“Maxwell’s equations and Boundary conditions”;
1.6 Maxwell’s equations and Boundary conditions
The boundary conditions for the electromagnetic fields across material boundaries are given below:
where is a unit vector normal to the boundary pointing from medium-2 into medium-1.
The quantities are any external surface charge and surface current densities on the boundary surface and are measured in units of [coulomb/m2] and [ampere/m].
In words, the tangential components of the E-field are continuous across the interface.
The difference of the tangential components of the H-field are equal to the surface current density.
The difference of the normal components of the flux density D are equal to the surface charge density and the normal components of the magnetic flux density B are continuous.
The Dn boundary condition may also be written in a form that brings out the dependence on the polarization surface charges:
The total surface charge density will be where the surface charge density of polarization charges accumulating at the surface of a dielectric is seen to be (n is the outward normal from the dielectric):
The relative directions of the field vectors are shown in figure.
Each vector may be decomposed as the sum of a part tangential to the surface and a part perpendicular to it that is E = Et + En.
Using the vector identity,
we identify these two parts as:
Field directions at boundary
Using these results we can write the first two boundary conditions in the following vector form where the second form is obtained by taking the cross product of the first with nˆ and noting that Js is purely tangential.
The boundary conditions can be derived from the integrated form of Maxwell’s equations if we make some additional regularity assumptions about the fields at the interfaces.
In many interface problems there are no externally applied surface charges or currents on the boundary. In such cases the boundary conditions may be stated as:
(source-free boundary conditions)
1.7 Poynting’s theorem
Poynting's theorem is a powerful statement of energy conservation.It can be used to relate power absorption in an object to incident fields but is often misunderstood and misinterpreted.
To avoid complicated mathematical expressions, Poynting's theorem requires a mathematical statement for a satisfactory description.According to Poynting's theorem, if S is any closed mathematical surface and V is the volume inside S, then
where,
Wc is the energy possessed by charged particles at a given point in V.
E · E is the energy stored in the E-field at a given point in V.
H · H is the energy stored in the H-field at a given point in V.
A closed surface is any surface that completely encloses a volume.
The integral over the volume V corresponds to a sum of the terms in the integrand over all points inside V.
Thus the integral over V corresponds to the total energy inside V possessed by all charged particles and that stored in the E- and H-fields. The term on the left is the time rate of change of the total energy inside V which is total power.
The term on the right is an integral over the closed mathematical surface enclosing V. For convenience, let
P = E x H
P which is called the Poynting vector,has units of watts per square meter and is interpreted as a power density.
The direction of the cross product of E and H is perpendicular to both E and H and the vector dot product of E x H and dS selects the component of E x H that is parallel to dS
P·dS is the power passing out through the differential surface element (dS) and the surface integral is the sum of the power passing through the dS elements over the entire surface (S) which is equal to the total power passing out through S.
Thus the Poynting's theorem is a statement of the conservation of energy: the time rate of change of the total energy inside V is equal to the total power passing out through S.
The Poynting vector E x H is very useful in understanding energy absorption but Poynting's theorem applies only to a closed surface and the volume enclosed by that surface.
1.8 Time harmonic (sinusoidal) fields
Maxwell’s Laws in Time-Harmonic Form:
To go to sinusoidal steady state, we assume a time variation of cost.
ω Phasor notation is a convenient way to work with sinusoidal waveforms.
We know that the definition of a phasor is,
where the phasor V˜ is a complex number.We want to express the components of the electric and magnetic fields as phasors.
Now, we often suppress the coordinate dependence of the fields but all of the fields are functions of space and time:
In the phasor domain, time derivatives become multiplication by jω:
Therefore Maxwell’s equations in time-harmonic (phasor) form are:
In point form,
1.9 Maxwells equations in phasor form
In case the field quantities are sinusoidally time varying then the electric field E can be expressed as,
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
If we remember the fields are real part of their exponential forms we may consider the fields are as follows:
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,