Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Electromagnetism → Maxwell's equations ↓


Faraday's law of induction


Faraday's law of induction is a fundamental principle of electromagnetism and plays a vital role in the science of electricity and magnetism. It was formulated by Michael Faraday in 1831 and is one of four equations known as Maxwell's equations, which form the foundation of classical electromagnetism. This law describes how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF), a phenomenon known as electromagnetic induction.

In simple terms, Faraday's law states that a changing magnetic field within a closed loop of wire induces a current in the wire. This property of nature underlies the functioning of electric generators, transformers and many types of electric motors and is a key concept in understanding how electricity can be generated from mechanical motion.

Concept and formulation

Mathematically, Faraday's law of induction can be expressed as:

EMF = -dΦB/dt

Where:

  • EMF is the electromotive force in volts.
  • ΦB (Phi sub B) is the magnetic flux flowing through a circuit, measured in webers (Wb).
  • t is time, measured in seconds.
  • The negative sign indicates the direction of the induced EMF (and hence the induced current in the closed loop), as specified by Lenz's law, which states that the direction of the induced EMF will oppose the change in magnetic flux that produces it.

Magnetic flux

To understand Faraday's Law properly, we first need to understand the concept of magnetic flux. Magnetic flux is a measure of the amount of magnetism, taking into account the strength and extent of the magnetic field. It is defined as the product of the average magnetic field times the perpendicular area it penetrates:

ΦB = B · A · cos(θ)

Where:

  • B is the magnetic field strength, measured in Tesla (T).
  • A is the area through which the magnetic field lines pass, measured in square metres (m²).
  • θ is the angle between the magnetic field lines and the perpendicular (normal) to the surface A.
Magnetic field (B) Area (A)

In the illustration, the rectangle represents the area (A) and the red line represents the direction of the magnetic field (B). If the magnetic field is perpendicular to the surface (θ = 0), the magnetic flux is maximum. Conversely, if the magnetic field is parallel to the field (θ = 90 degrees), the magnetic flux is zero.

Visualization of Faraday's law

Consider a simple experimental setup with a conducting loop in a magnetic field. Let us explain how changes in this magnetic field affect the induced electromotive force and hence the current in the loop.

Magnetic field (B) Circuit

In this example, the green line represents the magnetic field passing through a circular wire (circuit). Faraday's Law tells us that if the magnetic field changes - whether it intensifies, weakens, increases, decreases, or changes direction - the magnetic flux through the circuit changes, and an EMF is induced.

Examples of changes that induce EMF

  • The magnetic field (B) becomes stronger or weaker with time.
  • The area of the loop (A) that is exposed to the magnetic field is changed, such as by pressing or pulling the loop.
  • The loop rotates or moves within the magnetic field, causing the value of cos(θ) to change.

Applications of Faraday's law

Faraday's law of induction is essential in many real-world applications. One of the most common uses is in power generation. Let's look at some situations where Faraday's law is important:

Electric generator

In an electric generator, mechanical energy is converted into electrical energy. When the coil rotates in a magnetic field, the magnetic flux through the coil changes with time, producing EMF and resulting in electric current in the circuit. This principle is used in power stations to produce electricity.

ROTATION Coil

Here, the coil rotates in the magnetic field, ensuring a constant change in flux, thus providing a constant EMF. When the coil is perpendicular to the magnetic field, the change in flux is maximum, thereby optimising power generation.

Transformers

Transformers use the principle of electromagnetic induction to increase or decrease voltage levels in alternating current (AC) circuits. A transformer consists of two coils, called primary and secondary coils, wrapped around a magnetic core. The alternating current in the primary coil generates a changing magnetic field, which induces a voltage in the secondary coil. This voltage can be increased or decreased depending on the number of turns in the coil.

For ideal transformer, the relation is given as:

Vp/Vs = Np/Ns

Where:

  • Vp and Vs are the voltages of the primary and secondary coils respectively.
  • Np and Ns are the number of turns in the primary and secondary coils respectively.

Lenz's law and the direction of induced EMF

Lenz's law is an integral part of Faraday's law, which gives the direction of the induced EMF. This law states that the induced current will flow in such a way that it will oppose the change in magnetic flux that produces it. This opposition is the reason for the negative sign in Faraday's law. It is a consequence of the conservation of energy.

In this scenario, if the magnetic field through the loop increases, the induced EMF will produce a current whose magnetic field opposes the increase. Similarly, if the magnetic field decreases, the induced EMF will create a magnetic field that attempts to maintain the original magnetic flux.

Example problems and solutions

Let's explore a sample problem that demonstrates Faraday's Law:

Crisis

A rectangular coil of wire having 50 turns (N = 50) and cross-sectional area of 0.1 m² is placed in a uniform magnetic field which changes from 0.5 T to 2.0 T in 4 seconds. Calculate the average EMF induced in the coil.

Solution

The change in magnetic flux, ΔΦB, can be calculated as:

ΔΦB = N * Δ(B · A) = 50 * (2.0 T - 0.5 T) * 0.1 m² 
ΔΦB = 50 * 1.5 * 0.1 = 7.5 Wb

Using Faraday's law, the induced average EMF is:

EMF = - ΔΦB / Δt = - 7.5 Wb / 4 s = - 1.875 V

The negative sign indicates the direction of the induced EMF according to Lenz's law, but the magnitude of the average EMF induced in the coil is 1.875 V.

Conclusion

Faraday's law of induction is the basis of electromagnetism that provides fundamental information about how electric currents can be generated by changing magnetic fields. Understanding this principle is not only important for physicists and engineers, but it is also integral to the design and functioning of a wide range of electrical devices, from simple generators and transformers to more complex machinery used in industries. Faraday's insights about the nature of induction continue to influence the modern world and lead to innovations in technology and energy solutions.


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Faraday's law of induction

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