Chapter 3 Review

Key Terms

electric dipole
system of two equal but opposite charges a fixed distance apart

electric dipole moment
quantity defined as

for all dipoles, where the vector points from the negative to positive charge

electric potential
potential energy per unit charge

electric potential difference
the change in potential energy of a charge

moved between two points, divided by the charge.

electric potential energy
potential energy stored in a system of charged objects due to the charges

electron-volt
energy given to a fundamental charge accelerated through a potential difference of one volt

electrostatic precipitators
filters that apply charges to particles in the air, then attract those charges to a filter, removing them from the airstream

equipotential line
two-dimensional representation of an equipotential surface

equipotential surface
surface (usually in three dimensions) on which all points are at the same potential

grounding
process of attaching a conductor to the earth to ensure that there is no potential difference between it and Earth

ink jet printer
small ink droplets sprayed with an electric charge are controlled by electrostatic plates to create images on paper

photoconductor
substance that is an insulator until it is exposed to light, when it becomes a conductor

Van de Graaff generator
machine that produces a large amount of excess charge, used for experiments with high voltage

voltage
change in potential energy of a charge moved from one point to another, divided by the charge; units of potential difference are joules per coulomb, known as volt

xerography
dry copying process based on electrostatics

Key Equations

Potential energy of a two-charge system	$U(r)=k\frac{qQ}{r}$
Work done to assemble a system of charges	$W_{12\ldots N}=\frac{k}{2}\sum_i^N\sum_j^N\frac{q_iq_j}{r_{ij}}~\mathrm{for}~i\neq j$
Potential difference	$\Delta V=\frac{\Delta U}{q}$ or $\Delta U=q\Delta V$
Electric potential	$V=\frac{U}{q}=-\int_R^P\vec{\mathbf{E}}\cdot d\vec{\mathbf{l}}$
Potential difference between two points	$\Delta V_{AB}=V_B-V_A=-\int_A^B\vec{\mathbf{E}}\cdot d\vec{\mathbf{l}}$
Electric potential of a point charge	$V=\frac{kq}{r}$
Electric potential of a system of point charges	$V_P=k\sum_1^N\frac{q_i}{r_i}$
Electric dipole moment	$\vec{\mathbf{p}}=q\vec{\mathbf{d}}$
Electric potential due to a dipole	$V_P=k\frac{\vec{\mathbf{p}}\cdot\hat{\mathbf{r}}}{r^2}$
Electric potential of a continuous charge distribution	$V_P=k\int\frac{dq}{r}$
Electric field components	$E_x=-\frac{\partial V}{\partial x},~E_y=-\frac{\partial V}{\partial y},~E_z=-\frac{\partial V}{\partial z}$
Del operator in Cartesian coordinates	$\vec{\nabla}=\hat{\mathbf{i}}\frac{\partial}{\partial x}+\hat{\mathbf{j}}\frac{\partial}{\partial y}+\hat{\mathbf{k}}\frac{\partial}{\partial z}$
Electric field as gradient of potential	$\vec{\mathbf{E}}=-\vec{\nabla}V$
Del operator in cylindrical coordinates	$\vec{\nabla}=\hat{\mathbf{r}}\frac{\partial}{\partial r}+\hat{\pmb{\upphi}}\frac{\partial}{\partial\phi}+\hat{\mathbf{z}}\frac{\partial}{\partial z}$
Del operator in spherical coordinates	$\vec{\nabla}=\hat{\mathbf{r}}\frac{\partial}{\partial r}+\hat{\pmb{\uptheta}}\frac{\partial}{\partial\theta}+\hat{\pmb{\upphi}}\frac{1}{r\sin\theta}\frac{\partial}{\partial\phi}$

Summary

3.1 Electric Potential Energy

The work done to move a charge from point $A$ to $B$ in an electric field is path independent, and the work around a closed path is zero. Therefore, the electric field and electric force are conservative.
We can define an electric potential energy, which between point charges is $U(r)=k\frac{qQ}{r}$ , with the zero reference taken to be at infinity.
The superposition principle holds for electric potential energy; the potential energy of a system of multiple charges is the sum of the potential energies of the individual pairs.

3.2 Electric Potential and Potential Difference

Electric potential is potential energy per unit charge.
The potential difference between points $A$ and $B$ , $V_B-V_A$ , that is, the change in potential of a charge $q$ moved from $A$ to $B$ , is equal to the change in potential energy divided by the charge.
Potential difference is commonly called voltage, represented by the symbol $\Delta V$ :

$\Delta V=\frac{\Delta U}{q}~\mathrm{or}~\Delta U=q\Delta V.$
An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of $1~\mathrm{V}$ . In equation form,

$1~\mathrm{eV}=(1.60\times10^{-19}~\mathrm{C})(~\mathrm{V})=(1.60\times10^{-19}~\mathrm{C})(~\mathrm{J/C})=1.60\times10^{-19}~\mathrm{J}.$

3.3 Calculations of Electric Potential

Electric potential is a scalar whereas electric field is a vector.
Addition of voltages as numbers gives the voltage due to a combination of point charges, allowing us to use the principle of superposition: $V_P=k\sum_1^N\frac{q_i}{r_i}$ .
An electric dipole consists of two equal and opposite charges a fixed distance apart, with a dipole moment $\vec{\mathbf{p}}=q\vec{\mathbf{d}}$ .
Continuous charge distributions may be calculated with $V_P=k\int\frac{dq}{r}$ .

