Electromagnetic Induction
State and show understanding of Faraday’s law of electromagnetic induction.
State and show understanding of Lenz’s law.
Discuss construction and working of A.C. generators.
Define eddy currents, explain how they arise and give a few examples where eddy currents are useful and where they are a nuisance.
Describe self-inductance and mutual inductance and understand their uses.
State the expression for energy stored in an inductor and use it wherever needed.
Discuss the construction, working principle, and importance of transformers.
Discuss the sources of energy loss in a practical transformer.
Introduction
Michael Faraday demonstrated the reverse effect of Oersted’s experiment. He explained the possibility of producing emf across the ends of a conductor when the magnetic flux linked with the conductor changes. This was termed as electromagnetic induction. The discovery of this phenomenon brought about a revolution in the field of power generation.
The phenomenon in which electric current is generated by varying magnetic fields.
Whenever the magnetic flux linked to a conductor or a coil changes, an emf is induced in it. This phenomenon is called electromagnetic induction. The emf thus produced is called induced emf and current due to this emf is called induced current.
If the magnetic flux through a circuit changes, an emf and a current are induced.
Magnetic Flux
The magnetic flux (Φ) linked with a surface held in a magnetic field (B) is defined as the number of magnetic lines of force crossing a closed area (A).
The magnetic flux through a surface of area A when placed in a magnetic field B is given by:
Φ = B · A · cosθ
Where, θ is the angle between the magnetic field vector B and the area vector A.
The direction of the area vector is given by the normal to the plane of the surface.
Magnetic flux is a scalar quantity; its SI unit is Weber (Wb).
If the magnetic field is non-uniform, then magnetic flux is calculated as:
Φ = ∫ B · dA
Cases of Magnetic Flux
Case 1: θ = 0°
If the direction of the magnetic field is parallel to the direction of the area vector (i.e., when the surface lies perpendicular to the direction of the field), the magnetic flux crossing the surface is maximum.
Case 2: θ = 90°
If the direction of the magnetic field is perpendicular to the direction of the area vector (i.e., when the surface lies parallel to the direction of the field), the magnetic flux crossing the surface is zero.
Case 3: θ = 180°
If the direction of the magnetic field is anti-parallel to the area vector, the maximum flux is linked with the surface but in the opposite direction.
Flux Linkage
If a coil has N number of turns, the same flux passes through every turn. So, flux linkage is the total flux passing through N turns, i.e.:
Flux Linkage = N × Φ
Faraday’s Laws of Electromagnetic Induction
Faraday’s First Law (Qualitative Statement):
Whenever the amount of magnetic flux linked with a closed circuit changes, an emf is induced in the circuit. The induced emf lasts as long as the change in magnetic flux continues.
Faraday’s Second Law (Quantitative Statement):
The magnitude of emf induced in a closed circuit is directly proportional to the rate of change of magnetic flux linked with the circuit, i.e.:
e = -dΦ/dt
The negative sign indicates the opposition to the change of flux by the induced emf (Lenz’s law).
If the closed loop is a coil of N turns, induced emf appears in every turn. The total emf induced is the sum of these individual emfs:
e_total = -N × (dΦ/dt)
Ways of Changing Magnetic Flux
- Changing the magnitude of the magnetic field within the closed loop.
- Changing either the total area or the portion of the area that lies within the magnetic field.
- Changing the angle between the direction of the field and the plane of the closed loop (e.g., by rotating it).
Lenz’s Law and Direction of Induced Current
Lenz’s law states that, “The direction of the induced current is such that it always opposes the cause which produced it.”
The induced emf (or current) has a direction such that the magnetic field due to this current opposes the cause (change in magnetic flux) that induces the current.
This principle is consistent with the conservation of energy. Any work done against the opposing force is converted into electrical energy.
Fleming’s Right-Hand Rule
This rule is used to determine the direction of the induced emf or current, especially in a generator. If the thumb, forefinger, and middle finger of the right hand are extended mutually perpendicular to each other:
- The thumb points in the direction of motion of the conductor.
- The forefinger points in the direction of the magnetic field.
- The middle finger points in the direction of the induced current.
Motional Electromotive Force (Motional EMF)
Consider a straight conductor moving in a uniform magnetic field. If a conductor of length L moves with velocity v in a magnetic field B perpendicular to its plane:
emf = B × L × v
The direction of the current is determined using Fleming’s Right-Hand Rule. This induced emf is known as motional emf.
The general form for motional emf can be expressed as:
emf = ∫ (v × B) · dl
Emf Induced in a Rotating Coil
Consider a rectangular coil of area A and N turns rotating in a uniform magnetic field B with angular velocity ω. At any time t, if the angle between the normal to the plane of the coil and the magnetic field is θ, the magnetic flux is:
Φ = B × A × cos(θ)
The induced emf is given by Faraday’s law:
e = -N × (dΦ/dt) = N × B × A × ω × sin(ωt)
This results in an alternating emf as it varies sinusoidally with time.
AC Generator (Alternator)
An AC generator is a device that converts mechanical energy into sinusoidally varying electrical energy based on electromagnetic induction.
Principle
When a closed conducting coil rotates in a uniform magnetic field, the flux linked with the coil changes continuously, inducing an emf.
Construction
- Armature: A rectangular coil wound with insulated copper wire on a non-magnetic metallic frame.
