# Chapter 2 Review

#### Key Terms

**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