NCERT Class 12 Books Physics Chapter 7- Alternating Current

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The Chapter Goes like this-
 

INTRODUCTION

We have so far considered direct current (dc) sources and circuits with dc sources. These currents do not change direction with time. But voltages and currents that vary with time are very common. The electric mains supply in our homes and offices is a voltage that varies like a sine function with time. Such a voltage is called alternating voltage (ac voltage) and the current driven by it in a circuit is called the alternating current (ac current)*. Today, most of the electrical devices we use require ac voltage. This is mainly because most of the electrical energy sold by power companies is transmitted and distributed as alternating current. The main reason for preferring use of ac voltage over dc voltage is that ac voltages can be easily and efficiently converted from one voltage to the other by means of transformers. Further, electrical energy can also be transmitted economically over long distances. AC circuits exhibit characteristics which are exploited in many devices of daily use. For example, whenever we tune our radio to a favourite station, we are taking advantage of a special property of ac circuits – one of many that you will study in this chapter.

AC VOLTAGE APPLIED TO A RESISTOR

Figure 7.1 shows a resistor connected to a source ε of ac voltage. The symbol for an ac source in a circuit diagram is . We consider a source which produces sinusoidally varying potential difference across its terminals. Let this potential difference, also called ac voltage, be given by
v v t = m sinω (7.1)
where vm is the amplitude of the oscillating potential difference and ω is its angular frequency. To find the value of current through the resistor, we apply Kirchhoff’s loop rule ∑ε( )t = 0 (refer to Section 3.13), to the circuit shown in Fig. 7.1 to get
v t i R m sinω =
or sin vm i t R = ω
Since R is a constant, we can write this equation as
i i t = m sinω 
where the current amplitude im is given by
m m v i R = (7.3) Equation (7.3) is Ohm’s law, which for resistors, works equally well for both ac and dc voltages. The voltage across a pure resistor and the current through it, given by Eqs. (7.1) and (7.2) are plotted as a function of time in Fig. 7.2. Note, in particular that both v and i reach zero, minimum and maximum values at the same time. Clearly, the voltage and current are in phase with each other.

We see that, like the applied voltage, the current varies sinusoidally and has corresponding positive and negative values during each cycle. Thus, the sum of the instantaneous current values over one complete cycle is zero, and the average current is zero. The fact that the average current is zero, however, does not mean that the average power consumed is zero and that there is no dissipation of electrical energy. As you know, Joule heating is given by i 2R and depends on i 2 (which is always positive whether i is positive or negative) and not on i. Thus, there is Joule heating and dissipation of electrical energy when an ac current passes through a resistor.

REPRESENTATION OF AC CURRENT AND VOLTAGE BY ROTATING VECTORS — PHASORS

In the previous section, we learnt that the current through a resistor is in phase with the ac voltage. But this is not so in the case of an inductor, a capacitor or a combination of these circuit elements. In order to show phase relationship between voltage and current in an ac circuit, we use the notion of phasors.

The analysis of an ac circuit is facilitated by the use of a phasor diagram. A phasor* is a vector which rotates about the origin with angular speed ω, as shown in Fig. 7.4. The vertical components of phasors V and I represent the sinusoidally varying quantities v and i. The magnitudes of phasors V and I represent the amplitudes or the peak values vm and im of these oscillating quantities. Figure 7.4(a) shows the voltage and current phasors and their relationship at time t 1 for the case of an ac source connected to a resistor i.e., corresponding to the circuit shown in Fig. 7.1. The projection of voltage and current phasors on vertical axis, i.e., vm sinω t and im sinω t, respectively represent the value of voltage and current at that instant. As they rotate with frequency ω, curves in Fig. 7.4(b) are generated. From Fig. 7.4(a) we see that phasors V and I for the case of a resistor are in the same direction. This is so for all times. This means that the phase angle between the voltage and the current is zero.

AC VOLTAGE APPLIED TO AN INDUCTOR
 

Figure 7.5 shows an ac source connected to an inductor. Usually, inductors have appreciable resistance in their windings, but we shall assume that this inductor has negligible resistance. Thus, the circuit is a purely inductive ac circuit. Let the voltage across the source be v = vm sinω t.

Using the Kirchhoff’s loop rule, ∑ε ( ) t = 0 , and since there is no resistor in the circuit,
d 0 d i v L t − = (7.10) where the second term is the self-induced Faraday emf in the inductor; and L is the self-inductance of the inductor. The negative sign follows from Lenz’s law (Chapter 6). Combining Eqs. (7.1) and (7.10), we have d sin d i v vm t t L L = = ω (7.11)

Equation (7.11) implies that the equation for i(t), the current as a function of time, must be such that its slope di/dt is a sinusoidally varying quantity, with the same phase as the source voltage and an amplitude given by vm /L. To obtain the current, we integrate di/dt with respect to time:

d d d d i t t v L t t m ∫ ∫ = sin( ) ω
and get,
cos( ) constant vm i t L = − ω + ω

The integration constant has the dimension of current and is timeindependent. Since the source has an emf which oscillates symmetrically about zero, the current it sustains also oscillates symmetrically about zero, so that no constant or time-independent component of the current exists. Therefore, the integration constant is zero.

Using − = −     
cos( ) sin ω ω  t t π 2 , we have
i i t = − m      sin ω  π 2 (7.12)
where m m v i L = ω is the amplitude of the current. The quantity ω L is analogous to the resistance and is called inductive reactance, denoted by XL :
XL = ω L (7.13) The amplitude of the current is, then m m L

The dimension of inductive reactance is the same as that of resistance and its SI unit is ohm (Ω). The inductive reactance limits the current in a purely inductive circuit in the same way as the resistance limits the current in a purely resistive circuit. The inductive reactance is directly proportional to the inductance and to the frequency of the current. A comparison of Eqs. (7.1) and (7.12) for the source voltage and the current in an inductor shows that the current lags the voltage by π/2 or one-quarter (1/4) cycle. Figure 7.6 (a) shows the voltage and the current phasors in the present case at instant t 1 . The current phasor I is π/2 behind the voltage phasor V. When rotated with frequency ω counterclockwise, they generate the voltage and current given by Eqs. (7.1) and (7.12), respectively and as shown in Fig. 7.6(b).
We see that the current reaches its maximum value later than the voltage by one-fourth of a period T 4 2 =       π/ ω . You have seen that an inductor has reactance that limits current similar to resistance in a dc circuit. Does it also consume power like a resistance? Let us try to find out.

The instantaneous power supplied to the inductor is
p i v i t v t L = = m − m      sin ω  sin(ω ) π 2 ×
= −i v t t m m cos sin (ω ω ) ( ) sin 2( ) 2 i vm m = − ωt
So, the average power over a complete cycle is L sin 2( ) 2 i vm m P = − ωt sin 2( ) 2 i vm m = − ωt = 0, since the average of sin (2ωt) over a complete cycle is zero.

Thus, the average power supplied to an inductor over one complete cycle is zero. Figure 7.7 explains it in detail.
 
NCERT Books Class 12 Physics Chapter 7- Alternating Current- PDF Download

Chapter 7- Alternating Current

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