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What is the value of inductance L for which the current is a maximum in series LCR circuit with C = 10 μF and ω = 1000s−1 ?
- a)100mH
- b)1mH
- c)10 mH
- d)Cannot be calculated unless R is known
Correct answer is option 'A'. Can you explain this answer?
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Understanding the Series LCR Circuit
In a series LCR (Inductor-Capacitor-Resistor) circuit, the current reaches its maximum value when the circuit is in resonance. Resonance occurs when the inductive reactance equals the capacitive reactance.
Key Formula for Resonance
At resonance, the following condition holds:
XL = XC
Where:
- XL = Inductive Reactance = ωL
- XC = Capacitive Reactance = 1/(ωC)
Given Values
- Capacitance (C) = 10 µF = 10 x 10^-6 F
- Angular Frequency (ω) = 1000 s^-1
Calculating Capacitive Reactance
To find the capacitive reactance:
XC = 1/(ωC)
XC = 1/(1000 * 10 x 10^-6)
XC = 1/(0.01) = 100 Ω
Finding Inductance (L)
Now, we set XL equal to XC to find L:
XL = ωL = XC
1000L = 100
Solving for L:
L = 100/(1000) = 0.1 H = 100 mH
Conclusion
Thus, the inductance L for which the current is a maximum in the series LCR circuit is 100 mH, confirming that the correct answer is option 'A'.
This analysis shows that to achieve maximum current, the inductive and capacitive reactances must balance, leading to the derived value of inductance.
In a series LCR (Inductor-Capacitor-Resistor) circuit, the current reaches its maximum value when the circuit is in resonance. Resonance occurs when the inductive reactance equals the capacitive reactance.
Key Formula for Resonance
At resonance, the following condition holds:
XL = XC
Where:
- XL = Inductive Reactance = ωL
- XC = Capacitive Reactance = 1/(ωC)
Given Values
- Capacitance (C) = 10 µF = 10 x 10^-6 F
- Angular Frequency (ω) = 1000 s^-1
Calculating Capacitive Reactance
To find the capacitive reactance:
XC = 1/(ωC)
XC = 1/(1000 * 10 x 10^-6)
XC = 1/(0.01) = 100 Ω
Finding Inductance (L)
Now, we set XL equal to XC to find L:
XL = ωL = XC
1000L = 100
Solving for L:
L = 100/(1000) = 0.1 H = 100 mH
Conclusion
Thus, the inductance L for which the current is a maximum in the series LCR circuit is 100 mH, confirming that the correct answer is option 'A'.
This analysis shows that to achieve maximum current, the inductive and capacitive reactances must balance, leading to the derived value of inductance.
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