faraday’s laws of electromagnetic induction are related to formula derivation 12th 11 class physics

find faraday’s laws of electromagnetic induction are related to formula derivation 12th 11 class physics ?

Magnetic Flux

The magnetic flux through any surface placed in a magnetic field is determined by the number of field lines that cut through that surface. The magnetic flux through a coil of area A in a uniform magnetic field B is defined as

Φ = B x A = B A cosθ

where θ is the angle between the normal to the plane of the coil and the magnetic field. If the coil has N turns, the magnetic flux through the coil is given by Φ = N B A cosθ

The SI unit of flux is called weber (Wb). For a curved surface, Φ = ∫ B.dA

Faraday’s Laws of Electromagnetic Induction

The magnitude and direction of induced emf can be determined by the application of two laws of electromagnetic induction: (i) Faraday’s law, and (ii) Lenz’s law.

Faraday’s Law of Electromagnetic Induction

On the basis of various experiments, Faraday found that

1. whenever magnetic flux linked with a circuit changes, an induced emf is produced in the circuit,

2. the induced emf lasts as long as the change in the magnetic flux is taking place, and

3. the magnitude of the induced emf is directly proportional to the rate of change of magnetic flux, i.e.

Lenz’s Law

According to Lenz’s law, the direction of the induced emf is such that it always opposes the cause that has produced it. Thus

where k is a positive constant whose value depends on the system of units. In SI system of units, k = 1 and one can write

If Φ is the flux through one turn of a coil, then for a coil of N turns

The magnitude of the induced current is given by

The direction of induced current is obtained by Lenz’s law.

Flow of Induced Charge

When a current is induced in a circuit due to change in magnetic flux, induced charge q flows through the circuit.

Heat Dissipation
Heat dissipated due to induced current is

= induced current x change in flux

Fleming’s Right Hand Rule

This rule gives the direction of the induced emf when a conductor moves at right angles to a magnetic field. Hold the thumb and the first two fingers of your right hand mutually perpendicular to each other. Then, if the first finger points in the direction of the magnetic field and the thumb points in the direction of the motion of the conductor, then the second finger gives the direction of the induced emf (and hence of the induced current).

Applications of Lenz’s Law

(i) If the magnet is moved towards the coil or coil is moved towards the magnet, the induced current i is anticlockwise Fig. below. The current is clockwise if the magnet is moved away from the coil.

(ii) The induced current i in the coil is anticlockwise if fig; below

(a) the coil is moved towards the long wire carrying current I
or
(b) the current I increases with time. The current I is clockwise if
(a) the coil is moved away from wire
or
(b) the current I decreases with time.
(iii) Two coils carrying currents I1 and I2 placed with their planes parallel approach each other (Fig. below)

(a) If I₁ and I₂ are both clockwise (or anticlockwise), then both I₁ and I₂ will decrease.

(b) If the currents I₁ and I₂ are in opposite sense, both the currents will increase.

Expression for Induced EMF

(i) Change in flux due to change in magnetic field

(B). If B increases with time, the induced current i is anticlockwise so that it produces a magnetic field pointing outwards (opposite to B). The induced emf is (Fig. below)

If B decreases with time, I will be clockwise.
If B remains constant but the radius of the coil

(ii) Change in flux due to change in area (A) If a rectangular coil PQRS is moved out of a region of uniform magnetic field B with a velocity v, the emf induced is (Fig. below (a))

|e| = Blv where l = PS = QR

Induced current i is clockwise. If R is the resistance of the coil,

Force F required to pull the coil out with constant velocity v is

The current will be anticlockwise, if the coil is pushed into the region of magnetic field.

(a)  If the coil is moved within the region of uniform magnetic field, no change in flux takes place and hence no emf is induced. (b) If the magnetic field is non-uniform and the coil is kept stationary in it, no change in flux occurs and hence no emf is induced.

The above results also hold in the case of rod XY sliding on metallic rails PQRS to the right as shown in Fig. 14.5(b).

(iii) Change in flux due to change in orientation (θ ) (A.C. generator)

If a coil of area A, consisting of N turns is rotated in a magnetic field B with angular velocity w, the emf induced in it is given by e = e₀ sin θ = e₀ sin wt where e₀ = NBAw is the amplitude (peak value). Thus an alternating emf is produced.

