The phase difference between the current and voltage in L-C-R circuit at resonance is:

CONCEPT:

• The ac circuit containing the capacitor, resistor, and the inductor is called an LCR circuit.
• For a series LCR circuit, the total potential difference of the circuit is given by:

\(V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)

Where VR = potential difference across R, VL =  potential difference across L and VC =  potential difference across C

• For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)
Where R = resistance, XL =induvtive reactance and XC = capacitive reactive

• The resonant frequency of a series LCR circuit is given by

\(⇒ \nu =\frac{1}{2\pi\sqrt{LC}}\)

If ϕ is the phase difference between the current and voltage in L-C-R circuit, then,

\(⇒ tanϕ=\frac{X_L-X_C}{R}=\frac{V_L-V_C}{V_R}\)

​Reactance: 

• It is basically the inertia against the motion of the electrons in an electrical circuit.

• There are two types of reactance:

  1. Capacitive reactance (XC) (Ohms is the unit)

  2. Inductive reactance (XL) (Ohms is the unit)​

​CALCULATION:

We know that for a series LCR circuit, the resonating condition is given by:

⇒ X_L = X_C = X     -----(1)

If ϕ is the phase difference between the current and voltage in L-C-R circuit, then,

\(⇒ tanϕ=\frac{X_L-X_C}{R}\)     -----(2)

By equation 1 and equation 2,

\(⇒ tanϕ=\frac{X_L-X_C}{R}\)

\(⇒ tanϕ=\frac{X-X}{R}\)

⇒ tanϕ = 0

⇒ ϕ = 0

• Hence, option 1 is correct.