Electromagnetic Waves

8.1 Introduction

Displacement Current: Maxwell introduced the concept of displacement current to resolve inconsistencies in Ampere's circuital law.
Maxwell's Equations: A set of equations that express the basic laws of electromagnetism, including:

+ Electric and magnetic fields

+ Charge and current densities

+ Lorentz force formula

Electromagnetic Waves: Time-varying electric and magnetic fields that propagate in space.
Speed of Electromagnetic Waves: Approximately 3 × 10^8 m/s (close to the speed of light).
Light as an Electromagnetic Wave: Maxwell's work unified electricity, magnetism, and light.
Existence of Electromagnetic Waves: Predicted by Maxwell's equations.
Experimental Demonstration: Hertz (1885) experimentally confirmed the existence of electromagnetic waves.
Technological Impact: Revolution in communication (e.g., radio waves).

Electromagnetic Spectrum: Range: From γ-rays (wavelength ~10^-12 m) to long radio waves (wavelength ~10^6 m).

As per Maxwell's Hypothesis, a changing electric field produces a magnetic field. This causes a contradiction in Ampere's Circuital Law.

Applying Ampere's law to a capacitor with a time-varying current leads to a contradiction.

Displacement Current:

Maxwell introduced the concept of displacement current to resolve the contradiction.

Displacement Current Formula:

$i_d = \epsilon_0 \frac{d\Phi_E}{dt}$, where $\Phi_E$ is the electric flux.

Generalized Ampere's Circuital Law:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 (i_c + i_d)$, where $i_c$ is the conduction current and $i_d$ is the displacement current.

Displacement current has the same physical effects as conduction current.
Displacement current can be present in different regions of space, including regions with no conduction current.
Displacement current is a source of electromagnetic waves.

Sources of Electromagnetic Waves: Accelerated charges radiate electromagnetic waves.

Nature of Electromagnetic Waves: Electric and magnetic fields in an electromagnetic wave are perpendicular to each other and to the direction of propagation.
Electromagnetic Wave Propagation: Electromagnetic waves propagate through space with a speed of $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8$ m/s.

Frequency and Wavelength: $\nu \lambda = c$, where $\nu$ is the frequency and $\lambda$ is the wavelength.
Electric and Magnetic Field Relationship: $B_0 = \frac{E_0}{c}$.
Electromagnetic Waves in Material Media:
Velocity of Electromagnetic Waves: $v = \frac{1}{\sqrt{\mu \epsilon}}$, where $\mu$ and $\epsilon$ are the magnetic permeability and electric permittivity of the medium.

Self-sustaining oscillations: Electromagnetic waves are self-sustaining oscillations of electric and magnetic fields in free space or vacuum.
No material medium required: Electromagnetic waves do not require a material medium to propagate.
Technological Importance of Electromagnetic Waves:
Energy transfer: Electromagnetic waves can carry energy from one place to another.
Applications: Radio and TV signals, light, and other forms of electromagnetic radiation have numerous technological applications.