Download App

AC Circuits — Physics STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsAC Circuits4 Marks
Question
When an AC source is connected to an ideal inductor show that the average power supplied by the source over a complete cycle is zero.
✓

Answer

In an AC circuit containing only an ideal inductor, the current i lags behind the emf e by a phase angle of $\frac{\pi}{2}$ rad. Here, for $e=e_0 \sin \omega t$, we have, $i=i_0 \sin \left(\omega t-\frac{\pi}{2}\right)$
Instantaneous power, $P=$ ei
$=\left(e_0 \sin \omega t\right)\left[i_0\left(\sin \omega t \cos \frac{\pi}{2}-\cos \omega t \sin \frac{\pi}{2}\right)\right]$
$=-e_0 i_0 \sin \omega t \cos \omega t$ as $\cos \frac{\pi}{2}=0$ and $\sin \frac{\pi}{2}=1$.
Average power over one cycle,
$ P_{ av }=\frac{\text { work done in one cycle }}{\text { time for one cycle }}$
$=\frac{\int_0^T P d t}{T}$
$=\frac{\int_0^T\left[e_0 i_0 \cos \phi \sin ^2 \omega t \pm e_0 i_0 \sin \phi \sin \omega t \cos \omega t\right) d t}{T}$
$=\frac{e_0 i_0}{T}\left[\cos \phi \int_0^T \sin ^2 \omega t d t \pm \sin \phi \int_0^T \sin \omega t \cos \omega t d t\right]$
$\text { Now, } \int_0^T \sin ^2 \omega t d t=\int_0^T\left(\frac{1-\cos 2 \omega t}{2}\right) d t$
$=\int_0^T \frac{1}{2} d t-\int_0^T \frac{\cos 2 \omega t}{2} d t=\frac{T}{2}-\frac{1}{2}\left(\frac{\sin 2 \omega t}{2 \omega}\right)_0^T$
$=\frac{T}{2}-\frac{1}{4 \omega}(\sin 2 \omega T-\sin 0)$
$=\frac{T}{2}-\frac{1}{4 \omega}\left[\sin 2\left(\frac{2 \pi}{T}\right) T-0\right]$
$=\frac{T}{2}-\frac{1}{4 \omega}[0-0]=\frac{T}{2}$
Also,
$ \int_0^T \sin \omega t \cos \omega t d t=\frac{1}{2} \int_0^T \sin 2 \omega t d t=\frac{1}{2}\left[\frac{-\cos 2 \omega t}{2 \omega}\right]_0^T$
$=-\frac{1}{4 \omega}\left[\cos 2\left(\frac{2 \pi}{T}\right) T-\cos 0\right]=-\frac{1}{4 \omega}[1-1]=0 $
Hence, $P_{ av }=\frac{e_0 i_0}{T} \cos \phi \times \frac{T}{2}=\frac{e_0 i_0}{2} \cos \phi$
$ =\frac{e_0}{\sqrt{2}} \cdot \frac{i_0}{\sqrt{2}} \cos \phi $
$=  e_{ rms } i _{ rms } \cos \Phi= e _{ rms } i _{ rms }\left(\frac{R}{Z}\right) \text {, where the impedance } Z =\sqrt{R^2+\left(X_{ L }-X_{ C }\right)^2} .$
$\therefore P _{ av }=0$, i.e., the circuit does not dissipate power.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Answer the following in Detail" from AC Circuits→AC Circuits — all question types→Physics STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Show that the energy of a charged capacitor is $\frac{1}{2} CV ^2$. Also, express this in other forms.
OR
Derive an expression for the energy stored in a charged capacitor. Express it in different forms.
→
Derive an expression for the period of motion of a simple pendulum. On which factors does it depend?
→
An AC circuit consists of only an inductor of inductance 2 H. If the current is represented by a sine wave of amplitude 0.25 A and frequency 60 Hz, calculate the effective potential difference across the inductor (π = 3.142)
→
A body oscillates in SHM according to the equation $x=5 \cos \left(2 \pi t+\frac{\pi}{4}\right)$, where $x$ and $t$ are in Sl units. Calculate the
(i) displacement and
(ii) speed of the body at $t =1.5 s$.
→
A substance may be athermanous or diathermanous for certain wavelength ranges while good absorber for other wavelength ranges. Explain.
→
Define ideal simple pendulum and obtain an expression for its periodic time.
→
Determine the angle to be made with the vertical by a two-wheeler while turning on a horizontal track?
→
Calculate the rise of water inside a clean glass capillary tube of radius $0.1 mm$, when immersed in water of surface tension $7 \times 10^{-2} N / m$. The angle of contact between water and glass is zero, density of water $=1000 kg / m ^3, g =9.8 m / s ^2$.
 
→
Obtain an expression for conservation of mass starting from the equation of continuity.
→
State and explain Lenz’s law in the light of principle of conservation of energy.
→