Topic covered

`color{blue}{star}` INTRODUCTION
`color{blue}{star}` MAGNETIC FLUX
`color{blue}{star}` FARADAY’S LAW OF INDUCTION


`color{blue} ✍️` In the early decades of the nineteenth century, experiments on electric current by Oersted, Ampere and a few others established the fact that electricity and magnetism are inter-related. They found that moving electric charges produce magnetic fields.

`color{blue} ►`For example, an electric current deflects a magnetic compass needle placed in its vicinity.

`color{blue} ✍️` In this chapter, we will study the phenomena associated with changing magnetic fields and understand the underlying principles. The phenomenon in which electric current is generated by varying magnetic fields is appropriately called electromagnetic induction.


`color{blue} ✍️` The discovery and understanding of electromagnetic induction are based on a long series of experiments carried out by Faraday and Henry. We shall now describe some of these experiments.

`color{purple}{bb ul"Experiment 6.1"}`
`color {blue}{➢➢}` Figure 6.1 shows

`color{blue} ✍️` A coil `C_(1)^(*)` connected to a galvanometer G. When the North-pole of a bar magnet is pushed towards the coil, the pointer in the galvanometer deflects, indicating the presence of electric current in the coil.

`color{blue} ✍️` The deflection lasts as long as the bar magnet is in motion. The galvanometer does not show any deflection when the magnet is held stationary.

`color{blue} ✍️` When the magnet is pulled away from the coil, the galvanometer shows deflection in the opposite direction, which indicates reversal of the current’s direction.

`color{blue} ✍️` Moreover, when the South-pole of the bar magnet is moved towards or away from the coil, the deflections in the galvanometer are opposite to that observed with the North-pole for similar movements.

`color{blue} ✍️` Further, the deflection (and hence current) is found to be larger when the magnet is pushed towards or pulled away from the coil faster. Instead, when the bar magnet is held fixed and the coil `C_1` is moved towards or away from the magnet, the same effects are observed. It shows that it is the relative motion between the magnet and the coil that is responsible for generation (induction) of electric current in the coil.

`color{purple}{bb ul"Experiment 6.2"}`
`color{blue} ✍️` In Fig. 6.2 the bar magnet is replaced by a second coil `C_2` connected to a battery.
The steady current in the coil `C_2` produces a steady magnetic field. As coil `C_2` is moved towards the coil `C_1`, the galvanometer shows a deflection.

`color{blue} ✍️` This indicates that electric current is induced in coil `C_1`. When `C_2` is moved away, the galvanometer shows a deflection again, but this time in the opposite direction. The deflection lasts as long as coil `C_2` is in motion.

`color{blue} ✍️` When the coil `C_2` is held fixed and `C_1` is moved, the same effects are observed. Again, it is the relative motion between the coils that induces the electric current.

`color{purple}{bb ul"Experiment 6.3"}`
`color{blue} ✍️` The above two experiments involved relative motion between a magnet and a coil and between two coils, respectively.
Through another experiment, Faraday showed that this relative motion is not an absolute requirement.

`color{blue} {➤➤}` Figure 6.3 shows two coils `C_1` and `C_2` held stationary. Coil `C_1` is connected to galvanometer `G` while the second coil `C_2` is connected to a battery through a tapping key K.

`color{blue} ✍️` It is observed that the galvanometer shows a momentary deflection when the tapping key K is pressed. The pointer in the galvanometer returns to zero immediately.

`color{blue} ✍️` If the key is held pressed continuously, there is no deflection in the galvanometer. When the key is released, a momentory deflection is observed again, but in the opposite direction.

`color{blue} ✍️` It is also observed that the deflection increases dramatically when an iron rod is inserted into the coils along their axis.


`color{blue} ✍️` Faraday’s great insight lay in discovering a simple mathematical relation to explain the series of experiments he carried out on electromagnetic induction. However, before we state and appreciate his laws, we must get familiar with the notion of magnetic flux, `Φ B.`

`color{blue} ✍️` Magnetic flux is defined in the same way as electric flux is defined in Chapter 1. Magnetic flux through a plane of area `A` placed in a uniform magnetic field B (Fig. 6.4) can be written as

`color{purple}(Φ_B = B . A = BA cos θ)`


`color {blue}{➢➢}` where `θ` is angle between `B` and `A`. The notion of the area as a vector has been discussed earlier in Chapter 1. Equation (6.1) can be extended to curved surfaces and nonuniform fields.

`color{blue} ✍️` If the magnetic field has different magnitudes and directions at various parts of a surface as shown in Fig. 6.5, then the magnetic flux through the surface is given by

`color{purple}(Φ B =B_1•dA_1 + B_2• dA_2+... sum_("all") B_i•dA_i)`


`color {blue}{➢➢}` where ‘all’ stands for summation over all the area elements `dA_i` comprising the surface and `B_i` is the magnetic field at the area element `dA_i.` The SI unit of magnetic flux is weber `(Wb)` or tesla meter squared `(T m^2).` Magnetic flux is a scalar quantity.


`color{blue} ✍️`From the experimental observations, Faraday arrived at a conclusion that an emf is induced in a coil when magnetic flux through the coil changes with time. Experimental observations discussed in Section 6.2 can be explained using this concept.

`color{blue} ✍️`The motion of a magnet towards or away from coil `C_1` in Experiment 6.1 and moving a current-carrying coil `C_2` towards or away from coil `C_1` in Experiment 6.2, change the magnetic flux associated with coil `C_1`.

