Electromagnetic Waves and Wave Optics 

Electromagnetic waves and wave optics form a crucial part of the JEE Physics syllabus under Modern Physics and Optics. These topics explain the dual behavior of light—sometimes behaving as a wave and other times as a particle.

While electromagnetic waves describe the oscillating electric and magnetic fields propagating through space, wave optics deals with the phenomena that can only be explained by treating light as a wave, such as interference, diffraction, and polarisation.

1.0What Are Electromagnetic Waves?

Electromagnetic waves are oscillating electric and magnetic fields that propagate through space, carrying energy without the need for a medium. Unlike mechanical waves (such as sound), EM waves can travel in a vacuum.

• They are transverse waves: the electric field and magnetic field  oscillate perpendicular to each other and to the direction of wave propagation.
• All electromagnetic waves travel at the speed of light (c ≈ 3 × 10⁸ m/s) in vacuum.
• They transport energy and momentum.
• The velocity of electromagnetic wave in a medium is given by

In medium : $v=\frac{1}{\sqrt{\mu ϵ}}=\frac{1}{\sqrt{({\mu }_0{\mu }_r)({ϵ}_0{ϵ}_r)}}=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}\frac{1}{\sqrt{{\mu }_r{ϵ}_r}}=\frac{c}{\sqrt{{\mu }_r{ϵ}_r}}$

• Refractive index, $n=\frac{c}{v}=\sqrt{{\mu }_r{ϵ}_r}$

Note: 

1. Like any other wave, $c=f\lambda$ is applicable for EMW as well.
2. The velocity of electromagnetic waves of different frequency in vacuum is the same but in a medium is different. It is more for red light and less for violet light.
3. It has been found that the velocity (c) of electromagnetic wave in free space is equal to the ratio of amplitude of electric field vector (E₀) and magnetic field vector (B₀), i.e.

$c=\frac{E_0}{B_0}$

Examples: Light waves, radio waves, X-rays, gamma rays.

2.0How Are Electromagnetic Waves Formed?

Electromagnetic waves are generated when charged particles accelerate. This produces a time-varying electric field, which induces a time-varying magnetic field, and together they sustain each other.

• A changing electric field generates a magnetic field.
• A changing magnetic field generates an electric field.
• This mutual generation leads to a self-sustaining wave that propagates outward.

Example: In a radio antenna, oscillating electrons create oscillating electric and magnetic fields that propagate as radio waves.

Maxwell's Equations: The Foundation

The existence and properties of EM waves were predicted by James Clerk Maxwell based on a set of four fundamental equations, known as Maxwell's Equations. These equations unify electricity, magnetism, and optics.

1. Gauss's Law for Electricity: Relates the electric field to the charge distribution.

$\oint \vec{E}\cdot \overrightarrow{dA}=\frac{Q_{\text{enc}}}{{ϵ}_0}$

2. Gauss's Law for Magnetism: States that magnetic monopoles do not exist.

$\oint \vec{B}\cdot \overrightarrow{dA}=0$

3. Faraday's Law of Induction: Describes how a changing magnetic field creates an electric Field.

$\oint \vec{E}\cdot d\vec{l}=-\frac{d{\varphi }_B}{dt}$

4. Ampere-Maxwell Law: Describes how both an electric current and a changing electric field create a magnetic field.

$\oint \vec{B}\cdot d\vec{l}={\mu }_0\left(I_c+{\epsilon }_0\frac{d{\varphi }_E}{dt}\right)$

These equations show that a changing electric field creates a changing magnetic field, which in turn creates a changing electric field, and this self-perpetuating process leads to the propagation of an EM wave.

Properties of EM Waves

• Transverse Nature: The electric and magnetic fields are perpendicular to each other and to the direction of propagation.
• Speed in Vacuum: All EM waves travel at the speed of light (c) in a vacuum.
• c = 1 / √(μ₀ ε₀) ≈ 3 × 10⁸ m/s, where μ₀ and ε₀ are the permeability and permittivity of free space.
• Energy and Momentum: EM waves carry both energy and momentum. The energy density is equally shared between the electric and magnetic fields.
• No Medium Required: They can travel through a vacuum, which is a key difference from mechanical waves.
• Relationship between E and B fields: The magnitudes of the electric and magnetic fields are related by E=cB.
• ${\mu }_0{\epsilon }_0=\frac{{\mu }_0}{4\pi }(4\pi {\epsilon }_0)=\frac{10^{-7}}{9\times 10^9}=\frac{1}{9\times 10^{16}}=\frac{1}{(3\times 10^8{)}^2}=\frac{1}{c^2}.$

