Electromagnetic Induction Class 12 Physics Chapter 6 Notes

Electromagnetic Induction Class 12 Physics Chapter 6 Notes

Electromagnetic Induction

1. Electricity and Magnetism: In the early 19th century, electricity and magnetism were considered separate and unrelated phenomena. However, experiments conducted by scientists like Oersted, Ampere, and others revealed that moving electric charges, such as those in an electric current, could produce magnetic fields. This discovery suggested a fundamental connection between electricity and magnetism.
2. Reciprocal Effect: Scientists began to investigate whether the reciprocal effect was possible, i.e., whether moving magnets could produce electric currents. The experiments conducted by Michael Faraday in England and Joseph Henry in the USA around 1830 provided conclusive evidence that changing magnetic fields could induce electric currents in closed coils.
3. Electromagnetic Induction: Electromagnetic induction refers to the generation of electric current in a conductor when exposed to varying magnetic fields.
4. Practical Significance: Faraday’s discovery of electromagnetic induction was met with curiosity about its practical utility. When asked about the use of this phenomenon, Faraday responded by comparing it to the birth of a baby, emphasizing its significance and potential. Indeed, electromagnetic induction has played a crucial role in the development of generators and transformers, which are fundamental to modern electricity generation and distribution.
5. Impact on Civilization: The introduction underscores that the progress and conveniences of today’s civilization, including electric lighting, trains, telephones, and personal computers, owe a great deal to the discovery of electromagnetic induction. It highlights how this scientific breakthrough has had a profound impact on society and technology.

Experiments conducted by Faraday and Henry to illustrate the fundamental principles of electromagnetic induction

Experiment 6.1: In this experiment, a coil (C1) is connected to a galvanometer (G). When a North-pole of a bar magnet is moved toward the coil, the galvanometer’s pointer deflects, indicating the presence of an induced electric current in the coil. The deflection is temporary and occurs only while the magnet is in motion. When the magnet is stationary, there is no deflection. Moving the South-pole of the magnet toward or away from the coil produces opposite deflections. The deflection is also larger when the magnet is moved faster. The key takeaway is that relative motion between the magnet and the coil induces the current.

Experiment 6.2: Here, the bar magnet is replaced by a second coil (C2) connected to a battery, creating a steady current in C2, which in turn produces a steady magnetic field. As coil C2 is moved toward or away from coil C1, the galvanometer shows deflections, indicating induced electric currents in C1. Again, the relative motion between the coils induces the current.

Experiment 6.3: This experiment demonstrates that relative motion is not an absolute requirement for electromagnetic induction. Two coils, C1 and C2, are held stationary. C1 is connected to a galvanometer (G), while C2 is connected to a battery via a tapping key (K). When the key is pressed momentarily, the galvanometer shows a brief deflection that returns to zero when the key is released. If the key is continuously held down, there is no deflection. Releasing the key results in another momentary deflection in the opposite direction. Additionally, inserting an iron rod along the axis of the coils enhances the deflection dramatically.

Magnetic flux

Magnetic flux, denoted by Φ_B, plays a crucial role in understanding electromagnetic induction. It is defined as the magnetic field (B) passing through a given surface area (A). The mathematical expression for magnetic flux is:

Φ_B = B · A = BA cos θ

Where:

• Φ_B is the magnetic flux.
• B is the magnetic field strength (in teslas, T).
• A is the area vector (in square meters, m^2).
• θ is the angle between the magnetic field vector (B) and the surface area vector (A).

This definition allows us to calculate the magnetic flux through a plane or a surface exposed to a uniform magnetic field. The angle θ is important because it represents the orientation of the surface area with respect to the magnetic field. When θ is 0 degrees (cos θ = 1), the magnetic field is perpendicular to the surface, and the flux is maximized. When θ is 90 degrees (cos θ = 0), the magnetic field is parallel to the surface, and there is no magnetic flux through it.

For more complex cases where the magnetic field varies across a surface or if the surface is curved, you can calculate the total magnetic flux Φ_B by summing up the contributions from each small area element dA_i. The formula for this is:

Φ_B = ∫(B · dA)

Where the integral symbol (∫) represents the sum over all the area elements dA_i, each with its associated magnetic field strength B_i. This formula extends the concept of magnetic flux to non-uniform fields and curved surfaces.

