# Homework Solutions

## Ch 23: Problems 5, *, 9, 12, 17, **, 18, 19, ***, 27, 30, 42

Questions
1. Sparks are often observed (or heard) on a dry day when clothes are removed in the dark. Explain.

Clothes rub against your body and electrons are transferred so a charge builds up. When the resulting electric field gets large enough, it breaks down the air -- it rips electrons from the air molecules -- and a bunch of electrons are transferred from the object having net negative charge to the object having net positive charge. As the electrons go back to the air molecules light is given out and a sound is heard.

2. Explain from an atomic viewpoint why charge is usually transferred by electrons.

Atoms are held in place by surrounding atoms. Most of the atom -- including the positively charged nucleus -- can not move. But an electron may be free to travel about or an electron may be torn away from the rest of the atom.

3. A ballloon is negatively charged by rubbing and then clings to a wall. Does this mean that the wall is positively charged? Why does the balloon eventually fall?

The balloon is attracted to and held in place by charge polarization or charge induction -- even tho' the wall is electrically neutral. We talked about this -- and demonstrated it -- in class. Figure 23.?? is a good illustration of this.

The balloon eventually falls because some electrons do move from the balloon to the wall or vice versa and eliminate the excess negative charge.

9. What are the similiarities and differences between Newton’s law of gravity,

F = G m1m2/r2, and Coulomb’s law, F = k q1q2/r2?

The inverse-square nature of both these force laws is their greatest similiarity.

The masses in the Law of Gravity can only be positive so Gravity is only an attractive force. The charges in Coulomb's Law can be either positive or negative so the electric force can be either positive or negative.

23. Consider two equal point charges separated by some distance d. At what point (other than infinity) would a third test charge experience no net force?

At the midpoint, exactly half way between the two equal charges, the electric field will be zero. At that point, the force on a third test charge will be zero.

Problems
23.5 Suppose that 1.00 g of hydrogen is separated into electrons and protons. Suppose also that the protons are placed at Earth’s North Pole and the electrons are placed at the South Pole. What is the resulting compressional force on Earth?

23.*. (a) What magnitude of charge must be placed equally on Earth and the Moon to make the magnitude of the electrical force between these two bodies equal the gravitational force between them?
(b) What would be the electric field on the Moon due to Earth’s charge?

23.17. Three point charges are located at the corners of an equilateral triangle as in Figure P23.7. Calculate the net electric force on the 7.0 microCoulomb charge.

23.9. Four identical point charges (q = + 10.0 microCoulombs) are located on the corners of a rectangle as shown in Figure P23.9. The dimensions of the rectangle are L = 60.0 cm and W = 15.0 cm. Calculate the magnitude and direction of the net electric force exerted on the charge at the lower left corner by the other three charges.

23.12. An object having a net charge Q is placed in a uniform electric field of magnitude E directed vertically. What is the mass of this object if it “floats” in the field?

23.**. On a dry winter day, if you scuff your feet across a carpet, you build up a charge and get a shock when you touch a metal doorknob. In a dark room you can actually see a spark about 2.0 cm long. Air breaks down at a field strength of 3.0 x 106 N/C. Assume that just before the spark occurs, all the charge is in your finger, drawn there by induction due to the proximity of the doorknob. Approximate your fingertip as a sphere of diameter 1.5 cm, and assume that there is an equal amount of charge on the doorknob 2.0 cm away.
(a) How much charge have you built up?
(b) How many electrons does this correspond to?

23.18. Two 2.0-microCoulomb charges are located on the x-axis. One is a x = 1.0 m and the other is at x = - 1.0 m.
(a) Determine the electric field on the y-axis at y = 0.50 m.
(b) Calculate the electric force on a - 3.0 microCoulomb charge placed on the y-axis at y = 0.50 m.

23.19. Four point charges are at the corners of a square of side a as in Figure P23.25.
(a) Determine the magnitude and direction of the electric field at the location of charge q.
(b) What is the resultant force on q?

23.***. A charge of - 4.0 microCoulombs is located at the origin and a charge of - 5.0 microCoulombs is located along the y-axis at y = 2.0 m. At what point along the y-axis is the electric field zero?

23.27. A uniformly charged ring of radius 10 cm (0.10 m) has a total charge of 75 microCoulombs. Find the electric field on the axis of the ring at
(a) 1.0 cm = 0.01 m
(b) 5.0 cm = 0.05 m
(c) 30 cm = 0.30 m
(d) 100 cm = 1.0 m
from the ring.

23.30. Example 23.9 derives the exact expression for the electric field at a point on the axis of a uniformly charged disk. Consider a disk of radius R = 3.0 cm = 0.03 m, having a uniformly distributed charge of + 5.2 microCoulombs.
(a) Using the result of Example 23.12, compute the electric field at a point on the axis and 3.0 mm (0.003 m) from the center. Compare this answer with the field computed from the near-filed approximation.
(b) Using the result of Example 23.12, compute the electric field at a point on the axis and 30 cm (0.30 m) from the center. Compare this answer with the field obtained be treating the disk as a _ 5.2 microCoulomb point charge at a distance of 30 cm.

23.42. Protons are projected with an initial speed vo = 9.55 x 105 m/s into a region where the uniform electric field E = ( - 720 j) N/C is present, as in Figure P23.53. The protons are to hit a target that lies at a horizontal distance of 1.27 mm (0.00127 m) from the point where the protons are laugched. Find
(a) the two projection angles theta that would result in a hit and
(b) the total time of flight for each trajectory.