So, in vacuum, the speed of electromagnetic wave (EMW) is $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=3\times 10^8\text{m/s}$

In a medium:

$v=\frac{1}{\sqrt{\mu \epsilon }}=\frac{1}{\sqrt{({\mu }_0{\mu }_r)({\epsilon }_0{\epsilon }_r)}}=\frac{1}{\sqrt{({\mu }_0{\epsilon }_0)({\mu }_r{\epsilon }_r)}}=\frac{c}{\sqrt{{\mu }_r{\epsilon }_r}}.$

• Refractive index:

$n=\frac{c}{v}=\sqrt{{\mu }_r{\epsilon }_r}$

Note: Like any other wave, c=fλ is applicable for electromagnetic waves (EMW) as well.

The Electromagnetic Spectrum

The electromagnetic spectrum is the range of all types of EM radiation, based on their frequency or wavelength. The spectrum is continuous and ranges from low-frequency radio waves to high-frequency gamma rays.

• Radio waves: Used in communication.
• Microwaves: Used in radar and ovens.
• Infrared (IR): Emitted by warm objects.
• Visible light: The small portion of the spectrum visible to the human eye.
• Ultraviolet (UV): Can cause sunburn.
• X-rays: Used in medical imaging.
• Gamma rays: Produced by nuclear reactions, highly energetic.

Sources of EM Waves

EM waves are produced by accelerated charges. A stationary charge produces an electric field, and a charge moving at a constant velocity produces both electric and magnetic fields. However, only an accelerating charge can radiate energy in the form of EM waves.

3.0Wave Optics

Introduction to Wave Optics

Wave optics, or physical optics, is the branch of optics that studies phenomena where the wave nature of light is significant. This includes interference, diffraction, and polarization, which cannot be explained by ray optics (geometrical optics).

Huygens' Principle

Huygens' Principle is a geometric construction that explains how waves propagate. It states:

1. Every point on a wavefront acts as a source of secondary wavelets.
2. These wavelets spread out in all directions with the speed of the wave.
3. The new wavefront at any instant is the envelope of all these secondary wavelets.

This principle can be used to derive the laws of reflection and refraction and to explain interference and diffraction.

Interference of Light

Interference is the superposition of two or more waves to form a new wave pattern. For light, this results in a pattern of bright and dark fringes.

Young's Double-Slit Experiment (YDSE)

YDSE is a classic experiment that demonstrates the wave nature of light. A monochromatic light source illuminates two narrow slits, which act as coherent sources. The waves from these slits interfere to produce a pattern of alternating bright and dark fringes on a screen.

• Path Difference (Δx): The difference in the distance traveled by light from the two slits to a point on the screen.
• Constructive Interference (Bright Fringes): Occurs when Δx=nλ, where n=0,±1,±2,…. The waves are in phase.
• Destructive Interference (Dark Fringes): Occurs when $\Delta x=\left(n+\frac{1}{2}\right)\lambda$, where n=0,±1,±2,…. The waves are out of phase.

Conditions for Sustained Interference

1. Coherent Sources: The sources must have a constant phase difference.
2. Monochromatic Light: The light should have a single wavelength.
3. Sources must be close to each other: To get a clear and visible fringe pattern.

Fringe Width and Intensity

The distance between two consecutive bright or dark fringes is called the fringe width (β).

$\beta =\frac{\lambda D}{d}$

• λ is the wavelength of light.
• D is the distance between the slits and the screen.
• d is the distance between the two slits.

The intensity of the interference pattern varies from maximum (constructive) to minimum (destructive).

$I_{net}=I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi$

Where ϕ is the phase difference. If $I_1=I_2=I_0,then{I}_{max}=4I_{0}and{I}_{min}=0$

Diffraction of Light

Diffraction is the bending of waves around obstacles or through openings. It's the reason we can hear sounds around corners. For light, diffraction is significant when the size of the aperture or obstacle is comparable to the wavelength of light.

Fraunhofer and Fresnel Diffraction

• Fraunhofer Diffraction: The source and screen are far from the obstacle (or lens is used), resulting in a parallel beam of light. This is the case for single-slit diffraction.
• Fresnel Diffraction: The source and screen are close to the obstacle, resulting in spherical wavefronts.

Diffraction at a Single Slit

When a monochromatic plane wave passes through a narrow single slit, it diffracts to produce a central bright maximum surrounded by alternating dark and bright fringes of decreasing intensity.

