Electromagnetic Waves -EC501- Module1( MAKAUT-Syllabus)

Today Our communication elements: -

Basics of Vectors: - Introduction to the vector, Scalars vs Vectors, Vector representation in EM waves, Vector operation in electromagnetics, Unit vector and direction, Polarization and vector, The importance of vectors in electromagnetic analysis.

Vector calculus: - Scalar, vector, and field; main operations in vector calculation; application.

Maxwell’s Equations: Gauss’s Law for Electricity, Gauss’s Law for Magnetism, Faraday’s Law of Electromagnetic Induction, Ampère–Maxwell Law, Application.

Basic Laws of Electromagnetics, Coulomb’s Law, Gauss’s Law, Biot–Savart Law, Ampère’s Circuital Law, Faraday’s Law of Electromagnetic Induction, Gauss’s Law for Magnetism.

Poynting Vector: - Definition, Application.

Boundary Conditions at Media Interface, Tangential Electric Field Continuity,  Tangential Magnetic Field Discontinuity, Normal Electric Flux Density Discontinuity, Normal Magnetic Flux- Continuity, Summary Table.

1. Introduction to the vector:-

A vector is a mathematical quantity that contains both a size (magnitude) and a direction. When it comes to electromagnetic waves, the vector is used to describe:

• Electric field vector (𝐸) – represents the strength and direction of the electric field.
• Magnetic field vector (𝐻 or 𝐵) – represents the power and direction of the magnetic field.
• Wave spread vector (𝑘) – represents the direction that the wave travels.

Mathematically, it can be expressed as a vector in a three-dimensional space:

2. Scalars vs Vectors: -

Before diving deep, it is important to separate the scalar and vector:

• Skalar: Only the order of magnitude is (eg, temperature, mass, speed).
• Vector: both size and direction (e.g., speed, power, electric field).

In electromagnetics:

• The permeability (ε) to a medium is a scalar in isotropic materials, but an anisotropic medium may have a tensor (as a vector).
• Electric field(s) and magnetic field (H) are always in vector volume.

3. Vector representation in EM waves: -

In electromagnetic waves, the vectors are important to represent:

• Electric field (𝐸) – Waves make vertical fluctuations in the direction of travel.
• Magnetic field (𝐻) – The electrical field and the direction of the waves are vertical for both.
• Wave vector (𝐤) – indicates the scattering direction, and the wavelength is closely related to the wavelength.

These conditions create an orthogonal coordination system in free space:

4. Vector operation in electromagnetics: -

To analyse electromagnetic fields, we use several vector operations:

A) vector joints and subtraction

• The vector can be added to or decreased by a combination of its components: r = a + b

In the EM waves, the superposition vector of electric fields from many sources uses the vector sum.

B) SCALLER (DOT) PRODUCTS

Dot products of two vectors A.B. gives a scale: A⋅B = ABcosθ

Used to find an angle between the field or to calculate the field components.

C) vector (cross) product

The cross product provides both an A × B vector and a vector: A × B = ABsinθn^

In the European Championships, the Poynting Vector (S = E × H) uses the cross product to represent the current flow direction.

5.  Unit vector and direction

In the EM wave analysis, the unit vector is used to indicate the direction without size. For example:

$\hat{\mathbf{a}}_x$ is the unit vector along the x-axis.

$\hat{\mathbf{a}}_y$ and $\hat{\mathbf{a}}_z$ for the y and z axes.

Disha Kosine is a vector and the cosine of angles between the coordinate axes, which are used to express vector direction.

6. Polarization and vector: -

Polarisation of an electromagnetic wave describes the orientation of the electric field vector:

• Linear polarisation – the electric field vector lives in a particular plane.
• Circular polarisation – the electric field vector rotates in a circular movement because the wave is scattered.
• Elliptical polarisation – the electric field vector detects an ellipse.

A good understanding of vector rotation and change is required to understand polarisation.

The importance of vectors in electromagnetic analysis: -

The vectors are needed because:

• They represent guidelines.
• They make it easy to use Maxwell's equations.
• They simplify the analysis of wave interactions such as reflections, refractions and diffractions.
• They allow the use of vector calculus tools to solve EM problems.

Without the vector, it would be almost impossible to describe the 3D nature of electromagnetic waves.

Vector calculation - 

The vector calculation is a branch of mathematics relating to the operations used for vector areas and their analysis. Unlike the regular calculus, which works with scalar functions (only volume of magnitude), the vector calculation on volumes focuses on both size and direction.

Scalar, Vector, and Field: -

Scalar field: Award a single value at each point (e.g., temperature distribution) in the room.

Vector field: assigns a vector at each point in the room (e.g., electric field, speed range).

In physics and engineering, especially in electromagnetics and fluid mechanics, vector calculation requires understanding how these areas change and interact.

Main operations in vector calculation: -

Gradient (∇φ)

The shield of a smaller area provides a vector indicating the largest growth rate in the region.

Example: In the case of heat transfer, ∇T, the hottest area, is indicated.

Divergence (∇ · A)

A vector area spreads from a point.

Example: In fluid flow, positive deviation means that the fluid leaves an area.

Curl (∇ × A)

The vector measures the rotation of the area.

Example: In electromagnetics, the curl of the electric field is related to changing magnetic fields.