Motional Emf

(i) When a rod (or wire) of length l is moved with a velocity v in a magnetic field B as shown in Fig. below (a), the emf induced between the ends P and Q of the rod is given by

e = Blv

If the rod is moved as shown in Fig. above (b), then e = Blv sin θ

(ii) When a semicircular rod (or wire) of radius R is moved with a velocity v in a magnetic field B as shown in Fig. below, the emf induced between the ends P and Q of the rod is given by e = Bv (2R) = 2BvR

(iii) When a rod PQ of length l pivoted at one end P is rotated with angular velocity w in a magnetic field B as shown in Fig. below (a), the emf induced between its ends is given by

(iv) When a disc of radius R is rotated about its centre with angular velocity w in a magnetic field B as shown in Fig. below (b), the emf induced between its centre O and a point P on its rim is given by

Electric Motor

When a current is passed through a coil placed in a magnetic field by connecting its end to a source of voltage V, it experiences a torque which rotates it. As a result, an emf e is induced in it. This emf is called back emf as it opposes the applied voltage V (from Lenz’s law). If R is the resistance of the coil, then current in it is

Some important points about a d.c. motor

(i) Back emf e and hence current i vary sinusoidally even if the source of voltage V is a d.c. (battery).

(ii) When output power is maximum, e = V/2 and η = 50%.

(iii) Initially, i.e. when the motor is switched on, e = 0 and initial current = V/R which is very large. So, for safety, a starter is used.

(iv) At full speed, back emf is maximum and current i is minimum.

Mutual Inductance

If the current in a coil is i then the flux linked with a neighbouring coil is Φ = Mi where M is the coefficient of mutual inductance. If current i is changing with time, the emf induced in the neighbouring coil is

Expressions of M in some situations

(i) A small coil of length l, number of turns N₁ wound closely on a long coil of N₂ turns.

A = common cross-sectional area
(ii) Two coplanar and concentric coils of radii R and r (R >> r) Fig. (below)

(iii) A small circular coil of radius r at the centre of a large rectangular coil of sides a and b with a, b >> r (Fig. below)

(iv) A rectangular loop of sides a and b placed at a distance x from a long straight wire (Fig. below)

note – If the medium is different from air, µ₀ in above expressions is replaced by µ = µ₀ µ_r, where µ_r is the relative permeability of the medium.

Self Inductance

If i is the instantaneous current in a coil, flux Φ = Li, where L is the self inductance of the coil. Induced emf

(i) The self inductance of a coil of N turns, cross sectional area A and length l is given by

(ii) Direction of induced emf is such that it opposes the change in current (Fig. below (a) and (b))

(iii) Energy stored in the inductor

(iv) Inductors in series (Fig. below)

Equivalent inductance is
(a) L = L1 + L2 (when the flux linked with one coil is not linked with the other, i.e. M = 0)

(b) L = L1 + L2 + 2M (when flux of one coil is in the same direction as that of the other coil) L = L1 + L2 – 2M (when the fluxes oppose each other)

(v) Inductors in parallel

Growth and Decay of Current in a d.c L—R Circuit (Fig. below)

If switch S1 is closed at t = 0, with switch S2 open, no current flows in the beginning (as the inductor behaves as open switch) [Fig. below]. The current starts increasing

After a long time (t = ∞), the current attains a steady value i₀ = E/R (now the ideal inductor behaves as a closed switch).

Decay of current: At time t = 0, let i₀ = E/R be the current in the circuit. If S2 is closed (with S1 open), the current decays as (see Fig. below)

Energy Stored in an Inductor

If the current in a coil of self inductance L is increased from zero to a steady value I, the energy stored in the magnetic field of the coil is

Transformer

The transformer is a device used for converting a low ac voltage into a high ac voltage and vice versa. The former is called the step-up transformer and the latter the step-down transformer. A transformer consists of two coils each of which is wound on an iron core. One of the coils is connected to a source of alternating emf. This coil is called the primary of the transformer while the other is called the secondary of the transformer. Any of the two coils can act as primary while the other as secondary. The alternating emf in one coil induces an alternating emf in the second coil. The presence of an iron core in the primary and secondary makes the flux linkage between the two coils very large. The alternating emf in the coil makes the magnetic flux in the iron also vary periodically. This varying magnetic flux in iron induces an alternating emf in the secondary. If the magnetic field lines remain confined to the core, then all the field lines threading the primary also go across the secondary. Then the magnetic fluxes across the secondary and primary will be simply proportional to the number of turns in them, i.e.

where Ns is the number of turns in the secondary and Np is the number of turns in the primary. Now from Faraday’s law the emf induced across the secondary is es = – (dΦs/dt) and that across the primary is ep = – (dΦp/dt). Thus

From this equation, it follows that if Ns > Np, then es > ep, i.e. the voltage across the secondary is greater than the input primary voltage. Such a transformer in which the number of turns in the secondary is more than in the primary is called a step-up transformer. But if Ns < Np, then es < ep. Such a transformer is called a step-down transformer. The former are used in TV, high-voltage power supplies and the latter in radio transmitter sets, battery eliminators, etc.

