Question 190: AC fundamentals

Question

In a series LCR circuit, a resistor of $300 \Omega$, a capacitor of 25 nF and an inductor of 100 mH are used. For maximum current in the circuit, the angular frequency of the ac source is _________ $	imes 10^4$ radians $\mathrm{s}^{-1}$

Accepted Answer

$$ \omega = \frac{1}{\sqrt{LC}} $$
For a series LCR circuit, the maximum current occurs at resonance, where the inductive reactance equals the capacitive reactance. Given the values:
Inductance: $$ L = 100 \, 	ext{mH} = 0.1 \, 	ext{H} $$
Capacitance: $$ C = 25 \, 	ext{nF} = 25 	imes 10^{-9} \, 	ext{F} $$
we first calculate the product $$ LC $$:
$$ LC = 0.1 	imes 25 	imes 10^{-9} = 2.5 	imes 10^{-9} $$
Next, compute the square root of the product:
$$ \sqrt{LC} = \sqrt{2.5 	imes 10^{-9}} $$
Recognize that:
$$ \sqrt{2.5 	imes 10^{-9}} = \sqrt{2.5} 	imes \sqrt{10^{-9}} \approx 1.581 	imes 10^{-4.5} $$
Since $$ 10^{-4.5} = 3.162 	imes 10^{-5} $$, we have:
$$ \sqrt{LC} \approx 1.581 	imes 3.162 	imes 10^{-5} \approx 5.0 	imes 10^{-5} $$
Now, the angular frequency at resonance becomes:
$$ \omega = \frac{1}{5.0 	imes 10^{-5}} = 2.0 	imes 10^{4} \, 	ext{radians/s} $$
Thus, the angular frequency of the AC source for maximum current in the circuit is:
$$ \omega = 2 	imes 10^4 \, 	ext{radians/s} $$

AC fundamentals

In a series LCR circuit, a resistor of 300 \Omega, a capacitor of 25 nF and an inductor of 100 mH are used. For maximum current in the circuit, the angular frequency of the ac source is _________ \times 10^4 radians \mathrm{s}^{-1}

Integer / Numeric Answer Type