Laplacian (∇²)

A scalar operator that states how a function varies from the average value around a point, often used in wave and spreading equations.

Application: -

Vector is widely used in calculus:

• Electromagnetics – Maxwell's tax vector is expressed in calculus form.
• Fluid Mechanics – describes the flow pattern.
• Engineering analysis – stress, stress, and field mapping.

Maxwell’s Equations: –

Maxwell's equations are a set of four basic equations that explain how electric and magnetic fields behave and interact. They make the spine of electromagnetics, integrating power, magnetism, and optics in the same principle.

1. Gauss’s Law for Electricity: -

• ∇⋅E=ρ/ε0

This law states that electric charges produce electric fields. The total electric flux through a closed surface is proportional to the net charge enclosed.​​

2. Gauss’s Law for Magnetism: -

• ∇⋅B=0

This states that there are no magnetic monopoles; magnetic field lines always form closed loops, meaning the net magnetic flux through a closed surface is zero.

3. Faraday’s Law of Electromagnetic Induction: -

• ∇×E=−∂t/∂B

A changing magnetic field induces an electric field. This principle is the basis of electric generators and transformers.​

4. Ampère–Maxwell Law: -

Magnetic fields are produced by electric currents and by changing electric fields. Maxwell’s addition of the displacement current term ($\mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

​) allows the equations to apply to time-varying fields.​

Application: -

Electromagnetic waves: The Maxwell equations estimate that the light is an electromagnetic wave.

Communication: Radio, TV and wireless systems work with principles taken from these equations.

Electric devices: motors, transformers and antennae are designed with Maxwell rules.

Basic Laws of Electromagnetics: -

Electromagnetics is the study of electric and magnetic fields and how they interact with fabric. It creates a foundation for technologies such as communication systems, electric machines, and microwave ovens. The subject is governed by a set of basic laws derived from experiments and is composed mathematically – these are the basic laws of electromagnetics.

1. Coulomb’s Law: -

Coulomb’s law describes the force between two stationary electric charges:

• F: Electric force
• q₁, q₂: Charges
• r: Distance between charges
• It states that like charges repel and unlike charges attract, with force proportional to the product of charges and inversely proportional to the square of the distance.

2. Gauss’s Law: -

Gauss’s law relates the electric flux through a closed surface to the total charge enclosed:

This simplifies the analysis of electric fields for symmetric charge distributions like spheres, cylinders, and planes.

3. Biot–Savart Law: -

The Biot–Savart law gives the magnetic field generated by a steady current:

It is essential for calculating magnetic fields in conductors with specific geometries.

4. Ampère’s Circuital Law: -

Ampère’s law relates the integrated magnetic field around a closed loop to the total current passing through it: ∇×B=μ0​J

Maxwell later added the displacement current term, enabling the law to apply to time-varying fields.

5. Faraday’s Law of Electromagnetic Induction: -

Faraday’s law states that a changing magnetic field induces an electric field:

This principle is used in electric generators, transformers, and inductive sensors.

6. Gauss’s Law for Magnetism: -

This law states that magnetic monopoles do not exist ∇⋅B=0

Magnetic field lines always form closed loops without a beginning or end.

Poynting Vector:-

Definition: -

The Poynting vector is given by:    S=E×H

Where:

• S: Poynting vector (W/m²) 
• E: Electric field vector (V/m) 
• H: Magnetic field vector (A/m)

The direction of S is the direction of wave propagation, determined by the right-hand rule (E × H).

Application:-

• Wireless communication – it determines how much power the antenna has.
• Microwave technique – calculates energy transfer in waves.
• Optitude – the light describes the light energy stream in the material.

Boundary Conditions at Media Interface: -

The range of boundaries states how electrical and magnetic fields behave at an interface between two different media. These conditions are necessary to solve the equations of Maxwell under real conditions, such as reflection, wrestling, and wave transfer.

When an electromagnetic wave faces an area (eg, air-to-glass, dielectric-to-conductor), some field components should satisfy continuity conditions. These rules are obtained directly from the equations to Maxwell.

1. Tangential Electric Field Continuity:-

From Faraday’s Law:

Meaning: The tangential component of the electric field is continuous across the interface. No sudden jump can occur unless a time-varying magnetic flux exists.

2. Tangential Magnetic Field Discontinuity: -

From Ampère–Maxwell Law:

Meaning: The tangential magnetic field changes by an amount equal to the surface current density at the boundary.

3. Normal Electric Flux Density Discontinuity: -

From Gauss’s Law for Electricity: ∇⋅D=ρv

This results in:   D1n​−D2n​=ρs​

​Meaning: The normal component of the electric flux density changes according to the surface charge density present at the interface.

4. Normal Magnetic Flux Continuity: -

From Gauss’s Law for Magnetism: ∇⋅B=0

We get:  B1n​=B2n

Meaning: The normal component of the magnetic flux density is always continuous—there are no magnetic monopoles.

Summary Table: -​

Field Component	Boundary Condition
Tangential Electric Field (Eₜ)	Continuous across the boundary
Tangential Magnetic Field (Hₜ)	Discontinuous by surface current density (Jₛ)
Normal Electric Flux (Dₙ)	Discontinuous by surface charge density (ρₛ)
Normal Magnetic Flux (Bₙ)	Continuous across the boundary

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