4 Marks Questions

Question 14 Marks

Draw a labelled diagram of an a.c. generator. Obtain the expression for the emf induced in the rotating coil of N turns each of cross-sectional area A in the presence of a magnetic field $\overrightarrow{ B }$.

Answer

The labelled diagram of an a.c. generator is given in adjacent figure :
Principle : An a.c. generator is based on the pheno-menon of electromagnetic induction. If a rectangular armature coil rotates about its axis in a uniform magnetic field, the magnetic flux linked with the coil changes and an induced emf is set up in the coil. The direction of induced current is given by Fleming's right hand rule.

Expression for induced emf: i.e.Working of Generator: When an armature coil ABCD of N turns and each turn enclosing area A is placed in a uniform magnetic field of strength B making an angle $\theta$ from normal to the direction of magnetic field, magnetic flux linked with the coil is$\Phi= NBA \cos \theta$.
As the coil is rotated about its own axis with an angular speed $\omega$ then value of angle $\theta=\omega t$ and hence, magnetic flux changes and an induced emf is developed across the ends of coil.
As $F = NBA \cos \omega t$
∴ Induced emf
$e=-\left(\frac{d \Phi}{d t}\right)=\frac{d}{d t}( NBA \cos \omega t)$
$=- NBA \frac{d}{d t}(\cos \omega t)$
$=- NBA \omega(-\sin \omega t)= NBA \omega \sin \omega t$
$=e_0 \sin \omega t$
thus $e=e_0 \sin \omega t$
where $e_0= NBA \omega$
= maximum (peak) value of induced emf
It is obvious that induced emf is sinusoidal in nature as shown in the following figure :

If coil is rotating at the rate of $f$ cycles per second then $\omega=2 \pi f \Rightarrow e=e_0 \sin (2 \pi f t)$, where $e_0=2 \pi f NBA$.

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Question 24 Marks

Establish the relation between mean value and peak value of AC.

Answer

Let the instantenous value of alternating current be
$i=I_{0} sin \omega t$
$I_{0}=$ peak value of current,
$t=$ time at any instant.
Also, $\omega=\frac{2\pi}{T}$, $T=$ Time period.
$i=\frac{dQ}{dt}\Rightarrow dQ=idt$
$\int_{0}^{Q}dQ=\int_{0}^{T/2}idt$
$Q|_{0}^{Q}=\int_{0}^{T/2}I_{0}sin \omega tdt$
$Q=\frac{-I_{0}}{\omega}|cos\frac{2\pi}{T}.t|_{0}^{T/2}$
$Q=-\frac{I_{0}}{\omega}|cos\frac{2\pi}{T}\times\frac{T}{2}-cos\frac{2\pi}{T}.0|$
$Q=-\frac{I_{0}}{\omega}(cos \pi-cos 0) = -\frac{I_{0}}{\omega}\times-2 = \frac{2I_{0}}{\omega}$
Also, $Q=I_{m}\times\frac{T}{2}$
from (1) and (2)
$I_{m}\times\frac{T}{2}=\frac{2I_{0}}{2\pi} \times T$
$I_{m}\times\frac{1}{2}=\frac{2I_{0}}{2\pi} \times T$

$I_{m}=\frac{2I_{0}}{\pi}= 0.637 I_{0}$
Mean value of current, $I_{m}=0.637$ times peak value of current.

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Question 34 Marks

Explain the use of chock coil.

Answer

Chock Coil: It is a device which is used in a.c. circuits for varrying the value of current in the circuit which involves little loss of energy. It consists of a high inductance coil which is made of thick insulated copper wire wound closely in a large number of turns over a laminated core made of soft iron.

Since thick copper wire is taken, its resistance R is almost zero, but due to large number of turns of the wire and high permeability of the iron-core, its inductance (L) is very high. The choke coil, therefore, offers a large inductive reactance $X_{L}=\omega L=2\pi fL$ and contributes to the impedance $Z=\sqrt{R^{2}+(\omega L)^{2}}$ of the circuit and current is reduced in the circuit. There is an arrangement to insert the core into the coil to any desired length. More is the length of the core in the coil greater will be inductance (L) of the coil, hence greater will be the impedance. In this way the choke coil can be used to vary the current in A.C. circuits.
The choke coil is put in series across an electrical appliance (say an electric bulb) of resistance R and is connected to an a.c. source. This forms an L-R circuit.
The average power dissipated per cycle in the circuit is given by
$P_{av}=V_{rms}I_{rms}cos \phi$ ...(1)
Hence power factor
$cos \phi=\frac{R}{Z}=\frac{R}{\sqrt{R^{2}+(\omega L)^{2}}}$ ...(2)
Ohmic resistance of the choke coil itself is nearly zero and its inductance L is very high, therefore $R<\omega L$ Hence $cos \phi\approx 0.$ Thus, the average power dissipated in choke coil will be nearly zero i.e. negligible.
As $Z=\sqrt{R^{2}+\omega^{2}L^{2}}$ is very large, so current is reduced without appreciable loss of energy i.e. without wastage of power. Thus working principle of choke coil is the principle of wattless current.
In practice the ohmic resistance of the choke coil is not exactly zero. That is why, in practice, some electrical energy is lost as heat. In addition to it, energy is also lost due to hysteresis-loss in the iron-core of the choke-coil. The loss of energy due to eddy currents is reduced by using laminated iron-core.

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