Andres La Rosa Lecture Notes
Portland State University PH-212
THE GAUSS' LAW
Q
Asymmetric charge distribution
However, in some cases, a SYMMETRIC CHARGE DISTRIBUTION allows us to guess the orientation of the corresponding electric field.
That is the case, for example, when we consider a INFINITELY-
LONG line of uniform charge distribution ( λ = charge per unit length.)
Symmetric charge distribution
+ + + + + + + + + + + + + + + + + + + + +
The electric field at any point away from the line (A and B, for instance) turns out to be perpendicular to the line.

2
Vertical line
Hypothetical ring on a horizontal plane
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Since the points P, Q, R and S are at the same distance from the chargedline, the corresponding electric fields should have the same magnitude:
It turns out, the Gauss's law will allow calculating the magnitude, as we will see below

3
The Gauss's law, to be described below) is a tool that allows to calculate (in a very simplified way) the electric field produced by symmetrically distributed charges
Note: But keep in mind that Gauss's law is valid for both, symmetric or asymmetric charge distributions.

4
Gauss's Law

5
Definition of the solid angle
ΔΩ
CASE:
Δ
S is parallel to r
Δ
S
ΔΩ
ΔΩ
Magnitude of
Δ
S
ΔΩ
Sphere of radius R
Δ
S Δ
S
ΔΩ
Total solid angle enclosed by a spherical surface

6
CASE:
Δ
S is not parallel to r
Δ
S
ΔΩ
Δ
S
ΔΩ
Δ
S

7
ΔΩ
ΔΩ =
Δ
S
ΔΩ =
Δ
S

8
Definition of the electric flux

9
Examples of electric flux

10
Question:
What is the electric flux that crosses the surface ABCD?

11
Questions

11
Electric flux through a closed surface
CASE: The electric field is produced by one point-charge
located inside the closed surface
E Δ S Cos θ
θ

13 q where
φ is the electrical flux crossing the mathematical surface
S
, and q is the point charge inside the surface
S q
CASE: More than one point-charge are
located inside the mathematical closed surface q
1 q
2
IN GENERAL net charge inside
S
Gauss' Law

14
Exercise
CONCLUSION:
Charges located outside the surface S do not contribute
to the electric flux

Exercise: Evaluate the electric flux across the surface S produced
by the four charges indicated in the figure q
1 q
3 q
4 q
2 charge inside
This result is valid for any arbitrary surface
S q
2
+ q
3 q
1
, q
2
, q
3
, q
4

17

18
Applying Gauss' Law to problems that present planar symmetry
See also textbook, page 617 S
S
cylindrical surface
(it has a circular base of area A)

19
σ
σ

20

2 1

2 2
Example
E
1
σ
1
E
2
E
E
A
= E
1
+ E
2
E
1
E
2
σ
2
E
2
E
1
E
1
E
2
E
C
= E
1
+ E
2
=
Example
E
1
σ
1
E
1
E
2
+
E
2
E
σ
2
E
1
-
E
2
E
1
=
E
2
=
X
This arrangement of charges is used to describe
(approximately) the working principle of a
CAPACITOR
X

23
Applying Gauss' Law to problems that present spherical symmetry
Q is the total charge on the sphere
Question:
Answer
E (r) =
E
E
S: Gaussian surface
A spherical surface of radius r
E

2 4
Question:
What is the electric field outside the sphere of radius R?
E
S: Gaussian surface
A spherical surface of radius r

2 5
E

2 6
CASE 2: Compact sphere of radius R
E
uniformly charged.
S: Gaussian surface
A spherical surface of radius r

27

2 8

2 9

30

31

32
Under electrostatic conditions, what is the electric field inside a conductor?

33
A compact conductor has a total charge Q. Under electrostatic conditions, where are those charges located?

34
What is the direction of the electric field near the surface of a conductor?