Saturday, January 16, 2010
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
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).
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 = σE plus the time derivative of the electric flux density through any surface bounded by the path.
E(x, y, z, t) = Ex (x, y, z, t)ax +Ey (x, y , z, t) ay + Ez (x, y, z, t) az
Where Ex = Exm cos(ωt+θx), Ey = Eym cos(ωt+θy), Ez = Ezm cos (ωt+θz). Here the magnitudes Exm, Eym, Ezm, and the phase angles θx, θy, θz, are independent of time but may depend on spatial coordinates, e. g., Exm (x, y, z), θx(x, y, z). Now Ex can be expressed as
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.
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.
If we define the power density vector as:
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.