Lecture 5: Time-varying EM Fields

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Lecture 5 Time-varying EM Fields
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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Introduction
Although Time-varying EM fields is the title of
the present lecture only, the rest of our course
will be devoted to time-varying fields! These
fields are generalizations of electrostatic and
magnetostatic fields studied earlier.
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Faradays law of induction
We consider an electric potential across an
inductor inserted in a circuit.
The voltage across the inductor can be expressed
as
(5.3.1)
where L is the inductance H, I(t) time
varying current passing through the inductor.
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Faradays law of induction
We have learned that a constant current induces
magnetic field and a constant charge (or a
voltage) makes an electric field. Similar
phenomena happen when currents/voltages are
changing in time. The relation between the
electric and magnetic field components is
governed by the Faradays law
(5.4.1)
Lentzs law
Voltage is induced in a closed loop that
completely surrounds the surface, through which
the magnetic flux passes.
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Faradays law of induction
Therefore, we consider an individual turn of a
coil (inductor) as a closed loop.
At a glance an electric current (an emf) is
generated in a closed loop when any change in
magnetic environment around the loop occurs
(Faradays law). When an emf is generated by a
change in magnetic flux according to Faraday's
Law, the polarity of the induced emf is such that
it produces a current whose magnetic field
opposes the change which produces it. The induced
magnetic field inside any loop of wire always
acts to keep the magnetic flux in the loop
constant.
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Faradays law of induction
The total magnetic flux passing through the loop
is
(5.6.1)
The terminals of the loop are separated by an
infinitely small distance, therefore, we assume
the loop as closed.
Recall from (3.20.2) that the electric potential
can be expressed as
(5.6.2)
Combining two last equations with (5.4.1) yields
the integral form of Faradays law
(5.6.3)
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Faradays law of induction
Application of the Stokess theorem to the
integral form of the Faradays law of induction
leads to
(5.7.1)
Equality of integrals implies an equality of
integrands, therefore
(5.7.2)
Is a differential form of the Faradays law of
induction.
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Faradays law of induction (Ex)
A good illustration of the Faradays law of
induction is an ideal transformer.
Find the voltage V2 at the loop 2 (secondary
coil) if the voltage V1 is applied to the loop 1
(primary coil).
The ideal assumption implies no loss in the
core and that the core has infinite permeability.
(5.8.1)
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Faradays law of induction
We assumed previously that the area of the loop
?s is not changing. It does not always hold.
Let us consider a conductive bar moving at a
speed v through a constant uniform magnetic field
B.
The Lorentz force will cause a charge separation
within the conductor.
Charge separation will create an electric field.
Since the net force on the bar is zero, the
electric and magnetic contributions must cancel
each other. Therefore
(5.9.1)
Electric field is an induced field acting along
conductor and producing
(5.9.2)
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Faradays law of induction
Example a Faradays disc generator consists of a
metal disc rotating with a constant angular
velocity ? 600 1/s in a uniform
time-independent magnetic field with a magnetic
flux density B B0uz, where B0 4 T. Determine
the induced voltage generated between the brush
contacts at the axis and the edge of the disc,
whose radius a 0.5 m.
An electron at a radius ? from the center has a
velocity ??.
Therefore, the potential generated is
(5.10.1)
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Faradays law of induction
We can evaluate a divergence for both sides of
the Faradays law
(5.11.1)
Since divergence of the curl is zero
(5.11.2)
Or in the integral form
(5.11.3)
Same as for MS fields, there is no magnetic
monopole and magnetic lines are continuous.
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Equation of Continuity
(5.12.1)
Current density is the movement of charge
density. The continuity equation says that if
charge is moving out of a differential volume
(i.e. divergence of current density is positive)
then the amount of charge within that volume is
going to decrease, so the rate of change of
charge density is negative. Therefore the
continuity equation amounts to a conservation of
charge.
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Equation of Continuity
Example charges are introduced into the interior
of a conductor during the time t lt 0. Calculate
the time needed for charges to move to the
surface of the conductor so the interior charge
density ?v 0 and interior electric field E 0.
Let us combine the Ohms law and the continuity
equation
(5.13.1)
(5.13.2)
(5.13.3)
(5.13.4)
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Equation of Continuity
The initial charge density ?v decays to 1/e ? 37
of its initial value in a relaxation time ? ?/?.
(5.14.1)
For copper, the relaxation time is
(5.14.2)
For dielectric materials, the relaxation time can
be quite big hours or even days.
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Displacement current
Maxwells experiment
Connect together an AC voltage source, an AC
ammeter reading a constant current I and an ideal
parallel plate capacitor

1. How can the ammeter read any values if the
   capacitor is an open circuit?
2. What happens to the time-varying magnetic field
   that is created by the current and surrounds the
   wire as we pass through the vacuum between the
   capacitors plates?

(5.15.1)
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Displacement current
Applying the divergence operation to both sides
of the Amperes law, we get
(5.16.1)
(5.16.2)
and
However, the last expression disagrees with the
equation of continuity
To overcome this contradiction, we introduce an
additional current called a displacement current,
whose current density will be Jd. The continuity
equation
(5.16.2)
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Displacement current
Therefore, the displacement current density is
(5.17.1)
This is the current passing between the plates of
the capacitor.
As a consequence, the postulate of Magnetostatic
we discussed previously must be modified as
follows
Differential form
(5.17.2)
Integral form
(5.17.3)
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Displacement current
Example 1 Verify that the conduction current in
the wire equals to the displacement current
between the plates of the parallel plate
capacitor in Figure. The voltage source supplies
Vc V0 sin ?t.
(5.18.1)
The conduction current in the wire is
(5.18.2)
(5.18.3)
(5.18.4)
(5.18.5)
Finally
(5.18.6)
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Displacement current
Example 2 Find the displacement current, the
displacement charge density, and the volume
charge density associated with the magnetic flux
density in a vacuum
(5.19.1)
There is no conductors, therefore, only the
displacement flux exists.
(5.19.2)
The displacement flux density
(5.19.3)
From the Gausss law
(5.19.4)
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Displacement current
Example 3 In a lossy dielectric with a
conductivity ? and a relative permittivity ?r,
there is a time-harmonic electric field E E0sin
?t. Compare the magnitudes of the conduction
current density and the displacement current
density.
Conduction current density
Displacement current density
The ratio of the magnitudes is
HW 4 is ready.