A series LCR circuit with inductance 10H, capacitance 10 μF, resistance 50Ω is connected to an ac source of voltage, V = 200 sin(100t) volt. If the resonant frequency of the LCR circuit is v_o and the frequency of the ac source is v, then:

Concept: 

The resonant frequency of LCR circuit = \(\upsilon_{0} = \frac{1}{2π }\sqrt{\frac{1}{LC}} \)     ----- (1)

The frequency of ac source is = v = ω/2π    ----- (2)

Here, V = 200 sin (100 t) = 200 sin (ωt)

Calculation:

Given: L = inductance = 10 H , resistance R = 50 Ω  , capacitance (c) = 10 μF, angular velocity = ω = 100 rad/s

V = 200 sin (100 t)

Resonant frequency = ?, the frequency of AC source = ?

From equation (1) we get: 

Resonant frequency, \(\upsilon_{0} = \frac{1}{2π }\sqrt{\frac{1}{LC}} \)  ⇒ \(\upsilon_{0} = \frac{1}{2\times 3/14}\sqrt{\frac{1}{10\times 10\times 10^{-6}}} = 50/π \)

ν_o = 50/π      ----- (3) 

The frequency of ac source = v = ω/2π = 100/2π = 50/π 

Hence, Option 3) is correct.