Chapter 23

Chapter 24

Chapter 26

Chapter 25

Chapter 30

Chapter 31

**Problem-solving methods**

Potential energy functions

Gauss' Law

Simple circuits

Multiloop circuits

12. Provide a vector diagram, define positive directions, and then write net E-field equations. This is much like a net force problem, but don't confuse force with field.

17. Start with a vector diagram, define positive direction, and then write the net E-field equation. Rewrite to express in terms of the ratio d/z. You can now apply the binomial theorem. You'll need to use 2nd-order terms.

19. You'll need to obtain a force equation and then apply an approximation to show that the force is a Hooke's Law type of restoring force. Hence, the motion is SHM. Show your work completely. (I showed you this technique in class for the Earth tunnel problem.)

25. This requires two integrals, one for each axis. You can use trig substitution for one of the integrals. See this example problem for a similar integral: 23-24.

43. This is a straightforward 2-dimensional dvat problem with an electric force producing the acceleration. Note that a quadratic equation has two solutions. Make it clear why you keep the one that you keep.

46. Keep it simple. Use the dot product. You need 2 vector diagrams--one for each of the initial and final states.

47. This requires an approximation. Explain why you can approximate. Explain your method for obtaining frequency. Now some hints:

The formula that you're expected to come up with for frequency should remind you of formulas we've had before for the frequency of oscillatory motion for various physical situations. I can think of at least four such situations, three from last semester and one from this semester, the Earth tunnel problem. In that problem, you had to find a frequency (actually period). Review the steps leading up to that result. The procedure for the oscillating dipole is analogous.

19 and 25. Problem 24.18 will be done in class and can serve as a guide problem for applying Gauss' Law to cylindrical charge distributions.

28. This is a simple problem once you recognize what fields you need to superimpose. Consider what you have to add to that central, circular region of the charged plane in order to give it a net charge of 0. Then superimpose the field of what you added onto the field of the entire charged plane. Justify your method in a sentence or two.

30. Before doing this problem, a careful rereading of section 24-8 is highly recommended.

33. For your Gaussian surface, choose a box centered on the central plane. Use ample sentences to explain how you determine the flux through each side. Of course, provide complete diagrams as always.

37. Extension: Integrate the electric field found in problem 24-37 to obtain the potential. That is, evaluate . Check your result by determining the value of r where V = 0.

40. Provide explanations using a combination of words and diagrams.

41. This requires an integration to find total enclosed charge.

43. Show your work as per the guidelines for Gauss' Law problems below with the exception of step 3. You no longer need to explicitly state your reasons for selecting a particular Gaussian surface. Show your derivation for parts a and b. Since parts b and d require similar applications of Gauss' Law, you need not show your derivation for part d. For part d, you may simply give your answer, stating why it is or isn't different from part b.

44. Numbers are given, so numbers and units must be substituted as a final step. The graph must be scaled.

8. Don’t forget a vector diagram. Tell how you deal with ds and dr.
Remember
that you find DV as an integral of a dot
product. See section 25-5 and your notes to review how to do this.

34. This requires an integral of the form . See example from notes for problem 23-25.

31. Hint: Express the energy density as a function of the radial distance in the region between the plates. Then integrate the energy density over the volume enclosed by the plates.

11. Start with a diagram that defines the coordinate system and all needed quantities and vectors. Set up the integral. Then integrate using a trigonometric substitution method. Don’t overlook the check as L approaches infinity. (Optional credit will be provided if you successfully integrate using trig substitution. For a review of a similar integral, see the solution to problem 23-24.)

19. Let di represent the current in an infinitesimal section of the wire. Relate this to the width, dx, of the current element and the linear current density. Then write the infinitesimal magnetic field, dB, due to the element di at point P. Set up the integral and evaluate. Once you have your result, show that it reduces to the expression for the field of a straight wire when d is much greater than w. See Appendix for a needed approximation.

26. In your solution, include two diagrams: one showing the magnetic fields at the location of one wire due to the other 3 wires and the other showing the forces due to those fields.

35. As this is your first demonstrated use of Ampere’s Law to solve a problem, provide justification in words for the steps in your solution. For example, explain how you choose the Amperian loop needed to apply Ampere’s Law and why this makes it possible to do the integral.

