Potential energy functions
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. Donít forget a vector diagram. Tell how you deal with ds and dr.
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. Donít 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 Ampereís 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 Ampereís 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 = 0į could be the position of the loop as shown in the diagram. q = 90į and 270į 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 donít 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, donít 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.
Solving Gauss' Law problems
Solving circuit problems
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.