659
C
H
A
P
T
E
R
2
3
Gauss’ Law
23-1 ELECTRIC FLUX
After reading this module, you should be able to . . .
23.01 Identify that Gauss’ law relates the electric field at points
on a closed surface (real or imaginary, said to be a Gaussian
surface) to the net charge enclosed by that surface.
23.02 Identify that the amount of electric field piercing a surface (not skimming along the surface) is the electric flux ⌽
through the surface.
23.03 Identify that an area vector for a flat surface is a vector
that is perpendicular to the surface and that has a magnitude equal to the area of the surface.
23.04 Identify that any surface can be divided into area elements (patch elements) that are each small enough and
flat enough for an area vector
to be assigned to it, with
the vector perpendicular to the element and having a magnitude equal to the area of the element.
dA
:
23.05 Calculate the flux ⌽ through a surface by integrating the
dot product of the electric field vector and the area vector
(for patch elements) over the surface, in magnitudeangle notation and unit-vector notation.
23.06 For a closed surface, explain the algebraic signs associated with inward flux and outward flux.
23.07 Calculate the net flux ⌽ through a closed surface, algebraic sign included, by integrating the dot product of the
electric field vector and the area vector
(for patch elements) over the full surface.
23.08 Determine whether a closed surface can be broken
up into parts (such as the sides of a cube) to simplify
the integration that yields the net flux through the
surface.
dA
:
E
:
dA
:
E
:
● The electric flux ⌽ through a surface is the amount of electric
field that pierces the surface.
● The area vector
for an area element (patch element) on
a surface is a vector that is perpendicular to the element and
has a magnitude equal to the area dA of the element.
● The electric flux d⌽ through a patch element with area
vector
is given by a dot product:
● The total flux through a surface is given by
d⌽ ϭ E
: ؒ dA
: .
dA
:
dA
:
Key Ideas
Learning Objectives
(total flux),
where the integration is carried out over the surface.
● The net flux through a closed surface (which is used in
Gauss’ law) is given by
(net flux),
where the integration is carried out over the entire surface.
⌽ ϭ ͶE
: ؒ dA
:
⌽ ϭ ͵E
: ؒ dA
:
What Is Physics?
In the preceding chapter we found the electric field at points near extended
charged objects, such as rods. Our technique was labor-intensive: We split the
charge distribution up into charge elements dq, found the field
due to an element, and resolved the vector into components. Then we determined whether the
components from all the elements would end up canceling or adding. Finally we
summed the adding components by integrating over all the elements, with several
changes in notation along the way.
One of the primary goals of physics is to find simple ways of solving such
labor-intensive problems. One of the main tools in reaching this goal is the use of
symmetry. In this chapter we discuss a beautiful relationship between charge and
dE
:
C
H
A
P
T
E
R
2
3
Gauss’ Law
23-1 ELECTRIC FLUX
After reading this module, you should be able to . . .
23.01 Identify that Gauss’ law relates the electric field at points
on a closed surface (real or imaginary, said to be a Gaussian
surface) to the net charge enclosed by that surface.
23.02 Identify that the amount of electric field piercing a surface (not skimming along the surface) is the electric flux ⌽
through the surface.
23.03 Identify that an area vector for a flat surface is a vector
that is perpendicular to the surface and that has a magnitude equal to the area of the surface.
23.04 Identify that any surface can be divided into area elements (patch elements) that are each small enough and
flat enough for an area vector
to be assigned to it, with
the vector perpendicular to the element and having a magnitude equal to the area of the element.
dA
:
23.05 Calculate the flux ⌽ through a surface by integrating the
dot product of the electric field vector and the area vector
(for patch elements) over the surface, in magnitudeangle notation and unit-vector notation.
23.06 For a closed surface, explain the algebraic signs associated with inward flux and outward flux.
23.07 Calculate the net flux ⌽ through a closed surface, algebraic sign included, by integrating the dot product of the
electric field vector and the area vector
(for patch elements) over the full surface.
23.08 Determine whether a closed surface can be broken
up into parts (such as the sides of a cube) to simplify
the integration that yields the net flux through the
surface.
dA
:
E
:
dA
:
E
:
● The electric flux ⌽ through a surface is the amount of electric
field that pierces the surface.
● The area vector
for an area element (patch element) on
a surface is a vector that is perpendicular to the element and
has a magnitude equal to the area dA of the element.
● The electric flux d⌽ through a patch element with area
vector
is given by a dot product:
● The total flux through a surface is given by
d⌽ ϭ E
: ؒ dA
: .
dA
:
dA
:
Key Ideas
Learning Objectives
(total flux),
where the integration is carried out over the surface.
● The net flux through a closed surface (which is used in
Gauss’ law) is given by
(net flux),
where the integration is carried out over the entire surface.
⌽ ϭ ͶE
: ؒ dA
:
⌽ ϭ ͵E
: ؒ dA
:
What Is Physics?
In the preceding chapter we found the electric field at points near extended
charged objects, such as rods. Our technique was labor-intensive: We split the
charge distribution up into charge elements dq, found the field
due to an element, and resolved the vector into components. Then we determined whether the
components from all the elements would end up canceling or adding. Finally we
summed the adding components by integrating over all the elements, with several
changes in notation along the way.
One of the primary goals of physics is to find simple ways of solving such
labor-intensive problems. One of the main tools in reaching this goal is the use of
symmetry. In this chapter we discuss a beautiful relationship between charge and
dE
: