1.3 Faraday's law

Faraday's law of induction explains how changing magnetic fields create electric currents. This fundamental principle underlies the operation of generators, transformers, and other electromagnetic devices, forming a crucial link between electricity and magnetism.

The law states that the induced electromotive force in a closed loop is proportional to the rate of change of magnetic flux through the loop. This relationship is central to understanding electromagnetic induction and its wide-ranging applications in modern technology.

Faraday's law of induction

• Fundamental law in electromagnetism describes the relationship between changing magnetic fields and induced electric fields
• States that a time-varying magnetic field induces an electromotive force (emf) in a conductor or circuit
• Forms the basis for understanding the operation of generators, transformers, and other electromagnetic devices

Magnetic flux and flux linkage

Magnetic flux through a surface

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• Quantifies the amount of magnetic field passing through a given surface area
• Depends on the strength of the magnetic field, the area of the surface, and the orientation of the surface relative to the field
• Mathematically expressed as the integral of the magnetic field over the surface area: $\Phi_B = \int \vec{B} \cdot d\vec{A}$
• Measured in units of weber (Wb) or tesla-square meter (T·m²)

Flux linkage in a coil

• Represents the total magnetic flux passing through all the turns of a coil or inductor
• Calculated by multiplying the magnetic flux through a single turn by the number of turns in the coil: $\lambda = N\Phi_B$
• Plays a crucial role in determining the induced emf in a coil according to Faraday's law
• Measured in units of weber-turns (Wb·turns)

Induced electromotive force (emf)

Faraday's experiments

• Faraday discovered that a changing magnetic field can induce an electric current in a conductor
• Demonstrated the phenomenon using a coil of wire and a moving magnet
• Observed that the induced current depends on the rate of change of the magnetic field and the number of turns in the coil
• Laid the foundation for the development of generators and transformers

Mathematical formulation of Faraday's law

• States that the induced emf in a closed loop is equal to the negative rate of change of the magnetic flux through the loop: $\mathcal{E} = -\frac{d\Phi_B}{dt}$
• For a coil with N turns, the induced emf is given by: $\mathcal{E} = -N\frac{d\Phi_B}{dt}$
• The negative sign indicates that the induced emf opposes the change in magnetic flux (Lenz's law)
• Measured in units of volts (V)

Lenz's law and conservation of energy

Direction of induced current

• Lenz's law states that the direction of the induced current is such that it opposes the change in magnetic flux that caused it
• The induced current creates a magnetic field that opposes the original change in magnetic flux
• Helps maintain the conservation of energy by preventing the creation of perpetual motion
• Determines the polarity of the induced emf in a coil or conductor

Energy considerations in induction

• The work done by the induced emf is equal to the change in magnetic energy in the system
• The induced current dissipates energy as heat due to the resistance of the conductor (Joule heating)
• In generators and motors, the mechanical work done is converted into electrical energy or vice versa
• The conservation of energy is always maintained in electromagnetic induction processes

Applications of Faraday's law

Generators and alternators

• Devices that convert mechanical energy into electrical energy using Faraday's law
• Consist of a rotating coil or conductor in a magnetic field, which induces an emf in the coil
• Alternators produce alternating current (AC) by using slip rings and brushes
• Generators can produce either AC or direct current (DC) depending on the commutator design

Transformers and power transmission

• Devices that change the voltage level of AC power using Faraday's law and mutual inductance
• Consist of two or more coils wound on a common magnetic core
• The primary coil is connected to the input voltage, while the secondary coil(s) provide the output voltage(s)
• Essential for efficient long-distance power transmission and distribution at high voltages

Eddy currents and magnetic braking

• Eddy currents are induced in conducting materials when exposed to changing magnetic fields
• Create a magnetic field that opposes the motion of the conductor, resulting in magnetic braking
• Used in applications such as braking systems, damping of oscillations, and induction heating
• Can cause energy losses in transformers and other electromagnetic devices

Maxwell's correction to Ampère's law

Displacement current

• Maxwell introduced the concept of displacement current to maintain the conservation of charge in time-varying electric fields
• Represents the rate of change of the electric flux density: $J_D = \frac{\partial \vec{D}}{\partial t}$
• Allows Ampère's law to be consistent with the continuity equation for electric charge
• Crucial for understanding the propagation of electromagnetic waves

Maxwell's equations in integral form

• Maxwell's equations summarize the fundamental laws of electromagnetism
• Gauss's law for electric fields: $\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0}$
• Gauss's law for magnetic fields: $\oint \vec{B} \cdot d\vec{A} = 0$
• Faraday's law: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
• Ampère-Maxwell law: $\oint \vec{B} \cdot d\vec{l} = \mu_0\left(I + \varepsilon_0\frac{d\Phi_E}{dt}\right)$