- Field Magnet: Provides the magnetic field. Permanent magnets are used in small generators, while electromagnets are used in larger ones.
- Slip Rings and Brushes: Slip rings are attached to the coil and rotate with it. Brushes are in contact with the rings to transfer the induced emf to the external circuit.
Working
When the armature rotates, the angle between the magnetic field and the normal to the coil changes, causing the flux linked with the coil to vary. This variation induces an alternating emf.
Eddy Currents
Eddy currents are loops of electric current induced in a conductor by a changing magnetic field. These currents flow in closed loops within the conductor, in planes perpendicular to the magnetic field.
Eddy currents can cause energy loss due to heating of the material (Joule heating). However, they have both useful applications and undesirable effects:
Applications of Eddy Currents
- Electromagnetic Damping: Used in galvanometers to dampen oscillations.
- Electric Brakes: Used in high-speed trains for braking systems.
- Induction Furnace: Used to heat metals for smelting.
- Metal Detectors: Operate based on eddy currents.
Minimizing Eddy Currents
Eddy currents can be minimized by using laminated cores in transformers and electrical machines. These laminations reduce the area available for eddy currents to flow, thus reducing losses.
Self-Induction
Self-induction is the property of a coil by which it opposes the change of current through it by inducing an emf in itself. This emf is called a “back emf.”
The self-induced emf is given by:
e = -L (di/dt)
Where L is the coefficient of self-induction or inductance.
Physical Significance
Self-induction opposes both the growth and decay of current in a coil. It is analogous to the inertia of a material body and is sometimes referred to as “electrical inertia.”
Mutual Induction
Mutual induction is the phenomenon in which a change in current in one coil induces an emf in a neighboring coil.
If two coils are placed close to each other, and the current in one coil changes, the magnetic flux linked with the second coil also changes, inducing an emf in the second coil.
The mutual inductance is given by:
e = -M (di/dt)
Where M is the mutual inductance, a measure of how effectively the two coils are coupled magnetically.
Energy Stored in an Inductor
When current flows through an inductor, energy is stored in its magnetic field. The energy stored is given by:
U = (1/2) L I²
Where:
- L is the inductance.
- I is the current through the inductor.
This energy can be recovered when the current decreases.
Transformer
A transformer is a device that converts low alternating voltage at high current into high alternating voltage at low current, and vice versa. It operates on the principle of mutual induction.
Construction
A transformer consists of two coils (primary and secondary) wound on a common laminated soft iron core. The primary coil is connected to the source, and the secondary coil is connected to the output circuit.
Working
When an alternating current flows through the primary coil, it creates a varying magnetic flux in the core. This varying flux links with the secondary coil, inducing an emf in it according to Faraday’s law.
The induced emf in the primary and secondary coils are given by:
e₁ = -N₁ (dΦ/dt), e₂ = -N₂ (dΦ/dt)
The ratio of emf in the secondary to the primary is determined by the turns ratio:
e₂ / e₁ = N₂ / N₁
If N₂ > N₁, the transformer is a step-up transformer, increasing voltage. If N₂ < N₁, it is a step-down transformer, decreasing voltage.
Efficiency of a Transformer
The efficiency of a transformer is defined as the ratio of output power to input power:
Efficiency = (Output Power / Input Power) × 100%
In an ideal transformer, the efficiency is 100%, but practical transformers have some losses.
Energy Losses in a Transformer
- Copper Loss: Due to resistance in the windings, heat is generated (H = I²R).
- Iron Loss: Caused by eddy currents and hysteresis in the iron core.
- Flux Leakage: Some magnetic flux may not link both primary and secondary coils.
- Humming Loss: Due to the vibration of the core caused by alternating magnetization.
Conceptual Questions
- Explain how Lenz’s law is consistent with the principle of conservation of energy.
- Why is the core of a transformer laminated?
- A copper ring is held horizontally, and a bar magnet is dropped through it. Explain why the magnet’s acceleration is less than g.
- What is self-induction? Why is an inductor coil typically made of copper?
Additional Conceptual Questions
- Explain why an induced current has no direction of its own.
- If the number of turns in a solenoid is doubled while keeping other factors constant, how does its self-inductance change?
- Two circular coils have the same number of turns, but one has twice the radius of the other. What is the ratio of their self-inductances?
- Can a DC voltage be stepped up or down using a transformer? Why or why not?
- Why is soft iron used as the core material in transformers?
- When a current is switched on in a circuit containing an inductor and a bulb in series, why doesn’t the bulb light up immediately?
Inductance
An inductor is a circuit element, typically in the form of a coil, that stores energy in its magnetic field when current passes through it. The ability of the coil to store this magnetic energy is called inductance and is denoted by L.
The self-induced emf is given by:
e = -L (di/dt)
Where L is the self-inductance of the coil.
Energy Stored in an Inductor
When a current I flows through an inductor, energy is stored in the magnetic field surrounding the coil. The energy stored is given by:
U = (1/2) L I²
This stored energy can be released back into the circuit when the current decreases.
Applications of Inductors
- Used in filters for audio and radio frequency signals.
- Used in energy storage applications (e.g., SMPS power supplies).
- Used in transformers for efficient energy transfer.
Conclusion
Electromagnetic induction forms the basis for various modern electrical devices such as transformers, generators, and motors. Understanding the principles of induction helps in the design and optimization of these devices, ensuring efficient and reliable operation.