18. Define an angle which specifies the orientation of the loop. For example, q = 0° could be the position of the loop as shown in the diagram. q = 90° and 270° could be the positions when the semicircular portion of the loop lies in a plane perpendicular to the page. Write an equation for the area enclosed by the loop as a function of time, etc. (Note that the flux integral is simply BA, since the magnetic field is uniform in space.)

21. The magnetic field has both a spatial and time dependence. When you integrate to find the magnetic flux, just integrate over the spatial variable. Take time as a constant in this part of the solution. That is, you're evaluating the flux at an instant of time. Once you've completed the flux integration, you can then treat time as a variable for the purpose of finding the induced emf.

29. For part c, use only electrical quantities (current and resistance) to find the rate of generation of thermal energy. For part e, use dynamical quantities. You'll need the force that you found in part d.

54: This is a 12-part problem, but don’t be dismayed. Each part is short. This problem really checks for fundamental understanding of LR circuits.

60: Solve independently for the three energies requested. That is, don’t use the law of conservation of energy to solve for the 3rd after you know the first two. Rather, use the law of conservation of energy to check that your 3 energies have the expected relationship. Carry 3 significant figures in your calculations. By the way, each energy calculation requires an integration, since current is a function of time.

**Solving for potential energy functions**

See Section 14.6, Proof of Equation 14-20, for an example.

- Draw a diagram. Label the origin, +r direction, dr vector, force vector, and relevant positions and distances. The force will be the force exerted by the field.
- Evaluate the work done by the force between the end points, starting with the integral dot product definition of work. In finding the dot product, evaluate F(dr)cosf. Substitute F as a magnitude, since the cosf takes care of signs.
- Find DU by setting it equal to the negative of the work done by the force.
- Substitute the value chosen for the 0 level of potential energy.
- Solve for U.

- Your sketch of the problem situation must include the Gaussian surface that you will use.
- On each different part of the Gaussian surface, draw and label the E-field and dA vectors. For example, for a cylindrical Gaussian surface, there are three parts: the two circular bases and the curved surface. In that case, you would have 3 pairs of E, dA vectors.
- Explain why you selected the surface that you did. Saying something like you selected it to match the symmetry of the situation isn't enough. You need to specifically say how your selection makes it possible to evaluate the flux integral. See Tactic 1 on p.550 of your text.
- Write the general statement of Gauss’ Law.
Use q
_{enc}to represent the charge so as not to lose sight of the fact that the charge is the charge enclosed by the surface. Evaluate the flux integral for each part of the surface. Show the result of the dot product and the integration for each part. Then sum the integrals to get the total flux over the surface. - Depending on the problem, you may need to re-express the enclosed charge in terms of the charge density and geometrical factors.
- Solve for the unknown.
- In addition, whenever you use a shell theorem, indicate how you apply it. Whenever you have to determine a charge distribution, justify with electrostatic principles.

- A circuit diagram with all components labeled and using distinguishing subscripts as necessary. List all given numerical values.
- Explicitly show your use of conservation laws (charge and energy).

**Solving multiloop circuit problems**

*See the examples in the text for how to determine the signs of emfs and
potential drops when traversing a circuit loop.*

- List all given values of currents, resistances, and emfs. Use subscripts to distinguish between multiple currents, resistances, and emfs. Show and label these on the circuit diagram.
- Show the directions you choose for the currents.
- Show the directions in which you choose to traverse the various circuit loops. These do not have to correspond to current directions.
- Label the loops in order to have a way to refer to them in your solution.
- Write the junction rule as it applies to one of the circuit junctions.
- Write all of the loop equations and identify which loop they go with.
- Substitute given numerical values into the loop equations. This is an exception to the rule that all numerical values are substituted in the final step.
- Select the minimum number of loop equations necessary to solve for the unknowns. Solve the equations simultaneously. Clearly display your results for the unknowns.
- As a check, substitute values into a loop equation that you didn't use above. If the equation doesn't check, go back and find your mistake(s). In particular, make sure that all your signs are correct.
- For greater confidence in your solution, substitute your values into all the loop equations and the junction equation in order to make sure they all check. There's no excuse to get a circuit problem wrong when you have available the means to check your results.