• Condition for minima (dark fringes): a sin θₙ = nλ, where n = ±1, ±2, …
• Condition for maxima (secondary bright fringes): $a\sin {\theta }_n=\left(n+\frac{1}{2}\right)\lambda$, where n=±1,±2,…
• a is the width of the slit.
• θₙ ​ is the angular position of the nᵗʰ  minimum or maximum.

The central maximum is significantly wider and brighter than the secondary maxima.

Resolving Power

Resolving power is the ability of an optical instrument to distinguish between two closely spaced objects. Due to diffraction, the images of two points are not sharp points but rather overlapping diffraction patterns.

• Rayleigh's Criterion: Two objects are just resolvable when the central maximum of the diffraction pattern of one object falls on the first minimum of the diffraction pattern of the other.

$$\begin{gathered} \text{Resolving Power of a Telescope:}\text{R.P.}=\frac{D}{1.22\lambda } \\ \text{Resolving Power of a Microscope: }\text{R.P.}=\frac{2\mu \sin \theta }{\lambda } \end{gathered}$$

Polarization of Light

Polarization is the phenomenon where the vibrations of the electric field vector in a light wave are restricted to a single plane. Unpolarized light vibrates in all directions perpendicular to its direction of propagation.

Unpolarized and Polarized Light

• Unpolarized Light: The electric field vector vibrates randomly in all directions perpendicular to the direction of propagation.
• Plane (Linearly) Polarized Light: The electric field vector vibrates in a single plane.

Polarization can be achieved using polaroids, which are special sheets that only allow vibrations in a specific direction (its pass axis).

Malus' Law

Malus' Law describes the intensity of light transmitted through a polarizer.

• I is the intensity of the transmitted light.
• $I_0$​ is the intensity of the incident polarized light.
• θ is the angle between the polarization direction of the incident light and the pass axis of the polarizer.

If unpolarized light of intensity I0​ is passed through a polarizer, the transmitted intensity is I0​/2.

Brewster's Law

When light is incident on a surface (like glass) at a specific angle, the reflected light is completely polarized. This angle is called Brewster's angle (θₚ).

• Brewster's Law: tan θₚ = μ , where is the refractive index of the medium.
• At this angle, the reflected ray and the refracted ray are perpendicular to each other.

Illustration-1

Light of wavelength 6000 A˚ is incident normally on a slit of width 24 × 10⁻⁵ cm. Find out the angular position of the second minimum from the central maximum.

Solution:

Given:

$$\begin{gathered} \lambda =6\times 10^{-7}\text{m},a=24\times 10^{-5}\times 10^{-2}\text{m} \\ a\sin \theta =n\lambda \\ \sin \theta =\frac{2\lambda }{a}=\frac{2\times 6\times 10^{-7}}{24\times 10^{-7}}=\frac{1}{2} \\ \boxed{\theta =30^{\circ }} \end{gathered}$$

Illustration-2

A beam of light consisting of wavelengths 6000 A˚ and 4500 A˚ is used in a YDSE with D=1 m and d=1 mm. Find the least distance from the central maximum where bright fringes due to the two wavelengths coincide.

Solution:

$$\begin{gathered} {\beta }_1=\frac{{\lambda }_{1}D}{d}=\frac{6000\times 10^{-10}\times 1}{10^{-3}}=0.6\text{mm} \\ {\beta }_2=\frac{{\lambda }_{2}D}{d}=0.45\text{mm} \end{gathered}$$

Let the n1th​ maxima of λ1​ and n2th​ maxima of λ2​ coincide at a position y.

Illustration-3

A beam of light consisting of wavelengths 6000 Å and 4500 Å is used in a YDSE with D = 1 m and d = 1 mm. Find the least distance from the central maxima, where bright fringes due to the two wavelengths coincide.

Solution:

$$\begin{gathered} {\beta }_1=\frac{{\lambda }_{1}D}{d}=\frac{6000\times 10^{-10}\times 1}{10^{-3}}=0.6\text{mm}; \\ {\beta }_2=\frac{{\lambda }_{2}D}{d}=0.45\text{mm} \\ Let{n}^{\text{th}}\text{maxima of}{\lambda }_{1}and{n}_2^{\text{th}}\text{maxima of }{\lambda }_2\text{coincide at a position of y.} \\ \text{then,}y=n_1{\beta }_1=n_2{\beta }_2=\text{LCM of }{\beta }_1\text{ and }{\beta }_2 \\ \Rightarrow y=\text{LCM of }0.6\text{mm and }0.45\text{mm} \\ y=1.8\text{mm} \\ \text{At this point 3rd maxima for 6000 A˚ and 4th maxima for 4500 A˚ coincide}. \end{gathered}$$