area vector vector with magnitude equal to the area of a surface and direction perpendicular to the surface
cylindrical symmetry system only varies with distance from the axis, not direction
electric flux dot product of the electric field and the area through which it is passing
flux quantity of something passing through a given area
free electrons also called conduction electrons, these are the electrons in a conductor that are not bound to any particular atom, and hence are free to move around
Gaussian surface any enclosed (usually imaginary) surface
planar symmetry system only varies with distance from a plane
spherical symmetry system only varies with the distance from the origin, not in direction
Key Equations
Definition of electric flux, for uniform electric field
Electric flux through an open surface
Electric flux through a closed surface
Gauss’s law
Gauss’s Law for systems with symmetry
The magnitude of the electric field just outside the surface of a conductor
Summary
2.1 Electric Flux
The electric flux through a surface is proportional to the number of field lines crossing that surface. Note that this means the magnitude is proportional to the portion of the field perpendicular to the area.
The electric flux is obtained by evaluating the surface integral
where the notation used here is for a closed surface .
2.3 Applying Gauss’s Law
For a charge distribution with certain spatial symmetries (spherical, cylindrical, and planar), we can find a Gaussian surface over which , where is constant over the surface. The electric field is then determined with Gauss’s law.
For spherical symmetry, the Gaussian surface is also a sphere, and Gauss’s law simplifies to .
For cylindrical symmetry, we use a cylindrical Gaussian surface, and find that Gauss’s law simplifies to .
For planar symmetry, a convenient Gaussian surface is a box penetrating the plane, with two faces parallel to the plane and the remainder perpendicular, resulting in Gauss’s law being .
2.4 Conductors in Electrostatic Equilibrium
The electric field inside a conductor vanishes.
Any excess charge placed on a conductor resides entirely on the surface of the conductor.
The electric field is perpendicular to the surface of a conductor everywhere on that surface.
The magnitude of the electric field just above the surface of a conductor is given by .
Answers to Check Your Understanding
2.1. Place it so that its unit normal is perpendicular to
.
2.2.
.
2.3 a.
; b.
; c.
; d.
.
2.4. In this case, there is only
. So, yes.
2.5.
; This agrees with the calculation of Example 1.5.1 where we found the electric field by integrating over the charged wire. Notice how much simpler the calculation of this electric field is with Gauss’s law.
2.6. If there are other charged objects around, then the charges on the surface of the sphere will not necessarily be spherically symmetrical; there will be more in certain direction than in other directions.
Conceptual Questions
2.1 Electric Flux
1. Discuss how would orient a planar surface of area
in a uniform electric field of magnitude
to obtain (a) the maximum flux and (b) the minimum flux through the area.
2. What are the maximum and minimum values of the flux in the preceding question?
3. The net electric flux crossing a closed surface is always zero. True or false?
4. The net electric flux crossing an open surface is never zero. True or false?
2.2 Explaining Gauss’s Law
5. Two concentric spherical surfaces enclose a point charge
. The radius of the outer sphere is twice that of the inner one. Compare the electric fluxes crossing the two surfaces.
6. Compare the electric flux through the surface of a cube of side length
that has a charge
at its centre to the flux through a spherical surface of radius a with a charge
at its centre.
7. (a) If the electric flux through a closed surface is zero, is the electric field necessarily zero at all points on the surface? (b) What is the net charge inside the surface?
8. Discuss how Gauss’s law would be affected if the electric field of a point charge did not vary as
.
9. Discuss the similarities and differences between the gravitational field of a point mass m and the electric field of a point charge
.
10. Discuss whether Gauss’s law can be applied to other forces, and if so, which ones.
11. Is the term
in Gauss’s law the electric field produced by just the charge inside the Gaussian surface?
12. Reformulate Gauss’s law by choosing the unit normal of the Gaussian surface to be the one directed inward.
2.3 Applying Gauss’s Law
13. Would Gauss’s law be helpful for determining the electric field of two equal but opposite charges a fixed distance apart?
14. Discuss the role that symmetry plays in the application of Gauss’s law. Give examples of continuous charge distributions in which Gauss’s law is useful and not useful in determining the electric field.