3.4 Determining Field from Potential

Just as we may integrate over the electric field to calculate the potential, we may take the derivative of the potential to calculate the electric field.
This may be done for individual components of the electric field, or we may calculate the entire electric field vector with the gradient operator.

3.5 Equipotential Surfaces and Conductors

An equipotential surface is the collection of points in space that are all at the same potential. Equipotential lines are the two-dimensional representation of equipotential surfaces.
Equipotential surfaces are always perpendicular to electric field lines.
Conductors in static equilibrium are equipotential surfaces.
Topographic maps may be thought of as showing gravitational equipotential lines.

3.6 Applications of Electrostatics

Electrostatics is the study of electric fields in static equilibrium.
In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink jet printers, and electrostatic air filters.

Answers to Check Your Understanding

3.1.

3.2. It has kinetic energy of

at point

and potential energy of

which means that as

approaches infinity, its kinetic energy totals three times the kinetic energy at

since all of the potential energy gets converted to kinetic.

3.3. positive, negative, and these quantities are the same as the work you would need to do to bring the charges in from infinity

3.4.

3.5.

electrons

3.6. It would be going in the opposite direction, with no effect on the calculations as presented.

3.7. Given a fixed maximum electric field strength, the potential at which a strike occurs increases with increasing height above the ground. Hence, each electron will carry more energy. Determining if there is an effect on the total number of electrons lies in the future.

3.8.

recall that the electric field inside a conductor is zero. Hence, any path from a point on the surface to any point in the interior will have an integrand of zero when calculating the change in potential, and thus the potential in the interior of the sphere is identical to that on the surface.

3.9. The x-axis the potential is zero, due to the equal and opposite charges the same distance from it. On the

-axis, we may superimpose the two potentials; we will find that for

again the potential goes to zero due to cancellation.

3.10. It will be zero, as at all points on the axis, there are equal and opposite charges equidistant from the point of interest. Note that this distribution will, in fact, have a dipole moment.

3.11. Any, but cylindrical is closest to the symmetry of a dipole.

3.12. infinite cylinders of constant radius, with the line charge as the axis

Conceptual Questions

3.1 Electric Potential Energy

1. Would electric potential energy be meaningful if the electric field were not conservative?

2. Why do we need to be careful about work done on the system versus work done by the system in calculations?

3. Does the order in which we assemble a system of point charges affect the total work done?

3.2 Electric Potential and Potential Difference

4. Discuss how potential difference and electric field strength are related. Give an example.

5. What is the strength of the electric field in a region where the electric potential is constant?

6. If a proton is released from rest in an electric field, will it move in the direction of increasing or decreasing potential? Also answer this question for an electron and a neutron. Explain why.

7. Voltage is the common word for potential difference. Which term is more descriptive, voltage or potential difference?

8. If the voltage between two points is zero, can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.

9. What is the relationship between voltage and energy? More precisely, what is the relationship between potential difference and electric potential energy?

10. Voltages are always measured between two points. Why?

11. How are units of volts and electron-volts related? How do they differ?

12. Can a particle move in a direction of increasing electric potential, yet have its electric potential energy decrease? Explain.

3.3 Calculations of Electric Potential

13. Compare the electric dipole moments of charges

separated by a distance

and charges

separated by a distance

14. Would Gauss’s law be helpful for determining the electric field of a dipole? Why?

15. In what region of space is the potential due to a uniformly charged sphere the same as that of a point charge? In what region does it differ from that of a point charge?

16. Can the potential of a nonuniformly charged sphere be the same as that of a point charge? Explain.

3.4 Determining Field from Potential

17. If the electric field is zero throughout a region, must the electric potential also be zero in that region?

18. Explain why knowledge of

is not sufficient to determine

What about the other way around?

3.5 Equipotential Surfaces and Conductors

19. If two points are at the same potential, are there any electric field lines connecting them?

20. Suppose you have a map of equipotential surfaces spaced

apart. What do the distances between the surfaces in a particular region tell you about the strength of the

in that region?

21. Is the electric potential necessarily constant over the surface of a conductor?

22. Under electrostatic conditions, the excess charge on a conductor resides on its surface. Does this mean that all of the conduction electrons in a conductor are on the surface?

23. Can a positively charged conductor be at a negative potential? Explain.

24. Can equipotential surfaces intersect?

3.6 Applications of Electrostatics

25. Why are the metal support rods for satellite network dishes generally grounded?

26. (a) Why are fish reasonably safe in an electrical storm? (b) Why are swimmers nonetheless ordered to get out of the water in the same circumstance?

27. What are the similarities and differences between the processes in a photocopier and an electrostatic precipitator?

28. About what magnitude of potential is used to charge the drum of a photocopy machine? A web search for “xerography” may be of use.