The SI unit of magnetic flux is the weber (Wb), which is equivalent to tesla meter squared (T m²). Magnetic flux is a scalar quantity, meaning it has magnitude but no specific direction. It is a fundamental concept in electromagnetism and is essential for understanding Faraday’s laws of electromagnetic induction.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law of Electromagnetic Induction is a fundamental principle in electromagnetism that describes how the electromotive force (emf) is induced in a circuit when there is a change in magnetic flux through the circuit. Faraday’s law is stated as follows:

“The magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.”

Mathematically, this law is expressed as:

ε = -dΦ_B/dt

Where:

• ε is the induced electromotive force (emf) in the circuit.
• dΦ_B/dt represents the rate of change of magnetic flux through the circuit with respect to time (t).
• The negative sign indicates the direction of the induced emf and, consequently, the direction of the induced current in a closed loop.

In the case of a closely wound coil with N turns, where each turn experiences the same change in magnetic flux, the total induced emf is given by:

ε = -NdΦ_B/dt

This equation highlights that the induced emf can be increased by increasing the number of turns (N) in the coil.

Faraday’s law demonstrates that there are various ways to change the magnetic flux through a circuit, thereby inducing an emf. This change can be achieved by altering any of the following factors:

1. Changing the magnetic field strength (B).
2. Varying the area (A) through which the magnetic field lines pass.
3. Adjusting the angle (θ) between the magnetic field direction and the plane of the coil.

In practical experiments, these changes can be realized by moving magnets or coils, varying the strength of the magnetic field, or altering the orientation of the coil with respect to the field. Faraday’s law is a foundational principle in the operation of electric generators, transformers, and many other devices that involve the conversion of magnetic energy into electrical energy or vice versa.

Lenz’s Law

Lenz’s Law is a fundamental principle in electromagnetism that describes the direction of the induced electromotive force (emf) in a circuit due to a changing magnetic flux. Lenz’s Law is stated as follows:

“The polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.”

In other words, Lenz’s Law dictates that the direction of the induced current is such that it creates a magnetic field that opposes the change in the magnetic field that caused the induction in the first place. This law is consistent with the law of conservation of energy, as it ensures that energy is not created or destroyed in the process of electromagnetic induction.

Let’s break down Lenz’s Law in the context of an example:

Suppose you have a coil of wire, and you move a permanent magnet toward the coil. As the magnet approaches the coil, it induces an emf and current in the coil. According to Lenz’s Law, the direction of this induced current will be such that it creates a magnetic field that opposes the increasing magnetic field of the approaching magnet. This opposing magnetic field exerts a repulsive force on the approaching magnet, making it more difficult to move the magnet toward the coil. The energy needed to overcome this repulsive force is provided by the external work done to move the magnet.

If, on the other hand, the induced current were to flow in the opposite direction, it would produce an attractive force on the approaching magnet, essentially allowing it to accelerate without any external work. This scenario would violate the law of conservation of energy, as it would suggest that energy is created without any source. Lenz’s Law ensures that such perpetual-motion machines, which violate the conservation of energy, cannot exist.

A scenario in which a straight conductor is moving in a uniform and time-independent magnetic field

1. Setup: Imagine a rectangular conductor PQRS where PQ is free to move. This rod PQ is being moved to the left with a constant velocity v. The conductor PQRS forms a closed circuit that encloses an area, and it is placed in a uniform magnetic field B perpendicular to its plane.
2. Changing Magnetic Flux: As the rod PQ moves, the length x of RQ changes with time. This means that the area enclosed by the loop PQRS changes as well. The magnetic flux FB through this loop is given by FB = Blx, where l is the length RS.
3. Induced EMF: Because x is changing with time, the rate of change of magnetic flux (dFB/dt) will induce an electromotive force (emf), denoted as ε. The emf ε is given by ε = -dFB/dt. Using the expression for FB, we can simplify this to ε = Blv, where v is the speed of the conductor PQ.
4. Motional EMF: The induced emf Blv, which is generated due to the motion of the conductor in the magnetic field, is called motional emf. It’s a form of induced emf that results from the motion of the conductor rather than from a changing magnetic field.
5. Explanation Using Lorentz Force: Another way to understand this motional emf is by considering the Lorentz force acting on the free charge carriers within the conductor PQ. As the rod moves with speed v, the charges experience a force of qvB in magnitude, directed towards point Q. The work done to move these charges across the conductor generates the induced emf, which is Blv.
6. Static Conductor vs. Changing Magnetic Field: It’s worth noting that when a conductor is stationary and the magnetic field is changing, an emf is also induced, as demonstrated by Faraday’s experiments. In this case, the force on the charges arises solely from the electric field (E), which is generated by the changing magnetic field. This phenomenon highlights the connection between electricity and magnetism.