Usually, there are a number of energy losses in actual transformers. These are: (i) Joule heating (I² R) losses in the primary and secondary coils due to their resistance (generally, these losses are minimized by using wires of large diameters so that resistance is low); and (ii) the losses in the iron core which include the heating of the core due to eddy currents and power loss due to hysteresis. The eddy currents can be minimized by using laminated iron. In an ideal transformer, the entire power in the primary is transferred to the secondary. For an ideal transformer,

input power = output power

where Ip and Is are the currents in the primary and the secondary of the transformer.

Alternating Current

If an alternating voltage V = V₀ sin w t is applied across a resistance R, the current I in the circuit is

where I0 = V₀ /R, is the maximum or peak value of the current. It is clear from Eq. (1) that the current I varies sinusoidally with time; its magnitude changes continuously with time and its direction is reversed periodically. A sinusoidally varying current whose magnitude changes continuously with time and whose direction reverses periodically is called an alternating current (or simply ac). The angular frequency w of an alternating current is related to its time period T and frequency n as

where w is expressed in radians per second (rad s^–1), T in seconds (s) and n in hertz. (Hz). In terms of T, Eq. (1) reads

Root Mean Square Voltage and Current

The mean value of a periodic function X (t) of time period T over one time period is defined as

(i) Mean or average value of alternating voltage V = V₀ sin (wt) is

Similarly mean value of alternating current I = I₀ sin (wt) over one time period is I = 0
(ii) Mean square value of alternating voltage
V = V₀ sin (wt) is

Root mean square (rms) value of the alternating voltage is

Similarly, root mean square (rms) value of alternating current I = I₀ sin (w t) is

Series LCR Circuit

The applied voltage V divides into three parts, VL (across L), VC (across C) and VR (across R) such that (Fig. below).

This is the case of resonance. Voltage and current are in phase. Z = R (minimum) and current is maximum.

Special Cases
(a) A.C. circuit containing only a pure resistor (Fig. below)

The voltage across R is always in phase with the current in the circuit.

(b) A.C. circuit containing only an ideal inductor (Fig. below)

X_L = wL is called inductive reactance. The voltage across the inductance leads the current in the circuit by a phase angle of π/2.

(c) A.C. circuit containing only an ideal capacitor (Fig. below)

X_C = 1 /wC is called capacitative reactance. The voltage across the capacitor lags behind the current in the circuit by a phase angle of π/2.

(d) A.C. circuit containing an ideal inductor and a pure resistor (Fig. below)

Power in LCR Circuit

In a series LCR circuit driven by an alternating voltage V = V0 sin wt, the current in the circuit is

Instantaneous power supplied to the circuit by the A.C. source is

Power Factor of an A.C. Circuit

The power supplied by the source depends not only on Vrms and Irms but also on cos f. The quantity cos f is called the power factor of the A.C. circuit. Now

Special Cases
(a) For an A.C. circuit containing only a resistor

(b) For an A.C. circuit containing only an inductor or a capacitor

Φ = 90°. Hence P = 0

(c) At resonance, Φ = 0 for an LCR circuit. Hence P = maximum, i.e. maximum power is delivered to the circuit form A.C. source.

Wattless Current

For an A.C. circuit containing only a pure inductor or an ideal capacitor Φ = 90°. Hence

Such an A.C. circuit consumes no power. The current flowing through the inductor or capacitor consumes no power and is called wattless current.

Bandwidth and Quality Factor of LCR Circuit

For an LCR circuit driven by an alternating voltage V = V₀ sin wt, the peak (amplitude) value of the current is given by

The positive root of this quadratic equation is

Q is a dimensionless number. Figure below shows the graph of P versus w for some values of Q

The power peak is sharp for high Q. The resonance is then said to be sharp. Higher the value of Q, the sharper is the resonance and greater is the power absorbed from the source.