Question

An inductor having some resistance is connected to an $AC$ source. Which of the following quantities has zero average value over a complete cycle?

• A. current
• B. induced emf
• C. Joule heat
• D. magnetic energy stored in the inductor

Answer 1

Answer: (A), (B)

The correct options are
A current
B induced emf

Let the $AC$ source be $V_{AC}=V\sin (\omega t+\varphi )$ with time period $\tau$ s.

Resistance $=R,$Inductance$=L$

Current in the circuit would be of the form $I_{AC}=Isin\left(\omega t\right)$
Average value over a complete cycle of:

Current $=\frac{1}{\tau }{\int }_0^{\tau }I\sin \left(\omega t\right)dt=\frac{I}{\tau }{\int }_0^{\tau }\sin \left(\omega t\right)dt=0$

Induced emf $=\frac{1}{\tau }{\int }_0^{\tau }-L\frac{d{I}_{AC}}{dt}dt=-\frac{L}{\tau }{\int }_0^{\tau }d{I}_{AC}=-\frac{L}{\tau }{\left[I\sin \right(\omega t\left)\right]}_0^{\tau }=0$

Joule heat $=\frac{1}{\tau }{\int }_0^{\tau }{I}_{AC}^{2}Rdt=\frac{I^{2}R}{\tau }{\int }_0^{\tau }{sin}^2\left(\omega t\right)dt=\frac{I^{2}R}{2\tau }\ne 0$

Magnetic energy stored in the inductor = $\frac{1}{\tau }{\int }_0^{\tau }\frac{1}{2}L{I}_{AC}^{2}dt=\frac{L{I}^2}{2\tau }{\int }_0^{\tau }{sin}^2\left(\omega t\right)dt=\frac{{LI}^2}{4\tau }\ne 0$