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AC Circuit Containing Inductance and Resistance in Series

Let an alternating emf E = E0ejωt be applied to a circuit having an inductance L and a non-inductive resistance R in series.
The potential drop across the inductance is
        VL = jωL I
The potential drop across resistance is
        VR = RI
Here, I is the current at any instant t.
. :         E = jωL I + RI
Current in the circuit,
        I = E/R + jωL                                    … (1)
But        I = E/Z                                        … (2)
Impedance of R-L circuit,
        Z = R + jωL                                    … (3)
. :        I = E0 ejωt/√(R2 + ω2L2) e                     (where tan θ = ωL/R)
        = E0 /√R2 + ω2L2) ej(ωt – 0)                            … (4)
        = I0 ej(ωt-θ)                                     … (5)
Here,        I0 = E0 /√R2 + ω2L2)                                 … (6)

It represents the peak value of the current through the circuit.

The impedance Z of the circuit is given by the term
        √(R2 + ω2L2).

The current lags in phase behind the emf by an angle θ = tan-1 ωL/R.
The variation of instantaneous values of emf and current with time are represented graphically.

Inductance and Resistance

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