Electromagnetic Waves: Basic Definitions and Fundamental Laws

BASIC DEFINITIONS AND FUNDAMENTAL LAWS

Definitions Governing Electric Field

Let us discuss some of the basic definitions which govern the electric field viz.

1. Electric field intensity ()

It is defined as the ratio of electrostatic Force (field exerted) to the electric charge.

Unit for electric intensity: newton/coulomb

2. Electric Displacement Vector ()

It is defined as the electric flux (Q) per unit area. It is also known as Electric flux density.

Electric displacement vector is a quantity which is used for analysing electrostatic fields in the presence of dielectrics, which is given by

We know that electric field intensity

Comparing equations (1) and (2), we get

3. Electrical permittivity (ε)

It is defined as the ratio of displacement current () to the electric field intensity ().

Electrical permittivity

Unit for electrical permittivity: C²N^‒1m^‒2

4. Dielectric constant (or) Relative permittivity (ε_r)

It is defined the ratio of permittivity of the medium (ε) to the permittivity of free space (or) vacuum (ε₀).

Relative permittivity ε_r = ε / ε₀

(or)

ε = ε₀ε_r

where ε₀ is the permittivity in free space or vacuum and its value ε₀ = 8.854×10^‒12 C²N^‒1m^‒2.

Definitions Governing Magnetic Field

Let us discuss some of the basic definitions which govern the magnetic field viz.,

1. Magnetic flux density ()

It is defined as the number of magnetic flux lines of force (ϕ_m) passing normally through unit area of cross section at that point as shown Fig. 6.1.

Unit for magnetic flux density: weber/m²

2. Magnetic field intensity ()

It is defined as the force experienced by a unit north pole placed at the given point in a magnetic field.

Unit for magnetic field intensity: ampere/m

3. Magnetic permeability (μ)

It is defined as ratio of magnetic flux density () to the magnetic field intensity ().

Magnetic permeability = μ = μ₀μ_r

It is the measure of degrees at which the lines of force can penetrate through the material.

Unit for magnetic permeability: Ns²C^‒2

4. Relative permeability (μ_r)

It is defined as the ratio of permeability of the medium (μ) to the permeability of free space (or) vacuum (μ₀).

Relative permeability μ_r = μ / μ₀

μ = μ₀μ_r

Where μ₀ is the permeability in free space or vacuum and its value is μ₀ = 4π×10^‒7 Ns²C^‒2.

Laws & Theorems Governing Electromagnetic Field

Let us discuss some of the important laws and theorems that govern the electromagnetic field viz.,

1. Gauss law for electric field

Gauss law for electric field states that, the total flux through any closed surface is equal to 1/ε₀ times of the total charge (Q) enclosed in the surface.

i.e., 

2. Faraday's law

Faraday's Law states that, the induced electromotive force (ɛ) in a coil is equal to the rate of change of the magnetic flux (ϕ) linking the coil.

i.e., ε = ‒ dϕ/dt

The negative sign implies the decrease in magnetic flux.

3. Ampere's circuit law

Ampere's circuit law states that, the line integral of magnetic field () surrounding any closed path is equal to μ₀ times of net current (I) passing through that path.

Since , we can write the above equation as

4. Gauss law for magnetic field

Gauss law for magnetic field states that the magnetic flux () passing through the closed surface is equal to zero.

5. Divergence

Divergence is the amount of spreading out of a vector from the point (source) as shown in Fig. 6.2.

Definition

The divergence of a vector  at a point 'P' is defined as the change of vector (expansion) per unit volume as volume shrinks to zero about P

6. Curl

The measure of twist of the vector with respect to a point is called curl.

Definition

The curl of a vector  at a point is defined as the amount of twisting (circulation) per unit area around that point as shown in Fig. 6.3.

Note: The Divergence and Curl together shall be represented by the vector identity .

7. Gauss divergence theorem

Theorem

It states that, "the volume integral of divergence of a vector  is equal to the vector  that spreads out through the surface which covers the volume (V).

Note: Gauss divergence theorem is used to convert surface integral to volume integral and vice‒versa.

8. Stoke's Theorem

Theorem

It states that, "the line integral of a vector  around a closed contour (line) is equal to the surface integral of curl over the region which is bounded by the contour."

Note: Stoke's theorem is used to convert line integral to surface integral and vice‒versa.

9. Equation of continuity

Let us consider a closed surface enclosing a charge q. There exists an outward flow of current given by

        ............(1)

From Gauss divergence theorem, surface integral is converted into volume integral

        ............(2)

Comparing eqn (1) and (2) we get

        ............(3)

If ρ is volume charge density, we can write the charge 'q' as

        ............(4)

Substituting equation (4) in equation (3), we get

The two volume integrals are equal only if their integrands are equal

Equation (5) becomes

        ............(6)

Equation (6) represents the equation of continuity.

10. Poynting vector

Poynting vector is defined as the amount of energy flow of electromagnetic wave per unit area per unit time along the wave propagation direction. It is denoted by  and is given by