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Electric Flux Density
Yikes! Things have gotten complicated!
In free space, we found that charge ρv ( r ) creates an electric
field E ( r ) .
Pretty simple!
ρv ( r )
E (r )
But, if dielectric material is present, we find that charge ρv ( r )
creates an initial electric field Ei ( r ) . This electric field in turn
polarizes the material, forming bound charge ρvp ( r ) .
This
bound charge, however, then creates its own electric field
Es ( r ) (sometimes called a secondary field), which modifies the
initial electric field!
Not so simple!
ρv ( r )
Ei ( r )
ρvp ( r )
Es ( r )
The total electric field created by free charge when dielectric
material is present is thus E ( r ) = Ei ( r ) + Es ( r ) .
Q: Isn’t there some easier way to account for the
effect of dielectric material??
A:
Jim Stiles
Yes there is! We use the concept of dielectric
permittivity, and a new vector field called the
electric flux density D ( r ) .
The Univ. of Kansas
Dept. of EECS
11/4/2004
Electric Flux Density.doc
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To see how this works, first consider the point form of Gauss’s
Law:
ρ (r )
∇ ⋅ E ( r ) = vT
ε0
where ρvT ( r ) is the total charge density, consisting of both the
free charge density ρv ( r ) and bound charge density ρvp ( r ) :
ρvT ( r ) = ρv ( r ) + ρvp ( r )
Therefore, we can write Gauss’s Law as:
ε 0∇ ⋅ E ( r ) = ρv ( r ) + ρvp ( r )
Recall the bound charge density is equal to:
ρvp ( r ) = −∇ ⋅ P ( r )
Inserting into the above equation:
ε 0∇ ⋅ E ( r ) = ρv ( r ) − ∇ ⋅ P ( r )
And rearranging:
ε 0∇ ⋅ E ( r ) + ∇ ⋅ P ( r ) = ρv ( r )
∇ ⋅ ⎡⎣ε 0E ( r ) + P ( r ) ⎤⎦ = ρv ( r )
Jim Stiles
The Univ. of Kansas
Dept. of EECS
11/4/2004
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Note this final result says that the divergence of vector field
ε 0E ( r ) + P ( r ) is equal to the free charge density ρv ( r ) . Let’s
define this vector field the electric flux density D ( r ) :
electric flux density D (r ) ε 0 E ( r ) + P (r )
⎡C ⎤
⎢⎣ m2 ⎥⎦
Therefore, we can write a new form of Gauss’s Law:
∇ ⋅ D ( r ) = ρv ( r )
This equation says that the electric flux density D ( r ) diverges
from free charge ρv ( r ) . In other words, the source of electric
flux density is free charge ρv ( r ) --and free charge only!
* The electric field E ( r ) is created by both free
charge and bound charge within the dielectric material.
* However, the electric flux density D ( r ) is created by
free charge only—the bound charge within the
dielectric material makes no difference with regard to
D (r ) !
Jim Stiles
The Univ. of Kansas
Dept. of EECS
11/4/2004
Electric Flux Density.doc
4/5
But wait! We can simplify this further. Recall that the
polarization vector is related to electric field by susceptibility
χe ( r ) :
P ( r ) = ε 0 χe ( r ) E ( r )
Therefore the electric flux density is:
D (r ) = ε 0E (r ) + ε 0 χ e (r ) E (r )
= ε 0 (1 + χ e ( r ) ) E ( r )
We can further simplify this by defining the permittivity of the
medium (the dielectric material):
permittivity ε ( r ) ε 0 (1 + χe ( r ) )
And can further define relative permittivity:
relative permittivity ε r ( r ) ε (r )
= 1 + χe ( r )
ε0
Note therefore that ε ( r ) = ε r ( r ) ε 0 .
Jim Stiles
The Univ. of Kansas
Dept. of EECS
11/4/2004
Electric Flux Density.doc
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We can thus write a simple relationship between electric flux
density and electric field:
D (r ) = ε (r ) E (r )
= ε 0 ε r (r ) E (r )
Like conductivity σ ( r ) , permittivity ε ( r ) is a fundamental
material parameter. Also like conductivity, it relates the
electric field to another vector field.
Thus, we have an alternative way to view electrostatics:
1. Free charge ρv ( r ) creates electric flux density D ( r ) .
2. The electric field can be then determined by simply
dividing D ( r ) by the material permittivity ε ( r ) (i.e.,
E ( r ) = D ( r ) ε ( r ) ).
ρv ( r )
Jim Stiles
D (r )
The Univ. of Kansas
E (r )
Dept. of EECS