Question

If an alternating emf E = E₀ sin ωt is applied to a series LCR circuit (Fig. 4.7) the resulting alternating current in the circuit is given by (steady-state)

i = I sin (ωt – ϕ )

(a) Find the current amplitude I and the phase constant ϕ . 

(b) Show that the current amplitude I has the maximum value (resonance)

when ω = ω₀, where ω₀ =1/√LC is the natural frequency.

(c) Show that the value of I and the phase angle ϕ at resonance are E₀/R and zero respectively.

Answer 1

(a) The equation for the current i in the LCR circuit can be written as

where the charge q on the capacitor is given by

We consider a steady-state solution of Eqn. (4.44) in the form (after the alternating emf has been applied for some time)

Substituting Eqn. (4.46) into Eqn. (4.44) and equating the coefficients of sin ωt and cos ωt from both sides, we get

(b) From Eqn. (4.49), we find that the maximum value of I occurs when

(c) The value of I at resonance is I₀ = E₀/R and the phase angle φ is zero at resonance.