Revision Notes for Class 12 Physics Chapter 7 Alternating Current

Here we are providing revision notes for class 12 physics chapter 7 alternating current. Students can use this article for quick revision of the chapter.

Revision Notes for Class 12 Physics Chapter 7 Alternating Current

1. Alternating current.

It is that current which varies in magnitude continuously and reverses its direction periodically. Its value at any instant is given by

I=I₀sinωt = I₀sin2πft

where I₀ is the peak value of a.c. or current amplitude. The frequency of a.c. supplied to our homes is 50 cps. The average value of a.c. over a complete cycle is zero.

2.   Direct current:

It is that current which flows with a constant magnitude in the same fixed direction.

3.   Average or mean value of a.c.

It is that value of direct current which sends the same charge in a circuit in the same time as is sent by the given alternating current in the same circuit in its half time period.

I_av = 2I₀ / π = 0.637 I₀

4.   Effective or rms or virtual value of a.c.

It is that value of direct current which produces the same heating effect in a given resistor as is produced by the given alternating current when passed for the same time.

I_rms or I_eff = I₀/√2 

5.   Alternating voltage.

It is that voltage whose magnitude varies continuously, and direction reverses periodically with time. Its instantaneous, average and root mean square values are respectively given by

ξ = ξ₀ sinωt =ξ₀ sin2πft and ξ_rms = ξ₀ / √2

6.  Phasors and phasor diagrams:

A rotating vector that represents a sinusoidally varying quantity is called a phasor. A diagram that represents alternating current and voltage of the same frequency as rotating vectors (phasors) along- with proper phase angle between them is called a phasor diagram or Argand diagram.

7.  A.C. circuit containing resistor only

An alternating voltage, applied to a resistor R drives a current I – I₀ sin ωt in the resistor. The current is in phase with the applied voltage.

Peak value of current,

I_{0 =} ξ₀ / R

8.  A.C. circuit containing only an inductor:

An alternating voltage, if applied to a pure inductor L drives a current, I = I₀ sin (ωt – π / 2) in the inductor. The current in the inductor lags behind the voltage in phase by π / 2 rad.

Peak value of current,

I_{0 =} ξ₀/ωL = ξ₀/χ_L 

Root mean square value of current,

I_{rms =} ξ_rms/√2 ωL

9.  Reactance:

The non-resistive opposition to the flow of a.c. is called reactance. It may be inductive reactance (X_L) or capacitive reactance (X_C).

10. Inductive reactance:

It is the effective resistance or opposition offered by an inductor to the flow of a.c. through it. It is given by X_L = ωL = 2 π/L

The SI unit of inductive reactance is ohm (Ω).

For a.c., X_L ∝ f

For d.c., f = 0, so X_L =0.      

11. A.C. circuit containing only a capacitor:

An alternating voltage, applied to a capacitor C drives a current, I = I₀ sin (ωt + π/2) in the capacitor. The current in the inductor is ahead of voltage in phase by π/2 rad.

Peak value of current,

I_{0 =} ξ₀ / (1/ωC) = ξ₀/χ_C

Root mean square value of current,

I_{rms =} ξ_rms/√2 (1/ωC)

12. Capacitive reactance:

It is the effective resistance or opposition offered by a capacitor to the flow of a.c. through it.

It is given by

χ_C = 1/ωC = 1/2πfC

The SI unit of capacitive reactance is ohm (Ω)

For a.c.,

χ_C α 1/f

For d.c.,

f = 0, so χ_C = ∞

Thus, a capacitor allows a.c. to flow through it easily but blocks d.c.

13. Impedance:

It is a quantity that measures the opposition of a circuit to the flow of current through it and so determines the magnitude of the current. In a d.c. circuit, this is the resistance R alone. In an a.c. circuit, the reactance X also has to be taken into account, according to the relation:

Z² = R² + X²

Where Z is the impedance. Impedance triangle is a right-angled triangle whose base represents resistance R, perpendicular represents reactance X and hypotenuse represents impedance Z of the circuit. From this triangle, the phase angle ϕ between voltage and current is given by

tanΦ = χ / R or, cosΦ = R / Z

14. A.C. through a series LR-circuit:

The alternating voltage leads the current by a phase angle ϕ

Instantaneous voltage,  

ξ = ξ₀ sinωt

Instantaneous current,

I = I₀ sin(ωt-Φ)

Peak current,

I₀ = ξ₀ / Z, where Z= √(R²+χ_L²)

Impedance,

Z= √(R²+χ_L²)

Phase angle,

Φ=tan^-1 (χ_L/R)

15. A.C. through a series CR-circuit:

The alternating voltage lags behind the voltage by phase angle ϕ

Instantaneous voltage,  

ξ = ξ₀ sinωt

Instantaneous current,

I = I₀ sin(ωt-Φ)

Peak current,

I₀ = ξ₀ / Z, where Z= √(R²+χ_C²)

Impedance,

Z= √(R²+χ_C²)

Phase angle,

Φ=tan^-1 (χ_C/R)

16. The series LCR-circuit:

For a series LCR-circuit connected across a source 

ξ = ξ₀ sinωt

the current ‘I’ is given by

I = I₀ sin(ωt-Φ) = (ξ₀ / Z) sin(ωt-Φ)

where Z is the total effective resistance of the LCR-circuit and is called its impedance.

Z = √[R²+ (X_L-X_C)²]

The phase angle ϕ between voltage and current is given by

Φ=tan^-1 (χ_L – χ_C)/R

The voltage leads the current if X_L > X_C and it lags behind the current if X_L <X_C.

17. Resonance condition of the LCR-circuit:

If X_L=X_C, the impedance of LCR circuit becomes Z = √[R²+ (X_L-X_C)²] = R

The impedance is minimum and hence current is maximum. The circuit is purely resistive, and voltage and current are in same phase. This is the resonance condition of LCR-circuit and is satisfied at the resonant frequency given by

f = 1/2π√LC

18. Q-Factor:

It indicates the sharpness of resonance. It is defined as the ratio of the resonant frequency to the difference in two frequencies taken on both sides of the resonant frequency such that at each frequency, the current amplitude becomes 1/√2 times the value at resonant frequency.

Q-factor = Resonant frequency / Band width