## Saturday, January 16, 2010

### Equation of Continunity

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Labels: Steady Electric Current

### Farady's Law of Electromagnetic Induction

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Labels: Time Varying Fields

### The Significance of Displacement Current

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 (10^{15} Hz).

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Labels: Time Varying Fields

### Maxwell's Equations and their Physical Significances

**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:

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 = σ**E** plus the time derivative of the electric flux density through any surface bounded by the path.

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:

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Labels: Time Varying Fields

### Maxwell's Equations in Frequency Domain

^{j}

^{ωt}, the Maxwell’s equations are given by

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### Maxwell's Equations in Phasor Form

**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**+ E

_{y}_{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

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:

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Labels: Time Varying Fields

### Faraday’s Law for time varying field

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### Electromagnetic Wave Equation or Helmholtz Equation

_{v}= 0) and that any currents in the region are conduction currents (

**J**=σ

**E**). 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:

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

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

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.

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Labels: Time Varying Fields

### 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.

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Labels: Time Varying Fields

### Field Equations of a Uniform Plane Wave

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Labels: Time Varying Fields

### Propagation Constant and Intrinsic Impedance of Lossless Media

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Labels: Time Varying Fields

### Propagation Constant and Intrinsic Impedance of Good Conducting Media

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Labels: Time Varying Fields

### The Group and Phase Velocity Relation

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### Pointing Theorem and Explains its Various Terms

**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/m^{2}.

** **

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.

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Labels: Time Varying Fields