Hz

Learn more about frequency here: brainly.com/question/27820465

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