Motional emf and Lorentz force

Moving conductors in magnetic fields

• A conductor moving in a magnetic field experiences a motional emf due to the Lorentz force on the charge carriers
• The magnitude of the motional emf is given by: $\mathcal{E} = Blv\sin\theta$, where B is the magnetic field strength, l is the length of the conductor, v is the velocity, and θ is the angle between the velocity and the magnetic field
• The direction of the induced current is determined by the right-hand rule
• Forms the basis for the operation of electric motors and generators

Hall effect and its applications

• The Hall effect occurs when a current-carrying conductor is placed in a magnetic field perpendicular to the current
• The Lorentz force deflects the charge carriers, creating a transverse electric field (Hall voltage) across the conductor
• The Hall voltage is proportional to the current, magnetic field strength, and the inverse of the carrier density
• Used in applications such as magnetic field sensors, current sensors, and semiconductor characterization

Inductance and mutual inductance

Self-inductance of a coil

• Self-inductance is the property of a coil that opposes changes in the current flowing through it
• Arises due to the magnetic field generated by the current in the coil
• The self-inductance of a coil is given by: $L = \frac{N\Phi_B}{I}$, where N is the number of turns, ΦB is the magnetic flux, and I is the current
• Measured in units of henry (H)

Mutual inductance between coils

• Mutual inductance occurs when the magnetic flux generated by one coil links with another coil
• The mutual inductance between two coils is given by: $M = \frac{N_2\Phi_{B1}}{I_1} = \frac{N_1\Phi_{B2}}{I_2}$, where N1 and N2 are the number of turns in each coil, ΦB1 and ΦB2 are the magnetic fluxes, and I1 and I2 are the currents
• Measured in units of henry (H)
• Forms the basis for the operation of transformers and coupled inductors

Energy stored in magnetic fields

• The energy stored in the magnetic field of an inductor is given by: $U_B = \frac{1}{2}LI^2$
• Represents the work done in establishing the current in the inductor
• Can be released back into the circuit when the current changes
• Plays a role in the transient behavior of inductive circuits and the operation of energy storage devices (superconducting magnetic energy storage)

AC circuits and resonance

RLC circuits and impedance

• RLC circuits contain resistors (R), inductors (L), and capacitors (C) connected in series or parallel
• The impedance (Z) is the total opposition to the flow of alternating current in an RLC circuit
• Depends on the resistance, inductance, and capacitance, as well as the frequency of the AC signal
• Expressed as a complex number: $Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$, where ω is the angular frequency

Resonance in AC circuits

• Resonance occurs when the inductive and capacitive reactances in an RLC circuit are equal in magnitude
• At resonance, the impedance is purely resistive, and the current and voltage are in phase
• The resonant frequency is given by: $\omega_0 = \frac{1}{\sqrt{LC}}$
• RLC circuits exhibit maximum power transfer and minimum impedance at resonance
• Used in applications such as radio and television tuning, wireless power transfer, and filters

Power in AC circuits

• In AC circuits, power consists of real (active) power and reactive power
• Real power (P) is the average power consumed by the resistive components, measured in watts (W)
• Reactive power (Q) is the power exchanged between the inductive and capacitive components, measured in volt-ampere reactive (VAR)
• Apparent power (S) is the vector sum of real and reactive power, measured in volt-ampere (VA)
• Power factor (PF) is the ratio of real power to apparent power: $PF = \frac{P}{S} = \cos\theta$, where θ is the phase angle between voltage and current

Electromagnetic oscillations and waves

LC oscillations

• LC circuits consist of an inductor and a capacitor connected in series or parallel
• Energy oscillates between the electric field of the capacitor and the magnetic field of the inductor
• The oscillation frequency is given by: $f = \frac{1}{2\pi\sqrt{LC}}$
• LC oscillations are the basis for the generation and detection of electromagnetic waves
• Used in applications such as radio and television broadcasting, wireless communication, and radar

Electromagnetic wave equation

• Maxwell's equations can be combined to form the electromagnetic wave equation: $\nabla^2 \vec{E} = \mu_0\varepsilon_0\frac{\partial^2 \vec{E}}{\partial t^2}$ and $\nabla^2 \vec{B} = \mu_0\varepsilon_0\frac{\partial^2 \vec{B}}{\partial t^2}$
• Describes the propagation of electromagnetic waves in free space and other media
• The speed of electromagnetic waves in free space is given by: $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3 \times 10^8 m/s$
• Electromagnetic waves are transverse waves with oscillating electric and magnetic fields perpendicular to each other and the direction of propagation

Properties of electromagnetic waves

• Electromagnetic waves can propagate through vacuum and do not require a medium for transmission
• They exhibit properties such as reflection, refraction, diffraction, and interference
• The wavelength (λ) and frequency (f) of an electromagnetic wave are related by: $\lambda = \frac{c}{f}$
• The electromagnetic spectrum consists of radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, distinguished by their wavelengths and frequencies
• Electromagnetic waves carry energy and momentum, which can be absorbed, emitted, or scattered by matter