Physics Standard 12 Vol I>Chapter -1 - Electromagnetic Induction and Alternating Current>EVALUATION>Q 21

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12th Tamil Nadu Board

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Find out the phase relationship between voltage and current in a pure inductive circuit.

Important Points to Remember in Chapter -1 - Electromagnetic Induction and Alternating Current from Tamil Nadu Board Physics Standard 12 Vol I Solutions

1. Magnetic flux:

The magnetic flux through a surface of area $A$ placed in a uniform magnetic field $B$ is defined as, $ϕ_{B} = B . A = B A \cos θ$ where $θ$ is the angle between $B$ and $A$ .

2. Electromagnetic Induction and Lenz's law:

(i) Faraday’s laws of induction imply that the emf induced in a coil of $N$ turns is directly related to the rate of change of flux through it, $ε = - N \frac{d ϕ_{B}}{d t}$ Here, $ϕ_{B}$ is the flux linked with one turn of the coil.

(ii) Lenz’s law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it. The negative sign in the expression for Faraday’s law indicates this fact.

3. Motional EMF:

(i) When a metal rod of length $l$ is placed normal to a uniform magnetic field $B$ and moved with a velocity $v$ perpendicular to the field, the induced emf (called motional emf) across its ends is $ε = B l v$

(ii) When a metal rod of length $l$ pivoted at one end, and is rotated with angular speed $ω$ normal to a uniform magnetic field $B,$ the induced emf across its ends is, $ε = \frac{1}{2} B ω l^{2}$ .

4. Eddy current:

Eddy current: Changing magnetic fields can set up current loops in any conductor bodies. Such currents are eddy currents. They dissipate electrical energy as heat.

5. Self induction:

(i) When a current in a coil changes, it induces a back emf in the same coil. The self-induced emf is given by, $ε = - L \frac{d I}{d t}$ where $L$ is the self-inductance of the coil. It is a measure of the inertia of the coil against the change of current through it.

(ii) The self-inductance of a long solenoid, the core of which consists of a magnetic material of permeability $μ_{r}$ is given by: $L = μ_{r} μ_{0} n^{2} A l$ where $A$ is the area of cross-section of the solenoid, $l$ its length and $n$ the number of turns per unit length.

6. Mutual induction:

(i) A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1). This relation is given by, $ε_{1} = - M_{12} \frac{d I_{2}}{d t}$ the quantity $M_{12}$ is called mutual inductance of coil 1 with respect to coil 2. One can similarly define $M_{21}$ There exists a general equality, $M_{12} = M_{21}$

7. AC generator:

(i) In an AC generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction.

(ii) If coil of $N$ turn and area $A$ is rotated at $v$ revolutions per second in a uniform magnetic field $B$ , then the motional emf produced is, $ε = N B A (2 π v) \sin (2 π v t)$ where we have assumed that at time $t = 0 s$ the coil is perpendicular to the field.

8. AC Voltage Applied to a Resistor:

(i) An alternating voltage $v = v_{m} \sin ω t$ applied to a resistor $R$ drives a current $i = i_{m} \sin ω t$ in the resistor, $i_{m} = \frac{v_{m}}{R}$ . The current is in phase with the applied voltage.

(ii) For an alternating current $i = i_{m} \sin ω t$ passing through a resistor $R$ , the average power loss $P$ (averaged over a cycle) due to joule heating is $(1 / 2) i_{m}^{2} R$ .

(iii) To express it in the same form as the dc power $(P = I^{2} R)$ a special value of current is used. It is called root mean square (rms) current and is denoted by $I$ : $I = \frac{i_{m}}{\sqrt{2}} = 0.707 i_{m}$

9. AC Voltage Applied to an Inductor:

(i) An ac voltage $v = v_{m} \sin ω t$ applied to a pure inductor $L$ , drives a current in the inductor $i = i_{m} \sin (ω t - π / 2)$ where $i_{m} = \frac{v_{m}}{X_{L}} \cdot X_{L} = ω L$ is called inductive reactance.

(ii) The current in the inductor lags the voltage by $π / 2$ . The average power supplied to an inductor over one complete cycle is zero.

10. AC Voltage Applied to a Capacitor:

(i) An ac voltage $v = v_{m} \sin ω t$ applied to a capacitor drives a current in the capacitor: $i = i_{m} \sin (ω t + π / 2)$ Here, $i_{m} = \frac{v_{m}}{X_{C}}, X_{C} = \frac{1}{ω C}$ is called capacitive reactance.

(ii) The current through the capacitor is $π / 2$ ahead of the applied voltage. The average power supplied to a capacitor over one complete cycle is zero.

11. Series LCR Circuit:

(i) For a series RLC circuit driven by voltage $v = v_{m} \sin ω t$ the current is given by $i = i_{m} \sin (ω t + ϕ)$ where, $i_{m} = \frac{v_{m}}{\sqrt{R^{2} + {(X_{C} - X_{L})}^{2}}}$ and
$ϕ = {t a n}^{- 1} \frac{X_{C} - X_{L}}{R}$

(ii) $Z = \sqrt{R^{2} + {(X_{C} - X_{L})}^{2}}$ is called the impedance of the circuit.

12. Power in AC Circuits:

(i) The average power loss over a complete cycle is given by, $P = V I \cos ϕ$ . The term $\cos ϕ$ is called the power factor.

(ii) In a purely inductive or capacitive circuit, $\cos ϕ = 0$ and no power is dissipated even though a current is flowing in the circuit. In such cases, current is referred to as a wattless current.

13. Phasor:

A phasor is a vector which rotates about the origin with angular speed $ω$ . The magnitude of a phasor represents the amplitude or peak value of the quantity (voltage or current) represented by the phasor.

14. Resonance:

(i) An interesting characteristic of a series RLC circuit is the phenomenon of resonance. The circuit exhibits resonance, i.e., the amplitude of the current is maximum at the resonant frequency, $ω_{0} = \frac{1}{\sqrt{L C}}$ .

(ii) The quality factor is defined by $Q = \frac{ω_{0} L}{R} = \frac{1}{ω_{0} C R}$ is an indicator of the sharpness of the resonance, the higher value of Q indicating sharper peak in the current.

15. LC oscillation:

A circuit containing an inductor $L$ and a capacitor $C$ (initially charged) with no ac source and no resistors exhibits free oscillations. The energy in the system oscillates between the capacitor and the inductor but their sum or the total energy is constant in time.

16. Transformer:

(i) A transformer consists of an iron core on which are bound a primary coil of $N_{p}$ turns and a secondary coil of $N_{s}$ turns. If the primary coil is connected to an ac source, the primary and secondary voltages are related by, $V_{s} = (\frac{N_{s}}{N_{p}}) V_{p}$ and the currents are related by, $I_{s} = (\frac{N_{p}}{N_{s}}) I_{p}$ .

(ii) If the secondary coil has a greater number of turns than the primary, the voltage is stepped-up $(V_{s} > V_{p})$ This type of arrangement is called a step up transformer. If the secondary coil has turns less than the primary, we have a step-down transformer.

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