Inductance

Inductance is a fundamental concept in electromagnetism, and it plays a crucial role in understanding the behavior of coils and circuits. Here are the key points regarding inductance:

1. Definition: Inductance is a property of an electrical circuit or coil that quantifies its ability to induce an electromotive force (emf) in response to a change in current. It represents the inherent opposition to changes in current flow within a circuit.
2. Proportional to Flux Change: The flux (magnetic flux) through a coil is directly proportional to the current passing through it. In other words, as the current changes, the magnetic flux through the coil also changes. Mathematically, FB ∝ I, where FB is the magnetic flux and I is the current.
3. Flux Linkage: In the case of a closely wound coil with N turns, each turn contributes to the induced emf when the flux through the coil changes. Flux linkage is a term used to describe the total magnetic flux linked with all the turns of the coil. For a closely wound coil, flux linkage (NFB) is directly proportional to the current (I).
4. Inductance: The constant of proportionality in the relation NFB ∝ I is called inductance. Inductance is denoted by the symbol L, and it depends on the geometry of the coil and intrinsic material properties. Just as capacitance depends on factors like plate area and separation for a capacitor, inductance depends on coil geometry and intrinsic properties.
5. Scalar Quantity: Inductance is a scalar quantity, meaning it has magnitude but no direction. It is measured in henrys (H), and 1 henry is equal to 1 weber per ampere (1 H = 1 Wb/A).
6. Unit and Name: The SI unit of inductance is the henry (H), named in honor of Joseph Henry, who made significant contributions to the discovery of electromagnetic induction independently of Michael Faraday.

Mutual Inductance

Mutual inductance is a fundamental concept in electromagnetism that describes the relationship between two coils or circuits. Here are the key points regarding mutual inductance:

1. Definition: Mutual inductance (denoted as M) is a measure of how one coil or circuit induces an electromotive force (emf) in another coil or circuit when the current in the first coil changes. It quantifies the extent to which the magnetic field generated by one coil links with the other coil and affects the induced emf.
2. Mutual Induction in Co-Axial Solenoids: The concept of mutual inductance is illustrated using the example of two long co-axial solenoids, S1 and S2. When a current (I2) flows through S2, it generates a magnetic field that links with S1, resulting in a magnetic flux (Φ1) through S1. The mutual inductance M12 of S1 with respect to S2 is given by: M12=μ0​⋅n1⋅n2⋅π⋅r1^2⋅l,where:
   • μ0​ is the permeability of free space.
   • n1 and n2 are the number of turns per unit length in S1 and S2, respectively.
   • r1 is the radius of S1.
   • l is the length of both solenoids.
3. Reciprocity: Mutual inductance is a reciprocal property. This means that if you swap the roles of S1 and S2 (i.e., consider the flux generated by S1 on S2), you will obtain the same value of mutual inductance (M21 = M12).
4. Dependence on Separation and Orientation: Mutual inductance depends on the physical separation between the coils and their relative orientation. For co-axial solenoids, the calculation is simplified due to the uniformity of the magnetic field.
5. Medium with Permeability: When a medium with relative permeability (mr) is present between the coils, the mutual inductance is modified as:
   • M=μr​⋅μ0​⋅n1⋅n2⋅π⋅r1^2⋅l
   • where μr​ is the relative permeability of the medium.
6. Applications: Mutual inductance is essential in understanding the behavior of transformers, inductors, and various electromagnetic devices. It plays a significant role in the design and analysis of electrical circuits and components.
7. Mutual Induction between Circuits: Mutual inductance is not limited to solenoids; it can occur between any two circuits or coils. When the current in one circuit changes, it induces an emf in the other circuit, and the magnitude of the induced emf depends on the rate of change of current and the mutual inductance between the circuits.

Self Inductance

Self-inductance is a crucial concept in electromagnetism that explains how a changing current in a coil can induce an electromotive force (emf) in the same coil. Here are the key points regarding self-inductance:

1. Definition: Self-inductance (denoted as L) is a property of a coil or circuit that quantifies the extent to which a changing current in that coil induces an emf in itself. It is also called the coefficient of self-induction.
2. Flux Linkage: When a current I flows through a coil with N turns, it generates a magnetic field. The magnetic flux (ΦB​) linked with the coil is directly proportional to the current:
   • N⋅I⋅ΦB​∝N⋅I
3. Expression for Self-Inductance: Self-inductance (L) is the constant of proportionality in the relationship between the magnetic flux and the current:
   • N⋅I⋅ΦB​=L⋅I.
   • The SI unit of self-inductance is the henry (H).
4. Induced Emf: When the current in the coil changes with time, it induces an emf (ε) within the same coil. The induced emf opposes the change in current and follows Faraday’s law of electromagnetic induction:
   • ε=−L⋅dtdI.
   • ​The negative sign indicates that the induced emf always opposes the change in current.
5. Physical Analogy: Self-inductance is often compared to inertia in mechanics. Just as inertia resists changes in velocity (acceleration) of a moving object, self-inductance resists changes in current in a circuit.
6. Energy Storage: Work needs to be done against the back emf (induced emf) to establish a current in the coil. This work done is stored as magnetic potential energy. The energy required to build up the current I is given by:
   • W=1/2​⋅L⋅I2.
   • This equation is analogous to the kinetic energy equation in mechanics (KE=1/2​⋅mv^2).
7. Dependence on Geometry and Medium: Self-inductance depends on the geometry of the coil (such as its number of turns and dimensions) and the permeability of the medium inside the coil. In the presence of a medium with relative permeability (μr​), self-inductance is modified.
8. Back Emf: The self-induced emf is also referred to as the “back emf” because it opposes any changes in current, acting as a counterforce.
9. General Case: When multiple coils are present, each coil contributes to the total self-inductance. The self-inductance of a coil due to its own changing current is called “self-inductance,” while the self-inductance of a coil due to the changing current in another nearby coil is called “mutual inductance.”

Self-inductance is a fundamental concept in the analysis of electrical circuits and plays a significant role in the behavior of inductors and electromagnetic devices. It is essential for understanding how circuits respond to changes in current and for designing and analyzing electrical circuits and components.

AC Generator

An AC generator, also known as an alternator, is a device that converts mechanical energy into electrical energy by utilizing the principles of electromagnetic induction. Here are the key points regarding the operation of an AC generator:

1. Basic Components: An AC generator consists of several essential components:
   • Armature: This is a coil of wire that is mounted on a rotor shaft. The coil is usually made up of multiple turns.
   • Magnetic Field: There is a uniform magnetic field (represented by B) in which the armature rotates. The magnetic field lines are directed perpendicular to the plane of the coil.
   • Rotor: The rotor is the rotating part of the generator that carries the armature.
   • External Circuit: The ends of the coil (armature) are connected to an external electrical circuit through slip rings and brushes. This connection allows the generated electrical energy to be transferred to the external circuit.
2. Principle of Operation: When the coil (armature) is mechanically rotated within the magnetic field, the effective area of the coil exposed to the magnetic field changes continuously with time as the coil rotates.
3. Flux Change: As a result of the changing orientation of the coil with respect to the magnetic field, the magnetic flux (ΦB​) through the coil changes over time.
4. Induced EMF: According to Faraday’s law of electromagnetic induction, a changing magnetic flux through a coil induces an electromotive force (emf or voltage) in the coil. The induced emf (ε) is proportional to the rate of change of magnetic flux and the number of turns in the coil (N). Mathematically, it can be expressed as: ε=−NdtdΦB​​
5. AC Output: The induced emf varies sinusoidally with time as the coil rotates. It is an alternating emf, which means its magnitude and direction change periodically. The equation for the instantaneous emf (ε) is: ε(t)=e0​sin(2πnt) where e0​ is the maximum emf, n is the frequency of rotation (related to the generator’s frequency), and t is time.
6. Frequency: The frequency of the alternating current (AC) generated by the generator depends on the speed of rotation of the armature. In India, the standard frequency is 50 Hz, while in some other countries, such as the USA, it is 60 Hz.
7. Mechanical Energy Source: AC generators require a source of mechanical energy to rotate the armature. Common sources include hydroelectric power (using falling water), thermal power (using steam produced by heating water with sources like coal or nuclear fuel), and even wind turbines (using wind energy to rotate the generator’s rotor).
8. Commercial Generators: Modern AC generators used in power plants can have a very high output capacity, often reaching hundreds of megawatts (MW). These generators can provide electrical power to millions of homes and businesses.
9. Rotation of Armature: In some generators, it is the armature that rotates within a stationary magnetic field, while in others, it is the magnetic field (electromagnets) that rotates around the stationary armature.

AC generators play a crucial role in generating electrical power for various applications, from residential electricity supply to industrial and commercial use. They provide the alternating current needed for most of our electrical devices and appliances. The conversion of mechanical energy into electrical energy through electromagnetic induction is a fundamental principle of electrical engineering.