Lecture 6: Maxwell’s Equations

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Lecture 6 Maxwells Equations
Instructor Dr. Gleb V.
Tcheslavski Contact gleb_at_ee.lamar.edu Office
Hours Room 2030 Class web site
www.ee.lamar.edu/gleb/em/Index.htm
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Maxwells equations
The behavior of electric and magnetic waves can
be fully described by a set of four equations
(which we have learned already).
Faradays Law of induction
(6.2.1)
Amperes Law
(6.2.2)
(6.2.3)
Gausss Law for electricity
Gausss Law for magnetism
(6.2.4)
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Maxwells equations
And the constitutive relations
(6.3.1)
(6.3.2)
(6.3.3)
They relate the electromagnetic field to the
properties of the material, in which the field
exists. Together with the Maxwells equations,
the constitutive relations completely describe
the electromagnetic field. Even the EM fields in
a nonlinear media can be described through a
nonlinearity existing in the constitutive
relations.
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Maxwells equations
Integral form
Faradays Law of induction
(6.4.1)
Amperes Law
(6.4.2)
(6.4.3)
Gausss Law for electricity
Gausss Law for magnetism
(6.4.4)
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Maxwells equations
Example 6.1 In a conductive material we may
assume that the conductive current density is
much greater than the displacement current
density. Show that the Maxwells equations can be
put in a form of a Diffusion equation in this
material.
We can write
(6.5.1)
and, neglecting the displacement current
(6.5.2)
Taking curl of (6.5.2)
(6.5.3)
Expanding the LHS
(6.5.4)
The first term is zero and
(6.5.5)
Is the diffusion equation with a diffusion
coefficient D 1/(??0)
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Maxwells equations
Example 6.2 Solve the diffusion equation for the
case of the magnetic flux density Bx(z,t) near a
planar vacuum-copper interface, assuming for
copper ? ?0 and ? 5.8 x 107 S/m. Assume that
a 60-Hz time-harmonic EM signal is applied.
Assuming ej?t time-variation, the diffusion
equation is transformed to the ordinary
differential equation
(6.6.1)
Where z is the normal coordinate to the boundary.
Assuming a variation in the z-direction to be
Bx(z) B0e-?z, we write
(6.6.2)
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Maxwells equations
The magnitude of the magnetic flux density decays
exponentially in the z direction from the surface
into the conductor
(6.7.1)
where
(6.7.2)
The quantity ? 1/? is called a skin depth -
the distance over which the current (or field)
falls to 1/e of its original value.
For copper, ? 8.5 mm.
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Maxwells equations
Example 6.3 Derive the equation of continuity
starting from the Maxwells equations
The Gausss law
(6.8.1)
Taking time derivatives
(6.8.2)
From the Amperes law
(6.8.3)
Therefore
(6.8.4)
The equation of continuity
(6.8.5)
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Poyntings Theorem
It is frequently needed to determine the
direction the power is flowing. The Poyntings
Theorem is the tool for such tasks.
We consider an arbitrary shaped volume
Recall
(6.9.1)
(6.9.2)
We take the scalar product of E and subtract it
from the scalar product of H.
(6.9.3)
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Poyntings Theorem
Using the vector identity
(6.10.1)
Therefore
(6.10.2)
Applying the constitutive relations to the terms
involving time derivatives, we get
(6.10.3)
Combining (6.9.2) and (6.9.3) and integrating
both sides over the same ?v
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Poyntings Theorem
(6.11.1)
Application of divergence theorem and the Ohms
law lead to the PT
(6.11.2)
Here
(6.11.3)
is the Poynting vector the power density and
the direction of the radiated EM fields in W/m2.
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Poyntings Theorem
The Poyntings Theorem states that the power that
leaves a region is equal to the temporal decay in
the energy that is stored within the volume minus
the power that is dissipated as heat within it
energy conservation.
