12/4/2003 Maxwell Equations Yet Again 1/8

( ) ( ) ( )

( ) ( )

( ) ( )

( )

0 0 0

0

r,x r, r,

r,x r,

r,r,

r, 0

v

tt tt

ttt

tt

t

µ µ ε

ρε

∂∇ = +

∂

∂∇ = −

∂

∇ ⋅ =

∇ ⋅ =

EB J

BE

E

B

Maxwell’s Equations (Yet Again)

Now let’s go back and again examine Maxwell’s Equations, which we first looked at in Chapter 3:

Now that we have introduced the concept of dielectrics and magnetic material, we can write these equations more generally as:

12/4/2003 Maxwell Equations Yet Again 2/8

( ) ( ) ( )

( ) ( )

( ) ( )

( )

r,x r, r,

r,x r,

r, r,

r, 0

v

tt tt

ttt

t t

t

ρ

∂∇ = +

∂

∂∇ = −

∂

∇ ⋅ =

∇ ⋅ =

DH J

BE

D

B

( ) ( )

( ) ( )

( )

( )

r,r,

r,r,

r,

r, 0

encC S

C S

encS

S

tt d I dst

tt d dst

t ds Q

t ds

∂⋅ = + ⋅

∂

∂⋅ = − ⋅

∂

⋅ =

⋅ =

∫ ∫∫

∫ ∫∫

∫∫

∫∫

DH

BE

D

B

These are the point form of Maxwell’s Equations; we can also write them in integral form:

12/4/2003 Maxwell Equations Yet Again 3/8

( ) ( )

( ) ( )

( ) ( )

r r

r r

r r

ε

µ

σ

=

=

=

D E

B H

J E

( ) ( ) ( )

( ) ( )

( ) ( )

( )

r,x r, r,

r,x r,

r, r,

r, 0

v

tt tt

ttt

t t

t

ρ

∂∇ = +

∂

∂∇ = −

∂

∇ ⋅ =

∇ ⋅ =

DH J

BE

D

B

But, we have a problem! Maxwell’s Equations now (i.e, in material) has too many unknowns and too few equations! To complete our electromagnetic knowledge, we must consider the constitutive equations, which are dependent on the material properties:

Now, let’s consider again Maxwell’s equations:

Sources Fields

We can interpret these equations as relating sources and the fields these sources create. The sources appear on right side of Maxwell’s equations, whereas the fields appear on the left.

12/4/2003 Maxwell Equations Yet Again 4/8

( )

( ) ( )

,r t

r r

t

ε

∇ ⋅ =

=

D

D E

( )

( ) ( )

x r,t

rr ε

∇ = −

=

E

D E

( ) )

( ) ( )

r,x r, tt

r rµ

∇ =

=

H

B H

For example, we know that an electric field and electric flux density is created from charge:

( )r,vρ

( )r,tt

∂∂

B

(J

But, we also know that an electric field and electric flux density can be created (induced) by a time varying magnetic flux density:

Likewise, we know that current is the source of a magnetic field and magnetic flux density:

But, note we have one source left! Note that it appears that a time-varying electric flux density can “induce” a magnetic field, much in the same way that a time-varying magnetic flux density induces and electric field.

12/4/2003 Maxwell Equations Yet Again 5/8

( )

( ) ( )

x r,t

r rµ

∇ =

=

H

B H

( )r,tt

∂∂

D

( ) ( ) ( )r,x r, r,

tt tt

∂∇ ⋅∇ ⋅∇ = ∇ ⋅ +

∂D

H J

( )x r, 0t∇ ⋅∇ =H

( ) ( )r,r,

ttt

∂∇ ⋅∇ ⋅ = −

∂D

J

( )rJ ( ) 0r⋅ =J

( )v rρ

( ) ( )v rrt

ρ∂∇ ⋅ = −

∂J

( )r,tt

∂∂

D

Q: What the heck is ??

A: Try taking the divergence of Ampere’s Law.

Since we know that the divergence of every curl is zero (i.e.,

), we find:

Recall that often we find that the divergence of current density is zero (i.e., ∇ ), as charge cannot be created or destroyed. The exception is when charge “pile up”, or diminish at some point. In this case, the charge density

must change as a function of time. Recall that this was expressed as the continuity equation:

12/4/2003 Maxwell Equations Yet Again 6/8

( )c r

( ) ( )vc

rrt

ρ∂∇ ⋅ = −

∂J

( ) ( )v r rρ = ∇ ⋅D

( ) ( )c

rrt

∂∇ ⋅∇ ⋅ = −

∂DJ

( ) ( )

displacement current

cr r

t∂

− =∂

=

D J

( ) ( ) ( )

( ) ( )

r,x r, r,

r, r,c

tt tt

t t

∂∇ = +

∂= −

DH J

J J

( ) ( )d v ti t Cdt

=

We called this type of current density—whose divergence is not zero—displacement current J . Therefore, we can state:

But, recall that , therefore:

Or, more specifically:

Therefore Ampere’s Law can be written as:

The most important application of displacement current is when considering capacitors. We know that:

12/4/2003 Maxwell Equations Yet Again 7/8

0σ ≈ ( )rJ( ) 0r =J

( )i t

( )i t

sρ

0σ =

0=J

( ) 0i t ≠

( ) 0i t ≠

( )v t+

−

Yet we also know that the conductors of a capacitor are typically separated by a dielectric with almost no conductance ( ). Thus, the current density in the dielectric is zero ( ).

Q: So how can current be flowing ??

A: Displacement current! The charge from current does not move through the capacitor, but instead “pile up” at each plate. This change in charge density at each plate is equivalent to a current—a displacement current.

12/4/2003 Maxwell Equations Yet Again 8/8

0σ =

0=J

( ) 0i t ≠

( ) 0i t ≠

( )v t+

−

( ) 0sd tdtρ + ≠

( ) 0sd tdtρ − ≠

++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++

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A capacitor is analogous to the “storage tank” that we discussed in chapter 3.