Fields and Waves I. Lecture 19 Maxwell’s Equations & Displacement Current K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, NY Y. Maréchal Power Engineering Department Institut National Polytechnique de Grenoble, France.
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Lecture 19
Maxwell’s Equations & Displacement Current
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Y. Maréchal
Power Engineering Department
Institut National Polytechnique de Grenoble, France
Materials from other sources are referenced where they are used. Those listed as Ulaby are figures from Ulaby’s textbook.
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Usual approximations
Maxwell’s equations Models all electromagnetism
Maxwell’s equations
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For Magneto quasistatics
Added term in curl E equation for time varying current or moving path that gives an electric field from a timevarying magnetic field.
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Added term in curl H equation for time varying electric field that gives a magnetic field.
For Electromagnetism
First introduced by Maxwell in 1873
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Displacement current
Ampere’s Law – Curl H Equation
(quasi) Static field
Time varying field
Displacement current density
Integral Form of Ampere’s Law for time varying fields
Displacement current
IC – Conduction Current [A] linked to a conductivity property
– Electric Flux Density (Electric Displacement) [in C/unit area]
– Conduction Current Density (in A/unit area)
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Total current
Conduction current density
Displacement current density
Connection between electric and magnetic fields under time varying conditions
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What are the meanings of these currents ?
Imaginary surface S1
++++++++++++++++++++++++++++
+
Imaginary surface S2
EField
                          

S1=cross section of the wire
S2=cross section of the capacitor
I1c, I1d : conduction and displacement currents in the wire
I2c, I2d : conduction and displacement currents through the capacitor
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The wire is considered as a perfect conductor
I1d = 0
+
From circuit theory:

Total current in the wire:
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The dielectric is considered as perfect (zero conductivity)
Electrical charges can’t move physically through a perfect dielectric medium
I2c= 0 no conduction between the plates
The electric field between the capacitors
d :spacing between the plates
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The displacement current I2d
Displacement current doesn’t carry real charge, but behaves like a real current
If wire has a finite conductivity σ then both wire and dielectric have conduction AND displacement currents
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Maxwell’s equations, boundary conditions
Note that the timevarying terms couple electric and magnetic fields in both directions. Thus, in general, we cannot have one without the other.
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Sources
Material property
Material property
Maxwell’s equations are fully coupled.
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Begin by taking the divergence of Ampere’s Law
where we have used the vector identity that the divergence of the curl of any vector is always equal to zero.
Now from Gauss’ Law,
or
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Now, integrate this equation over a volume.
Ulaby
From the divergence theorem, the left hand side is
For a fixed volume, we can move the derivative outside the integral on the right to obtain the final form of this equation.
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Differential and integral forms of the Continuity Equation (Equation for Charge and Current Conservation)
I3
I2
For statics, the current leaving some volume must sum to zero
If the charge is time varying, sum of currents is equal to this variation.
I1
I4
I5
A general form of the Kirchoff Current Law.
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Boundary conditions derived for electrostatics and magnetostatics
remain valid for timevarying fields:
 For instance, tangential Components of E
w
Material 1
h << w
h
Material 2
Note:
If region 2 is a conductor E1t = 0
Outside conductor E and D are normal to the surface
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Case 1:
REGIONS 1 & 2 are DIELECTRICS (Js = 0)
Material 1
dielectric
Material 2
dielectric
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REGIONS 1 is a DIELECTRIC
REGION 2 is a CONDUCTOR, D2 = E2 =0
Case 2:
Material 1
Material 2
conductor
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Quasi static
Because all four equations are coupled, in general, we must solve them simultaneously.
We will see a general way to do this in the next lecture, which will lead us to electromagnetic waves.
However, we will first look at the coupled equations as a perturbation of what we have done so far in electrostatics and magnetostatics.
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A parallel plate capacitor with circular plates and an air dielectric has a plate radius of 5 mm and a plate separation of d=10 mm. The voltage across the plates is where
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The electric field for a parallel plate capacitor driven by a timevarying source is
The timevarying electric field now produces a source for a magnetic field through the displacement current . We can solve for the magnetic field in the usual manner.
0
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The total displacement current between the capacitor plates
Using phasor notation for the voltage and current
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Applying Ampere’s Law to a circular contour with radius r < a, the fraction of the displacement current enclosed is
Ampere’s Law then gives us
Thus, we now have both electric and magnetic fields between the plates.
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2
3
1
?
In general, we should now use this magnetic field to find a correction to the electric field by plugging it into Faraday’s Law. However, under what we call quasistatic conditions, we only need to find this first term.
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Maxwell’s Equations.
Need a simultaneous solution for the electric and magnetic fields
Lead to a wave equation identical in form to the wave equation found for transmission lines
Quasi static approach
Valid if the system dimensions are small compared to a wavelength.
real meaning of low frequencies.
There is a reasonably complete derivation of this condition in Unit 9 of the class notes.
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The analysis of the capacitor under timevarying conditions assumed that the insulator had no conductivity. If we generalize our results to include both and we will have both a conduction and a displacement current.
The material will behave mostly like a dielectric when
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The material will behave mostly like a conductor when
Loss tangent of the material.
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