Equation of Continunity

Farady's Law of Electromagnetic Induction

Farady's Law of Electromagnetic Induction

The Significance of Displacement Current

The Ampere’s Circuital Law for static field is no longer useful in time varying fields as is shown below:

Taking the divergence of the Ampere’s Circuital Law give by ▼×H = J and using the continuity equation we have

That is, equation ▼×H = J leads to steady state conditions in which charge density is not time varying function. Therefore, for time dependent fields’ ▼×H = J needs some modifications. Maxwell suggested that the definition of total current density of Ampere’s Circuital Law is incomplete and advised to add something to J. If it is assumed to be J’, we have

Since J’ arises due to the variation of electric displacement (electric flux density) D with time, it is termed as displacement current density. The modified Ampere’s Circuital Law (Maxwell’s equation), therefore, for time varying field takes the following form

Applying Stokes’s Theorem in equation (1) can be given in integral form as

The important conclusion that can be drawn now is that, since displacement current is related to the electric field, it is not possible in case of time varying fields to deal separately with electric and magnetic fields but, instead, the two fields are interlinked giving rise to electromagnetic fields. It is to be noted that, in a good conductor J’ is negligible compared to J at frequency lower than light frequencies (10¹⁵ Hz).

Maxwell's Equations and their Physical Significances

By studying the physical properties of electric (E-field) and magnetic (B-field) fields we have been able to describe these properties by four, relatively simple, equations known as Maxwell’s equations.

These four fundamental equations of electromagnetism can be expressed in both an integral and differential form as tabulated below:

Coulomb’s and Gauss’s Laws

Equation (1) results from Coulomb’s and Gauss’s Laws and states that free charges act as sources or sinks of D. It suggests that the total electric flux density or total electric displacement through the surface enclosing a volume v is equal to the total charge within the volume.

Equation (2) arises from the application of Gauss’s law to magnetic fields and the non-existence of magnetic monopoles. There are no sources or sinks of B. This equation suggests that the net magnetic flux emerging through any closed surface is zero.

Equation (3) describes Faraday’s Law of electromagnetic induction and states that an electromotance is produced in a circuit when the magnetic flux through the circuit changes. It suggests that the electromagnetic force around a closed path is equal to the time derivative of the magnetic flux density through any surface bounded by the path.

Equation (4) describes Ampere’s Circuital Law (which is derived from the Biot-Savart Law) and states that the electromotive force around a closed path is equal to the conduction current J = σE plus the time derivative of the electric flux density through any surface bounded by the path.

Maxwell’s equations

In vacuum/free space ρ_v = 0, J = 0, ε = ε_o and μ = μ_o. Therefore, in vacuum the Maxwell’s equations take the following forms:

Maxwell's Equations in Frequency Domain

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Assuming the fields are varying harmonically with time as e^jωt, the Maxwell’s equations are given by

Maxwell’s equations

 

Maxwell's Equations in Phasor Form

In case the field quantities are sinusoidally time varying; the electric field E can be expressed as

E(x, y, z, t) = E_x (x, y, z, t)a_x +E_y (x, y , z, t) a_y + E_z (x, y, z, t) a_z

Where E_x = E_xm cos(ωt+θ_x), E_y = E_ym cos(ωt+θ_y), E_z = E_zm cos (ωt+θ_z). Here the magnitudes E_xm, E_ym, E_zm, and the phase angles θ_x, θ_y, θ_z, are independent of time but may depend on spatial coordinates, e. g., E_xm (x, y, z), θ_x(x, y, z). Now E_x can be expressed as

Maxwell's Equations

That is, the first derivative of a sinusoidal varying field is jω times the field. Therefore, the Maxwell’s equations in phasor form can be expressed as:

Maxwell's Equations in Phasor Form

Faraday’s Law for time varying field

Faraday’s Law for time varying field

Electromagnetic Wave Equation or Helmholtz Equation

We will consider a linear, isotropic, homogeneous medium. Moreover, the net free charge in the source free region is zero (ρ_v = 0) and that any currents in the region are conduction currents (J=σE). These types of regions are quite general ones and include the practical cases of free space (σ = 0) as well as most conductors and dielectrics. Maxwell’s equations for this region:

Electromagnetic Wave Equation

Taking curl of (3) and substituting (1) and (4) we have

Taking Curl

Similarly, by taking curl of (4) and substituting (2) and (3) we have

Taking Curl

The equations in (5) are called electromagnetic wave equations or Helmholtz equations. Equations (5a) and (5c) are given in time domain and equations (5b) and (5 d) are in terms of the phasor form. The wave equations in E and in H have exactly the same form.

Characteeristics of a Uniform Plane Wave

Uniform plan waves satisfy the following conditions:

1. At every point in space E and H are perpendicular to each other and to the direction of propagation. No fields, therefore, in the direction of wave propagation.

2. Everywhere in space, the fields vary harmonically with time and at the same frequency.

3. Each field has the same direction, magnitude and phase at every point in any plane perpendicular to the direction of wave propagation. The fields, therefore, are only function of the coordinate that represents the direction of wave propagation.

Field Equations of a Uniform Plane Wave

Propagation Constant and Intrinsic Impedance of Lossless Media

Propagation Constant and Intrinsic Impedance of Good Conducting Media

The Group and Phase Velocity Relation

Pointing Theorem and Explains its Various Terms

The unit of E is volt/m and that of H is A/m, therefore the product of their magnitudes have the unit of power density. The flow of power due to electromagnetic field in a particular direction is of prime importance, the vector product of E and H is used to determine the power of an electromagnetic wave.

If we define the power density vector as:

P=E×H Watt/m².

Using the Maxwell’s equations we have

Equation (1) is known as the point form of pointing theorem. Integrating both sides of (1) over some volume v and applying the divergence theorem, we obtain the integral form of pointing theorem as follows:

The term on the side of (2) is the net inward flux of P into the volume v. The First term on the right side of (2) is a power dissipation term in that it represents the rate of expenditure of energy by the electric field. The second term on the right side of (2) is given by

And represents the time rate of increase of energy stored in the magnetic and electric fields respectively in the volume v.

Therefore (2) states that the net inward flux of the pointing vector through some closed surface is the sum of the power dissipated in the volume enclosed by the surface and the rate of change of energy stored in the volume enclosed by the surface.