Question

An inductance of 2.0 H, a capacitance of 18μF and a resistance of 10 kΩ is connected to an AC source of 20 V with adjustable frequency.
(a) What frequency should be chosen to maximize the current in the circuit?
(b) What is the value of this maximum current?

Answer 1

Given:
The inductance of inductor, L = 2.0 H
The capacitance of capacitor, C = 18 μF
The resistance of resistor, R = 10 kΩ
The voltage of AC source, E = 20 V
(a) In an LCR circuit, the current is maximum when reactance is minimum, which occurs at resonance, i.e. when capacitive reactance becomes equal to the inductive reactance,i.e.
 X_L = X_C
`omegaL^2 = 1/{omegaC}`
`rArr omega^2 = 1/{LC} = 1/(2xx18xx10^-6)`

`rArr omega^2 = 10^6/6`

`rArr omega = 10^3/6`

`rArr 2pif = 10^3/6 `

`rArr = f = 1000/(6xx2pi) = 26.539` Hz = 27 Hz

(b) At resonance, reactance is minimum.
     Minimum Reactance, Z = R
     Maximum current (I) is given by,
  `"I" = "E"/"R"`

`rArr "I"=20/(10xx10)`

`rArr "I" = (2"A")/10^3  = 2 "mA"`