15. Discuss the restrictions on the Gaussian surface used to discuss planar symmetry. For example, is its length important? Does the cross-section have to be square? Must the end faces be on opposite sides of the sheet?
2.4 Conductors in Electrostatic Equilibrium
16. Is the electric field inside a metal always zero?
17. Under electrostatic conditions, the excess charge on a conductor resides on its surface. Does this mean that all the conduction electrons in a conductor are on the surface?
18. A charge
is placed in the cavity of a conductor as shown below. Will a charge outside the conductor experience an electric field due to the presence of
?
19. The conductor in the preceding figure has an excess charge of
. If a
point charge is placed in the cavity, what is the net charge on the surface of the cavity and on the outer surface of the conductor?
Problems
2.1 Electric Flux
20. A uniform electric field of magnitude
is perpendicular to a square sheet with sides
long. What is the electric flux through the sheet?
21. Calculate the flux through the sheet of the previous problem if the plane of the sheet is at an angle of
to the field. Find the flux for both directions of the unit normal to the sheet.
22. Find the electric flux through a rectangular area
between two parallel plates where there is a constant electric field of
for the following orientations of the area: (a) parallel to the plates, (b) perpendicular to the plates, and (c) the normal to the area making a
angle with the direction of the electric field. Note that this angle can also be given as
.
23. The electric flux through a square-shaped area of side
near a large charged sheet is found to be
when the area is parallel to the plate. Find the charge density on the sheet.
24. Two large rectangular aluminum plates of area
face each other with a separation of
between them. The plates are charged with equal amount of opposite charges,
. The charges on the plates face each other. Find the flux through a circle of radius
between the plates when the normal to the circle makes an angle of
with a line perpendicular to the plates. Note that this angle can also be given as
.
25. A square surface of area
is in a space of uniform electric field of magnitude
. The amount of flux through it depends on how the square is oriented relative to the direction of the electric field. Find the electric flux through the square, when the normal to it makes the following angles with electric field: (a)
, (b)
and (c)
. Note that these angles can also be given as
.
26. A vector field is pointed along the
-axis,
. (a) Find the flux of the vector field through a rectangle in the
-plane between
and
. (Leave your answer as an integral.)
27. Consider the uniform electric field
. What is its electric flux through a circular area of radius
that lies in the
-plane?
28. Repeat the previous problem, given that the circular area is (a) in the
-plane and (b)
above the
–plane.
29. An infinite charged wire with charge per unit length
lies along the central axis of a cylindrical surface of radius
and length
. What is the flux through the surface due to the electric field of the charged wire?
2.2 Explaining Gauss’s Law
30. Determine the electric flux through each surface whose cross-section is shown below.
31. Find the electric flux through the closed surface whose cross-sections are shown below.
32. A point charge
is located at the centre of a cube whose sides are of length
. If there are no other charges in this system, what is the electric flux through one face of the cube?33. A point charge of
is at an unspecified location inside a cube of side
. Find the net electric flux though the surfaces of the cube.
34. A net flux of
passes inward through the surface of a sphere of radius
. (a) How much charge is inside the sphere? (b) How precisely can we determine the location of the charge from this information?
35. A charge
is placed at one of the corners of a cube of side
, as shown below. Find the magnitude of the electric flux through the shaded face due to
. Assume
.
36. The electric flux through a cubical box
on a side is
. What is the total charge enclosed by the box?
37. The electric flux through a spherical surface is
. What is the net charge enclosed by the surface?
38. A cube whose sides are of length
is placed in a uniform electric field of magnitude
so that the field is perpendicular to two opposite faces of the cube. What is the net flux through the cube?
39. Repeat the previous problem, assuming that the electric field is directed along a body diagonal of the cube.
40. A total charge
is distributed uniformly throughout a cubical volume whose edges are
long. (a) What is the charge density in the cube? (b) What is the electric flux through a cube with
edges that is concentric with the charge distribution? (c) Do the same calculation for cubes whose edges are
long and
long. (d) What is the electric flux through a spherical surface of radius
that is also concentric with the charge distribution?
2.3 Applying Gauss’s Law
41. Recall that in the example of a uniform charged sphere,
. Rewrite the answers in terms of the total charge
on the sphere.
