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