Note on A.C. through an Inductor only and Capacitor only

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A.C. through an Inductance only

An alternative e.m.f applied to an ideal inductor of inductance L. Such circuit is known as purely inductive circuit.

\begin{align*} \text {The applied alternating e.m.f is given by} \\ E &= E_0 \sin \:\omega t \dots (i) \\ \text {The induced e.m.f across the inductor} = -L \frac {dI}{dt} \\ \text {which opposes the growth of current in the circuit.} \\ \text {As there is no potential drop across the circuit.} \\ \text{ so, we can write,} \\ E + \left (-L \frac {dI}{dt}\right ) = 0 \\ \text {or,} \: L \frac {dI}{dt} &= E \\ \text {or,} \: \frac {dI}{dt} &= \frac EL = \frac {E_0}{L} \sin \omega t \\ \text {or,}\: dI &= \frac {E_0}{L} \sin \: \omega t \: dt \\ \text {integrating both sides, we get} \\ \int dI &= \int \frac {E_0}{L} \sin \omega t \: dt \\ \text {or,} \: I &= \frac {E_0}{L} \int \sin\omega t \: dt \\ &= \frac {E_0}{L} \left (-\frac {\cos \omega t}{\omega } \right ) = \frac {E_0}{L\omega } (-\cos \: \omega t) \\ \text {or,} \: I &= \frac {E_0}{L\omega } \sin (\omega t - \pi /2) \\ [\because \sin(\omega t - \pi /2) = - \cos \omega t] \\ \therefore I &= I_0 \sin (\omega t - \pi /2) \dots (ii) \\ \text {where} \frac {E_0}{L\omega } = I_0 \: \text {is peak value of a.c.} \\ \end{align*}

Comparing equation (i) and (ii), we find that in an a.c. circuit containing L only, current I lag behind the voltage E by a phase angle of 90o.

Inductive Reactance (XL)

Comparing $$I_0 = \frac {E_0}{L\omega }$$ with $$I_0 = \frac {E_0}{R}$$, we conclude that $$(L\omega )$$ has the dimension of resistance. The term $$(L\omega )$$ is known as inductive reactance represented by XL.

The inductive reactance is the effective opposition offered by the inductor to the flow of current in the circuit.

Thus, $$X_L = \omega L = 2\pi fl$$

where f is frequency of a.c. supply.

In d.c. circuits, f = 0

$$\therefore X_L = 0$$

i.e A pure inductance offers zero resistance to d.c. further, $$X_L \propto f$$, i.e. higher the frequency of a.c., more is the inductive reactance.

\begin{align*} \text {i.e.} \: X_L &= \omega L \rightarrow \frac {1}{\text {sec}} \: (\text {henry}) \\ &= \frac {1}{\text {sec}} \frac {\text {volt}}{\text {amp}/\text {sec}} &= \text {ohm} \\ \end{align*}

A.C. through a Capacitor Only

The applied alternating e.m.f. across the capacitor is given by

\begin{align*} E &= E_0 \: \sin \omega t \dots (i) \\ \text {Let q be the charge on the capacitor at any instant.} \\ \text {So, potential difference across the capacitor,} \: V = \frac qc \\ \text {or,} \: E &= \frac qc [\therefore V = E] \\ \text {or,} q &= E.C. = E_0 \: \sin \: \omega t \: \\ \text {or,} I &= \frac {dq}{dt} = \frac {d}{dt} (E_0 \: \sin \: \omega t c ) \\ \text {or,} I &= E_0 \: c\: (\sin \omega t \: c) \\ \text {or,} \: I &= E_0 \: c (\cos \omega t) \omega \\ \text {or,} \: I &= \frac {E_0}{1/c\omega }\cos \omega t \\ \text {or,} \: I &= \frac {E_0}{1/c\omega} \sin\: (\omega t + \pi/2) \\ \therefore I &= I_0 \sin (\omega t + \pi/2) \dots (ii) \\ \text {where} I = \frac {E_0}{1/c\omega} = I_0 \: \text {is peak value of a.c.} \\ \end{align*}

Capacitive Reactance (Xc)

Comparing $$I_0 \frac {E_0}{1/c\omega }$$ with $$I_o = \frac {E_o}{R}$$, we conclude that $$\left (\frac {1}{c\omega } \right )$$ has the dimension of resistance. The term $$\left (\frac {1}{c\omega } \right )$$ is known as capacitive reactance $$X_c$$. The capacitive reactance is the effective opposition offered by a capacitor to the flow of current in the circuit. Its unit is ohm in SI-system.

\begin{align*} \text {Thus,} \: \: \: X_C &= \frac {1}{\omega c} = \frac {1}{2\pi fc} \\ \text {where f is frequency of a.c. supply} \\ \text {In a d.c. circuit, }\: f= 0, \\ \therefore X_c &= \infty \\ \text {i.e. a condenser will block d.c.} \\ \text {The unit of} \: X_c\: \text {can be deduced as:} \\ X_c &= \frac {1}{\omega c} = \text {sec} \frac {1}{\text {farad}} \\ &= \text {sec} \frac {1}{\text {coloumb}/ \text {volt}} = \frac {\text {volt sec}}{\text {Amp sec}} = \text {ohm} \\ \text {Hence} \: X_c \: \text {is measured in ohm.} \\ \end{align*}

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel.Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.

An alternative e.m.f applied to an ideal inductor of inductance L. Such circuit is known as a purely inductive circuit.

The inductive reactance is the effective opposition offered by the inductor to the flow of current in the circuit.

The capacitive reactance is the effective opposition offered by a capacitor to the flow of current in the circuit.

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