Week 3: Potential Energy and Potential 127
Homework for Week
Problem 1.
Physics Concepts
Make this week’s physics concepts summary as you work all of the problems in this week’s
assignment. Be sure to cross-reference each concept in the summary to the problem(s) they were
key to. Do the work carefully enough that you can (after it has been handed in and graded) punch
it and add it to a three ring binder for review and study come finals!
Problem 2.
Suppose you have chargeqat positionz=aon thez-axis and charge−qatz=−a– anelectric
dipoleas studied in the first chapter. a) Write an exact expression for the eletrostatic potential of
the dipole at~r= (r, θ, φ). Note that the potential must beφ-independent because of azimuthal
symmetry. b) Expand your answer to a) forr≫ato leading surviving order and express the answer
in terms of the magnitude of the (z-directed) dipole moment,pz= 2qa.
Bonus: Where is the potential of this arrangement identically zero?Right, thexy-plane. Suppose
one slides an (infinite) thingroundedconducting plane in between the two charges. This costs no
work (right?) and does not alter the fields or potentials in either half-space above or below it. Now
imagine removing the charge below this plane. Does doing so change the fields or potentials in the
upper half space (recall that the conductorscreensthe two spaces). Using the insight gained from
thinking about this, do you expect a bare charge of either sign to beattracted to or repelled by a
nearby grounded conducting sheet?
Problem 3.
Now let’s assume a charge−qatbothpositionsz=±aon thez-axis and a charge +2qat the origin.
Note that this is a pair ofopposedelectric dipoles. a) Write an exact expression for the eletrostatic
potential of the dipole at~r= (r, θ, φ). Note that the potential must beφ-independent because of
azimuthal symmetry. b) Expand your answer to a) forr≫ato leading (surviving) order. c) What
might we call this term? (Hint: Count the poles.)