`color{blue} ✍️`The change in magnetic flux induces emf in coil `C_1`. It was this induced emf which caused electric current to flow in coil `C_1` and through the galvanometer. A plausible explanation for the observations of Experiment 6.3 is as follows:

`color {blue}{➢➢}` When the tapping key K is pressed, the current in coil `C_2` (and the resulting magnetic field) rises from zero to a maximum value in a short time.

`color {blue}{➢➢}` Consequently, the magnetic flux through the neighbouring coil `C_1` also increases. It is the change in magnetic flux through coil `C_1` that produces an induced emf in coil `C_1`. When the key is held pressed, current in coil `C_2` is constant.
Therefore, there is no change in the magnetic flux through coil `C_1` and the current in coil `C_1` drops to zero.

`color{blue} ✍️`When the key is released, the current in `C_2` and the resulting magnetic field decreases from the maximum value to zero in a short time.

`color{blue} ✍️`This results in a decrease in magnetic flux through coil `C_1` and hence again induces an electric current in coil `C_(1)^(*)`. The common point in all these observations is that the time rate of change of magnetic flux through a circuit induces emf in it. Faraday stated experimental observations in the form of a law called Faraday’s law of electromagnetic induction. The law is stated below.

`color{green}{"The magnitude of the induced emf in a circuit is equal"}`
`color{ green}{" to the time rate of change of magnetic flux through the circuit."}`

`color{blue} {➢➢}`Mathematically, the induced emf is given by

`color{purple}(ε =(dΦ_B)/(dt))`


`color{blue} ✍️`The negative sign indicates the direction of `ε` and hence the direction of current in a closed loop. This will be discussed in detail in the next section.

`color {blue}{➢➢}` In the case of a closely wound coil of `N` turns, change of flux associated with each turn, is the same. Therefore, the expression for the total induced emf is given by

`color{purple}(ε =N(dΦ_B)/(dt))`


`color{blue} ✍️`The induced emf can be increased by increasing the number of turns `N` of a closed coil. From Eqs. (6.1) and (6.2), we see that the flux can be varied by changing any one or more of the terms `B, A` and `θ`.

`color {blue}{➢➢}` In Experiments 6.1 and 6.2 in Section 6.2, the flux is changed by varying `B`. The flux can also be altered by changing the shape of a coil (that is, by shrinking it or stretching it) in a magnetic field, or rotating a coil in a magnetic field such that the angle `θ` between `B` and `A` changes. In these cases too, an emf is induced in the respective coils.
Q 3158445304

Consider Experiment 6.2. (a) What would you do to obtain a large deflection of the galvanometer? (b) How would you demonstrate the presence of an induced current in the absence of a galvanometer?
Class 12 Chapter 6 Example 1

(a) To obtain a large deflection, one or more of the following steps can be taken:
(i) Use a rod made of soft iron inside the coil `C_2`,
(ii) Connect the coil to a powerful battery, and (iii) Move the arrangement rapidly towards the test coil `C_1.`

(b) Replace the galvanometer by a small bulb, the kind one finds in a small torch light. The relative motion between the two coils will cause the bulb to glow and thus demonstrate the presence of an induced current. In experimental physics one must learn to innovate. Michael Faraday who is ranked as one of the best experimentalists ever, was legendary for his innovative skills.
Q 3108445308

A square loop of side 10 cm and resistance `0.5 Ω` is placed vertically in the east-west plane. A uniform magnetic field of `0.10 T` is set up across the plane in the north-east direction. The magnetic field is decreased to zero in `0.70 s` at a steady rate. Determine the magnitudes of induced emf and current during this time-interval.
Class 12 Chapter 6 Example 2

The angle θ made by the area vector of the coil with the magnetic field is `45°.` From Eq. (6.1), the initial magnetic flux is
`Φ = BA cos θ`

`= (0.1xx10^(-2))/(sqrt2) wb`

Final flux, `Φ_min = 0`

The change in flux is brought about in 0.70 s. From Eq. (6.3), the magnitude of the induced emf is given by

`color{green} {ε= (|DeltaΦ_B|)/(Deltat) = (|Φ-0|)/(Deltat) }`

`= (10^(-3))/(sqrt2xx0.7)= 1.0mV`

And the magnitude of the current is

`color{orange} {I = ε/R }`

` = (10^(-3)V)/(0.5Omega) = 2mA`

Note that the earth’s magnetic field also produces a flux through the loop. But it is a steady field (which does not change within the time span of the experiment) and hence does not induce any emf.
Q 3128545401

A circular coil of radius `10 cm, 500` turns and resistance `2 Ω` is placed with its plane perpendicular to the horizontal component of the earth’s magnetic field. It is rotated about its vertical diameter through 180° in 0.25 s. Estimate the magnitudes of the emf and current induced in the coil. Horizontal component of the earth’s magnetic field at the place is `3.0 × 10^(–5)` T.
Class 12 Chapter 6 Example 3

Initial flux through the coil,

`color{green} {Φ_(B ("initial")) = BA cos theta}`

`= 3.0 × 10^(–5) × (π ×10^(–2)) × cos 0^o`

`= 3π × 10^(–7) Wb`

Final flux after the rotation,

`color{orange} {Φ_(B ("initial"))}`

` = 3.0 × 10^(–5) × (π ×10^(–2)) × cos 180^0`

`= 3π × 10^(–7) Wb`

Therefore, estimated value of the induced emf is,

`color{purple} {ε = N (DeltaΦ)/(Deltat)}`

`= 500 × (6π × 10^(–7))/0.25`

`= 3.8 × 10^(–3) V`

`I = ε//R = 1.9 × 10^(–3) A`

Note that the magnitudes of `ε` and I are the estimated values. Their instantaneous values are different and depend upon the speed of rotation at the particular instant.