42. Suppose that the charge density of the spherical charge distribution shown in Figure 2.3.3 is
for
and zero for
43. A very long, thin wire has a uniform linear charge density of
. What is the electric field at a distance
from the wire?
44. A charge of
is distributed uniformly throughout a spherical volume of radius
. Determine the electric field due to this charge at a distance of (a)
, (b)
, and (c)
from the centre of the sphere.
45. Repeat your calculations for the preceding problem, given that the charge is distributed uniformly over the surface of a spherical conductor of radius
.
46. A total charge
is distributed uniformly throughout a spherical shell of inner and outer radii
and
, respectively. Show that the electric field due to the charge is
47. When a charge is placed on a metal sphere, it ends up in equilibrium at the outer surface. Use this information to determine the electric field of
charge put on a
aluminum spherical ball at the following two points in space: (a) a point
from the centre of the ball (an inside point) and (b) a point
from the centre of the ball (an outside point).
48. A large sheet of charge has a uniform charge density of
. What is the electric field due to this charge at a point just above the surface of the sheet?
49. Determine if approximate cylindrical symmetry holds for the following situations. State why or why not. (a) A
long copper rod of radius
is charged with
of charge and we seek electric field at a point
from the centre of the rod. (b) A
long copper rod of radius
is charged with
of charge and we seek electric field at a point
from the centre of the rod. (c) A
wooden rod is glued to a
plastic rod to make a
long rod, which is then painted with a charged paint so that one obtains a uniform charge density. The radius of each rod is
, and we seek an electric field at a point that is
from the centre of the rod. (d) Same rod as (c), but we seek electric field at a point that is
from the centre of the rod.
50. A long silver rod of radius
has a charge of
on its surface. (a) Find the electric field at a point
from the centre of the rod (an outside point). (b) Find the electric field at a point
from the centre of the rod (an inside point).
51. The electric field at
from the centre of long copper rod of radius
has a magnitude
and directed outward from the axis of the rod. (a) How much charge per unit length exists on the copper rod? (b) What would be the electric flux through a cube of side
situated such that the rod passes through opposite sides of the cube perpendicularly?
52. A long copper cylindrical shell of inner radius
and outer radius
surrounds concentrically a charged long aluminum rod of radius
with a charge density of
. All charges on the aluminum rod reside at its surface. The inner surface of the copper shell has exactly opposite charge to that of the aluminum rod while the outer surface of the copper shell has the same charge as the aluminum rod. Find the magnitude and direction of the electric field at points that are at the following distances from the centre of the aluminum rod: (a)
, (b)
, (c)
, (d)
, and (e)
.
53. Charge is distributed uniformly with a density
throughout an infinitely long cylindrical volume of radius
. Show that the field of this charge distribution is directed radially with respect to the cylinder and that
54. Charge is distributed throughout a very long cylindrical volume of radius
such that the charge density increases with the distance
from the central axis of the cylinder according to
, where
is a constant. Show that the field of this charge distribution is directed radially with respect to the cylinder and that
55. The electric field
from the surface of a copper ball of radius
is directed toward the ball’s centre and has magnitude
. How much charge is on the surface of the ball?
56. Charge is distributed throughout a spherical shell of inner radius
and outer radius
with a volume density given by
, where
is a constant. Determine the electric field due to this charge as a function of
, the distance from the centre of the shell.
57. Charge is distributed throughout a spherical volume of radius
with a density
, where
is a constant. Determine the electric field due to the charge at points both inside and outside the sphere.58. Consider a uranium nucleus to be sphere of radius
with a charge of
distributed uniformly throughout its volume. (a) What is the electric force exerted on an electron when it is
from the centre of the nucleus? (b) What is the acceleration of the electron at this point?
59. The volume charge density of a spherical charge distribution is given by
, where
and
are constants. What is the electric field produced by this charge distribution?
2.4 Conductors in Electrostatic Equilibrium
60. An uncharged conductor with an internal cavity is shown in the following figure. Use the closed surface
along with Gauss’ law to show that when a charge
is placed in the cavity a total charge
is induced on the inner surface of the conductor. What is the charge on the outer surface of the conductor?
Figure 2.5.1 A charge inside a cavity of a metal. Charges at the outer surface do not depend on how the charges are distributed at the inner surface since
field inside the body of the metal is zero.
.
