Download Chapter 22 -Gauss`s Law

Chapter 22 -Gauss’s Law
I.
Introduction
Another method of determining the values of electric fields.
Also a guide to understanding how charges will be distributed
II. Define the Electric Flux, , through an area
A. Qualitatively:  is a measure of the number of lines of force passing through the area
On what should the electric flux,  , depend?
1.
Strength of the electric field – remember that the density of the number of lines of force is
proportional to the strength of the electric field, that is,   E
2.
Size of the area – the larger the area, the more lines of force passing through it,   A
3.
Orientation of the electric field to the area
4.
Digression: the area as a vector
B. Quantitatively:
1.

Uniform electric field, E = constant

area, A

A


E
A cos 
perspective view
side view
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
area, A

E
Define the electric flux through and area A placed in a uniform electric field E as:

 


 = EA cos  = E  A ,

Nm2
where the direction of A is perpendicular to the surface. The units are:
C
Example: uniform electric field

A uniform electric field, E  2000iˆ  3000kˆ N/C, is present in space. Find the electric flux
through each of the surfaces of the triangular block shown.
y
4m
B
C
3m
A
E
x
F
2m
5m
D
z
2.

Nonuniform electric field, E varies
Look at the electric flux due to a nonuniform electric

field passing through a small area  A The electric

field is relatively constant through the small area A ,
so the flux  passing through the area can be written
as an approximation:

E

A



  E  A
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Example: nonuniform electric field

An electric field, E  150yiˆ N/C, exists in space. Find the flux through each of the faces of the
cube whose sides are 2 meters long.
y
x
z
III. Electric Flux through a Gaussian Surface (a closed surface)
A. Gaussian or closed surface:
B. Signs of the electric flux, 
Gaussian or closed
surface
Note the signs of the electric flux in the examples, above.
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C. Find the value of the flux through a closed surface. Remember:
 
E  E  dA

1.
Qualitatively:
a.
no charges inside the closed surface
b.
a charge +Q inside the closed surface
c.
a charge –Q inside the closed surface
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d. net charge is zero inside the closed surface
2.
Quantitatively:
a.
Note the amount of charge inside the closed surfaces in examples 3 and 4.
b.
A point charge +q inside the closed surface.

dA
+q

r

dA

E
d

r
d
dA'
dA’
dA'  r 2 d
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
dA

E
3.
Gauss’s Law
 Q

E  E  dA  encl
o

IV. Applications of Gauss’s Law
Remember that Gauss’s Law can be used qualitatively to determine the location of charges and
quantitatively to determine the strength of the electric field for relatively simple charge distributions.
Remember also that in electrostatics, E = 0 inside a conductor. (Do you remember why?)
A. Location of excess charge
1.
Excess charge, -4Q, placed on a solid conductor
2.
A hollow conductor with a charge placed inside the hollow part.
+3Q
3.
A hollow conductor with a charge –2Q placed inside the hollow part and charge –5Q placed
on the conductor
-5Q
-2Q
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B. Quantitative calculation of the electric field using Gauss’s law
Construct a closed surface (Gaussian surface) in the region where you want to find the strength
of the electric field such that:
-
surfaces are perpendicular to the electric field, and the electric field has a constant value
on the surfaces (use ideas of symmetry to argue direction and magnitude of the electric
 

field and note then that the electric field E can be extracted from the integral E  dA .

-
1.
and, if necessary to close the surfaces, choose other surfaces that are parallel to the
 
electric field so that E  dA is zero
E-field of a point charge, +Q.
E
+Q
r
2.
Excess charge Q placed on a spherical conductor of radius R. Find the E-field both inside and
outside the sphere.
R
E
r
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3.
Charge Q uniformly distributed throughout the volume of a sphere of radius R. (Is the
sphere a conductor?) Find the E-field both inside and outside the sphere.
R
E
r
4.
A sphere of radius R has a charge density  = Ar, where A is a constant and r is the radial
distance from the center of the sphere. Field the E-field both inside and outside the sphere.
R
E
r
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5.
Charges uniformly distributed on a long, thin wire. The charge density is  Find the electric
at a perpendicular distance r from the wire.
6.
A coaxial cable with linear charge density  on the inner conductor and  on the outer
conductor. The radius of the inner conductor is a, the inside radius of the outer conductor is
b, and the outside radius is c. Find the E-field in all four regions.
b
c
a
end on view
22-9
7.
A large, flat surface with a single layer of charge uniformly distributed over the surface. The
surface charge density is  Find the E-field a distance z above the surface.
8.
Charge placed on a large, flat conductor. Find the E-field above, below and inside the
conductor.
9.
Charge is placed on an arbitrarily shaped conductor. Find the E-field close to the surface on
the conductor.
22-10