1. ELECTROSTATICS - GAUSS’ LAW
Fields and Waves
Lesson 3.2
ELECTROSTATICS - GAUSS’ LAW
Darryl Michael/GE CRD

2. MAXWELL’S 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. Use Gauss’ Law to find D and E in symmetric problems
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) x y z
Surface of infinite extent of charge
Can write down:

5. GAUSS’ LAW
Do Problem 2a
Problem 2b
,is constant.
For example a planar sheet of charge,
where z is constant
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)

6. GAUSS’ LAW
Use Gaussian surface to “pull” this out of integral
Integral now becomes:
Usually an easy integral for surfaces under consideration

7. Example of using GAUSS’ law to find
Z = a
-a < z < a
Z > a ; z< -a
Z = -a
“a slab of charge”
By symmetry:
From symmetry
If r0 > 0, then
Z=0

8. in region |z| < a and create a surface at arbitrary z
GAUSS’ LAW
First get
in region |z| < 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

9. GAUSS’ LAW
=0, since
Evaluate LHS:
These two integrals are equal

10. GAUSS’ LAW
Key Step: Take E out of the Integral
Computation of enclosed charge

11. GAUSS’ LAW
(drop the prime)
Do Problem 3a

12. Back to rectangular, slab geometry example…..
GAUSS’ LAW
Back to rectangular, slab geometry example…..
Need to find
, for |z| > a
Z = -a
Z = a r0
Z = -z’
Z = z’

13. GAUSS’ LAW
As before,
Computation of enclosed charge
Note that the z-integration is from -a to a ;
there is NO CHARGE outside |z|>a

14. GAUSS’ LAW
Once again,
For the region outside |z|>a

15. Note: E-field is continuous
GAUSS’ LAW -a z a
Note: E-field is continuous
Plot of E-field as a function of z for planar example