Download the gauss` law - Portland State University

Andres
La Rosa
1
Portland State University
Lecture Notes
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 INFINITELYLONG 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.
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THE GAUSS' LAW
Gauss's Law
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Definition of the solid angle ΔΩ
CASE:
ΔS is parallel to r
ΔS
Magnitude of ΔS
ΔΩ
ΔΩ
ΔΩ
Sphere of
radius R
ΔS
Total solid angle
enclosed by a
spherical surface
ΔS
ΔΩ
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CASE:
ΔS is not parallel to r
ΔS
ΔΩ
ΔS
ΔΩ
ΔS
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ΔS
ΔΩ
ΔΩ =
ΔΩ =
ΔS
8
Definition of the electric flux
9
Examples of electric flux
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Question:
What is the electric flux that crosses the surface ABCD?
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Questions
11
Electric flux through a closed surface
CASE: The electric field is produced by one point-charge
located inside the closed surface
E ΔS 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
q1
q2
IN GENERAL
net charge inside S
Gauss' Law
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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
q1
q4
q3
q2
charge inside
This result is
valid for any
arbitrary
surface S
q2
q1, q2, q3 , q4
+
q3
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)
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σ
σ
20
21
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Example
E1
σ1
E1
σ2
E1
E1
E2
E2
E2
E2
E
EC = E1 + E2 =
EA = E1 + E2
X
Example
E1
σ1
E2
E1
σ2
E1
E2
+
E2
-
E1 =
E2 =
E
X
This arrangement of
charges is used to
describe
(approximately) the
working principle of a
CAPACITOR
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Applying Gauss' Law to problems that present
spherical symmetry
CASE 1: Spherical
shell uniformly charged.
E
Q is the total charge on the sphere
Question:
E inside the sphere?
What is the electric field
E (r) =
Answer
E
E
E
S: Gaussian surface
A spherical surface of radius r
24
Question:
What is the electric field outside the sphere
of radius R?
E
S: Gaussian surface
A spherical surface of radius r
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E
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CASE 2: Compact sphere of radius R
E
uniformly charged.
S: Gaussian surface
A spherical surface of radius r
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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?