EM energy density is
(6.12.1)
Power loss density is
(6.12.2)
The differential form of the Poyntings Theorem
(6.12.3)
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Poyntings Theorem
Example 6.4 Using the Poyntings Theorem,
calculate the power that is dissipated in the
resistor as heat. Neglect the magnetic field that
is confined within the resistor and calculate its
value only at the surface. Assume that the
conducting surfaces at the top and the bottom of
the resistor are equipotential and the resistors
radius is much less than its length.
The magnitude of the electric field is
(6.13.1)
and it is in the direction of the current.
The magnitude of the magnetic field intensity at
the outer surface of the resistor
(6.13.2)
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Poyntings Theorem
The Poyntings vector
(6.14.1)
is into the resistor. There is NO energy stored
in the resistor. The magnitude of the current
density is in the direction of a current and,
therefore, the electric field.
(6.14.2)
The PT
(6.14.3)
(6.14.4)
The electromagnetic energy of a battery is
completely absorbed with the resistor in form of
heat.
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Poyntings Theorem
Example 6.5 Using Poyntings Theorem, calculate
the power that is flowing through the surface
area at the radial edge of a capacitor. Neglect
the ohmic losses in the wires, assume that the
radius of the plates is much greater than the
separation between them a gtgt b.
Assuming the electric field E is uniform and
confined between the plates, the total electric
energy stored in the capacitor is
(6.15.1)
The total magnetic energy stored in the capacitor
is zero.
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Poyntings Theorem
The time derivative of the electric energy is
(6.16.1)
This is the only nonzero term on the RHS of PT
since an ideal capacitor does not dissipate
energy.
We express next the time-varying magnetic field
intensity in terms of the displacement current.
Since no conduction current exists in an ideal
capacitor
(6.16.2)
Therefore
(6.16.3)
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Poyntings Theorem
The power flow would be
(6.17.1)
In our situation
(6.17.2)
and
(6.17.3)
Therefore
(6.17.4)
We observe that
(6.17.5)
The energy is conserved in the circuit.
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Time-harmonic EM fields
Frequently, a temporal variation of EM fields is
harmonic therefore, we may use a phasor
representation
(6.18.1)
(6.18.2)
It may be a phase angle between the electric and
the magnetic fields incorporated into E(x,y,z)
and H(x,y,z).
Maxwells Eqn in phasor form
(6.18.3)
(6.18.4)
(6.18.5)
(6.18.6)
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Time-harmonic EM fields
Power is a real quantity and, keeping in mind
that
(6.19.1)
complex conjugate
Since
(6.19.2)
Therefore
(6.19.3)
Taking the time average, we obtain the average
power as
(6.19.4)
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Time-harmonic EM fields
Therefore, the Poyntings theorem in phasors is
(6.20.1)
Total power radiated from the volume
The power dissipated within the volume
The energy stored within the volume
Indicates that the power (energy) is reactive
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Time-harmonic EM fields
Example 6.6 Compute the frequency at which the
conduction current equals the displacement
current in copper.
Using the Amperes law in the phasor form, we
write
(6.21.1)
Since
(6.21.2)
and
(6.21.3)
Therefore
(6.21.4)
Finally
(6.21.5)
At much higher frequencies, cooper (a good
conductor) acts like a dielectric.
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Time-harmonic EM fields
Example 6.7 The fields in a free space are
(6.22.1)
Determine the Poynting vector if the frequency is
500 MHz.
In a phasor notation
(6.22.2)
And the Poynting vector is
(6.22.3)
HW 5 is ready ?
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What is diffusion equation?
The diffusion equation is a partial differential
equation which describes density fluctuations in
a material undergoing diffusion.
Diffusion is the movement of particles of a
substance from an area of high concentration to
an area of low concentration, resulting in the
uniform distribution of the substance.
Similarly, a flow of free charges in a material,
where a charge difference between two locations
exists, can be described by the diffusion
equation.
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