A condenser of 250 µF is connected in parallel to a coil of inductance 0.16 mH and effective resistance 20 W. The resonant frequency will be ______ (give Answer in KHz)Correct answer is '80º KHz'. Can you explain this answer?

Question

A condenser of 250 &#181;F is connected in parallel to a coil of inductance 0.16 mH and effective resistance 20 W. The resonant frequency will be ______ (give Answer in KHz)Correct answer is '80º KHz'. Can you explain this answer?

Answer

Given information:
- Condenser of 250 F
- Coil of inductance 0.16 mH and effective resistance 20 W

To find:
Resonant frequency in KHz

Solution:

1. Resonant Frequency:
The resonant frequency of an RLC circuit can be calculated using the formula:

f_res = 1 / (2π√(LC))

Where:
f_res = Resonant frequency
L = Inductance
C = Capacitance

2. Capacitive Reactance:
The capacitive reactance (Xc) of the condenser can be calculated using the formula:

Xc = 1 / (2πfC)

Where:
Xc = Capacitive reactance
f = Frequency
C = Capacitance

3. Inductive Reactance:
The inductive reactance (Xl) of the coil can be calculated using the formula:

Xl = 2πfL

Where:
Xl = Inductive reactance
f = Frequency
L = Inductance

4. Impedance:
The impedance (Z) of the RLC circuit can be calculated using the formula:

Z = √(R^2 + (Xl - Xc)^2)

Where:
Z = Impedance
R = Resistance
Xl = Inductive reactance
Xc = Capacitive reactance

5. Resonance condition:
At resonance, the capacitive and inductive reactances are equal, i.e., Xl = Xc.

6. Calculation:
Using the given values, we can calculate the resonant frequency as follows:

Xc = 1 / (2πfC)
Xl = 2πfL

At resonance, Xl = Xc, so we can equate the above two equations:

2πfL = 1 / (2πfC)

Simplifying the equation further:

f^2 = 1 / (4π^2LC)

Taking the square root of both sides:

f = 1 / (2π√(LC))

Substituting the given values:

f = 1 / (2π√(0.16 mH * 250 F))
f ≈ 1 / (2π√(0.16 * 0.00025))
f ≈ 1 / (2π√(0.00004))
f ≈ 1 / (2π * 0.00632)
f ≈ 1 / (0.0398)
f ≈ 25.13 Hz

Converting the frequency to KHz:

f ≈ 25.13 * 10^-3 KHz
f ≈ 0.02513 KHz

Therefore, the resonant frequency is approximately 0.02513 KHz or 25.13 Hz.

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