Fields and Waves

1
Fields and Waves
Lesson 3.2
ELECTROSTATICS - GAUSS LAW
2
MAXWELLS FIRST EQUATION
Enclosed Charge
Differential Form
Integral Form
- dv integral over volume enclosed by ds
integral
For vacuum and air - think of D and E as being
the same
D vs E depends on materials
constant
3
GAUSS LAW - strategy
Do Problem 1
Use Gauss Law to find D and E in symmetric
problems
Get D or E out of integral
Always look at symmetry of the problem - and take
advantage of this
4
GAUSS LAW - use of symmetry
- charges are infinite in extent on say x,y plane
Example A sheet of charge
, is sum due to all charges
Arbitrary Point P
, points in

• all other components cancel

• only a function of z (not x or y)

Surface of infinite extent of charge
Can write down
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GAUSS LAW
Do Problem 2a
,is constant. For example a planar sheet of
charge, where z is constant
Problem 2b
Problem 2c
To use GAUSS LAW, we need to find a surface that
encloses the volume
GAUSSIAN SURFACE - takes advantage of symmetry
- when r is only a f(r)
- when r is only a f(z)
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GAUSS LAW
Use Gaussian surface to pull this out of
integral
Integral now becomes
Usually an easy integral for surfaces under
consideration
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GAUSS LAW
Z a
Z -a
a slab of charge
By symmetry
From symmetry
If r0 gt 0, then
Z0
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GAUSS LAW
First get
in region z lt a and create a surface at
arbitrary z
Use Gaussian surface with top at z z and the
bottom at -z
Note Gaussian Surface is NOT a material boundary
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GAUSS LAW
0, since
Evaluate LHS
These two integrals are equal
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GAUSS LAW
Key Step Take E out of the Integral
Computation of enclosed charge
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GAUSS LAW
(drop the prime)
Do Problem 3a
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GAUSS LAW
Back to rectangular, slab geometry example..
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GAUSS LAW
As before,
Computation of enclosed charge
Note that the z-integration is from -a to a
there is NO CHARGE outside zgta
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GAUSS LAW
Once again,
For the region outside zgta
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GAUSS LAW
-a
z
a
Note E-field is continuous
Plot of E-field as a function of z for planar
example