# Problems for Chapter 4

4-1. How many electrons are on a charged comb which attracts a 1 g piece of paper from a distance of 5 cm away, with an acceleration of 1 cm / s 2 ? Assume that the charge on the comb is equal in magnitude (but of opposite sign!) to that on the paper. Ignore gravity.

4-2. Compare the strengths of the electrostatic and gravitational forces between two isolated electrons. Will they be electrically repelled from each other or gravitationally attracted ?

4-3. The average lightning stroke has a charge of about 30 C. How many electrons does this represent?

4-4. Compute g.

4-5 through 4-9. Use the "Potential Workbook" Mathematica notebook to graph the following charge configurations:

• a proton at the origin
• a proton on the y axis at y = 1 and another at y = -1
• a string of 4 protons on the y axis at y = -1.5, -0.5, 0.5, 1.5
• alternating string of electrons and protons as in #3
• a string of 2 electrons in the middle and 2 protons on the ends as in #3
• 4 protons in a square at (1,1), (1, -1), (-1,-1) and (-1,1)
• 2 protons at (1,1) and (-1,-1), 2 electrons at other points as in #6
• 2 protons at (1,1) and (1,-1), 2 electrons at other points as in #6
• an electron at (1,1) and 3 protons at other points as in #6

For configurations with less than four charges, enter zero for the unused charges. For each configuration, answer the following questions:

4-5. What is the general shape of the equipotentials as seen from (2,0)?

4-6. What path would an electron take if it were placed at the origin?

4-7. What path would an electron take if it were placed at (0.5, 0)?

4-8. What path would an electron take if it were placed at (0, 0.5)?

4-9. What path would an electron take if it were placed at (0.5, 0.5)?

Now use the "Potential Simulation" Mathematica notebook to test your path answers. If the notebook tells you that something is "not a machine-representable real number", it is because the force between the test particle and one of the configuration charges is infinite. Why might that be? Note any symmetry in the charge configurations. What happens to a test charge which starts on a line of symmetry?

4-10. Molecular bonds can be of several types:

 ionic attraction of charged atoms covalent sharing of electrons among atoms van der Waals attraction of polarized molecules (with nonspherical charge distributions) Hydrogen a type of van der Waals attraction involving Hydrogen

The potentials for all but ionic are not Coulombic, but are more complex functions of the separation distances between atoms. Nonetheless, we will pretend that the energies are Coulombic for the purposes of this problem. For covalent bonds, we will assume that one electron is transferred to the other atom for each bond.

Compute the energies in kJ / mol of the following bonds:

Bound AtomsBond TypeSeparation Distance (Angstroms)
Na + to Cl - ionic2.76
H + to O -covalent (single)1.03
C - to H +covalent (single)1.14
C - to C +covalent (single)1.54
C -- to C ++covalent (double)1.34
C --- to C +++covalent (triple)1.2
H + to O -van der Waals2.6
OH + to O -Hydrogen2.76

4-11. The molecule Cyclohexane (an organic solvent) occurs in two configurations: the boat and the chair. From above, each looks like a hexagon. From the side, four carbons are coplanar in an approximate square for both configurations; for the boat, the end carbons are both on the same side of the planar square, while for the chair, the end carbons are on opposite sides. Assume that the locations of the carbons are (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), ( -.217,.5,.266) and either (1.217, .5, .266) or (1.217, .5, -.266) (these are relative coordinates which guarantee that the angles between each pair of bonds is 111 degrees). Compute the ratio of the chair energy to the boat energy, assuming that all atoms are equally charged. The energy of each configuration is the sum of the energies of each pair of atoms in the configuration.

If you have a VRML (Virtual Reality Modeling Language) -capable browser, or a VRML plug-in (like Live3D for Netscape 3.0), you can look at the chair and boat configurations in 3D! (both are 33K)

4-12 through 4-19:

We will see in the next section that such components can be eliminated in favor of an equivalent component; ie., two resistors in parallel can be replaced by a resistor of a different value, etc. Note that not all pairs of like components can be combined and that at least one circuit cannot be simplified at all.

4-20 through 4-27:

Refer to the diagrams above. Ignore all of the capacitors. Assume that the batteries are all 12 V, and all of the resistors are 5 W. What is the voltage drop and current across each of the resistors?

4-28. Normal household current (120 V, 60 Hz alternating current) produces the following reactions for 1 second of skin contact:

I (mA)Effect
1awareness threshold
5maximum harmless current
10-20sustained muscular contraction
50pain, possibly fainting and exhaustion
100-300ventricular fibrillation
6000temporary respiratory paralysis, possibly burns

What is the effect of a 1 second shock when

• you are barefoot on a dry floor (R = 10 KW to ground)?
• you are barefoot on a wet floor (R = 1 KW to ground)?
• you are wearing dry shoes (R = 1 MW to ground)?

4-29. It is very important NOT to ground a patient (which would allow current to flow from a medical instrument through their body to the ground), since even very small currents (20 mA) can cause ventricular fibrillation when applied directly to the heart (ie., through a catheter). Consider a patient who is inadvertently grounded and whose resistance to ground is 1 KW due to the insertion of a catheter. Assume that an EKG monitor wire is leaking 30 mV through the patient's skin. Is the patient in danger?

4-30. The space shuttle often flies in an approximately circular orbit 300 km above the surface of the Earth, orbitting once every hour and a half. Compute the acceleration due to gravity in the shuttle. Why do the astronauts appear to be weightless?

4-31. Ben Franklin is credited with determining that electric charge comes in two varieties, which he astutely called "positive" and "negative" because they acted as opposites; the addition of one to the other results in zero charge. How could he be sure that there were only two? (Hint: consider the results of applying first one charge then another; does it depend on the order of application?) If there were actually three types of charge, which were not opposites but instead acted as primary colors (such that only the sum of all three is colorless, or neutral), how would he have discovered them?

## How about problems about resistors and circuits?

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©1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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