1. LAPLACE’S EQUATION AND UNIQUENESS
WRITTEN BY: Steven Pollock (CU-Boulder)

2. A region of space contains no charges
A region of space contains no charges. What can I say about V in the interior?
3.1
Not much, there are lots of possibilities for V(r) in there
B) V(r)=0 everywhere in the interior.
C) V(r)=constant everywhere in the interior
=0
throughout
this interior
region
CORRECT ANSWER: A
USED IN: Fall 2008 (Dubson) and Spring 2008 and ‘13 (Pollock), Fall 2009 (Schibli)
LECTURE NUMBER: Dubson (Week 5, Lecture 12). Pollock (Lecture 12, or 13 in ‘13).
STUDENT RESPONSES: [[80%]] 0% 20% 0% 0% (FALL 2008)
n/a (SPRING 2008)
[[77%]] 0% 23% 0% 0% (FALL 2009)
[[47]], 10, 43, 0,0 (Sp ‘13)
INSTRUCTOR NOTES: Pollock didn’t click on this one, just talked it through. My answer is A, there's not enough information here to say much of anything (Except del^2V=0, of course)
-SJP
WRITTEN BY: Steven Pollock (CU-Boulder)

3. A region of space contains no charges. The boundary has V=0 everywhere
A region of space contains no charges. The boundary has V=0 everywhere. What can I say about V in the interior?
3.2
Not much, there are lots of possibilities for V(r) in there
B) V(r)=0 everywhere in the interior.
C) V(r)=constant everywhere in the interior
V=0
=0
CORRECT ANSWER: B
USED IN: Fall 2008 (Dubson) and Spring 2008 (Pollock), Fall 2009 (Schibli)
LECTURE NUMBER: Dubson (Week 5, Lecture 12). Pollock (Lecture 12).
STUDENT RESPONSES: 2% [[98%]] 0% 0% 0% (FALL 2008)
0% [[100%]] 0% 0% 0% (SPRING 2008)
52% [[38%]] 10% 0% 0% (FALL 2009)
3, [[76%]], 20, 0, 0 ‘Sp 13
INSTRUCTOR NOTES: Clicked on this one. 100% correct! I had JUST presented “V is not an extrema” on the board. Interestingly, we had not yet formally introduced the uniqueness theorem.
Apparently the difficulty depends a lot on what has just preceded it, though – very different results in different classes!
My answer is B, by uniqueness theorem. -SJP
WRITTEN BY: Steven Pollock (CU-Boulder)

4. Two very strong (big C) ideal capacitors are well separated.
3.3
If they are connected by 2 thin conducting wires, as shown, is this electrostatic situation physically stable? -Q Q -Q Q + +_ -
Yes No
??? + +_ -
CORRECT ANSWER: B
USED IN: Fall 2008 (Dubson) and Spring 2008 and ‘13 (Pollock), Fall 2009 (Schibli)
LECTURE NUMBER: Dubson (Week 5, Lecture 12). Pollock (Lecture 12, 13 in ‘13).
STUDENT RESPONSES: 18% [[82%]] 0% 0% 0% (FALL 2008)
25% [[ 75%]] 0% 0% 0% (SPRING 2008)
19% [[77%]] 0% 0% 4% (FALL 2009)
50, [[47]], 3, 0, 0 (Sp ‘13)
INSTRUCTOR NOTES: 25% voted for A. Interesting discussion. I explained it through “uniqueness”, (but they want to know WHY the charges will move - what provides the E field that “pushes” the charges through the conductors in that brief non-equilibrium instant…)
My answer is B no, it's not stable in the configuration shown.
The charges COULD run through the wires, making both plates have zero charge and zero fields everywhere. But the uniqueness theorem (Version II in griffiths) says this is the ONLY possible solution.
Perhaps it might be more physically “obvious” if you imagine that thin wire becomes just a giant conducting block. So now you have a capacitor with a giant conductor nearly filling the inner space. But wait, it’s not a capacitor, because of that upper wire – so picture a big “horseshoe shaped” conductor on the outside. Now it somehow seems easier to see that there is no reason for the charges to STAY separated like this… (?)
-SJP
WRITTEN BY: Steven Pollock (CU-Boulder)

5. Two very strong (big C) ideal capacitors are well separated.
3.4
What if they are connected by one thin conducting wire, is this electrostatic situation physically stable? -Q Q -Q + Q -
Yes No
??? + -
CORRECT ANSWER: A
USED IN: Fall 2008 (Dubson) and Spring 2008 and ‘13 (Pollock)
LECTURE NUMBER: Dubson (Week 5, Lecture 13). Pollock (Lecture 12). (Lecture ‘13 in ‘13)
INSTRUCTOR NOTES: Not voted on, but discussed it in the context of the previous one.
[[82]], 16, 2, 0, 0 (SP ‘13)
My answer is yes, the conducting wire changes nothing at all. This can be a little baffling for students, (but again, picture a capacitor with a big chunk of metal inside. The metal will fully polarize – that’s exactly what we see here)
WRITTEN BY: Steven Pollock (CU-Boulder)

6. Is this a stable charge distribution for two neutral, conducting spheres? (There are no other charges around) + - + -
CORRECT ANSWER: B
USED IN: Fall 2009 (Schibli), Sp 2013 (Pollock)
LECTURE NUMBER: 12 in ‘09, 14 in Sp 13
STUDENT RESPONSES: 12% [[86%]] 2% 0% 0% (FALL 2009)
36, [[64]], 0, 0, 0 (Sp ‘09)
INSTRUCTOR NOTES:
Just a little review of what was covered last class (did it as the “preclass” question in ‘13) One student said he was thinking that you could surely have two little dipoles sitting near each other – I pointed out that it said “conducting spheres”, which changed his mind! It takes no work to move charges through the conductor, so they CAN neutralize. Uniqueness theorem says that if Q is known for all conductors, then there is a unique solution (and the V=0 everywhere solution is the physical one, here) -SJP
WRITTEN BY: Thomas Schibli (CU-Boulder)
Yes No
C) ???

7. General properties of solutions of 2 V=0
V has no local maxima or minima inside. Maxima and minima are located on surrounding boundary.
V is boring. (I mean “smooth & continuous” everywhere).
V(r) = average of V over any surrounding sphere:
(4) V is unique: The solution of the Laplace eq. is uniquely determined if V is specified on the boundary surface around the volume.
CORRECT ANSWER:
USED IN: Fall 2009 (Schibli) and Pollock Sp ‘2013
LECTURE NUMBER: 14 in 2013 (earlier in 2009)
STUDENT RESPONSES: NA
INSTRUCTOR NOTES: Nice review from last class.
WRITTEN BY: Thomas Schibli (CU-Boulder)

8. If you put a + test charge at the center of this cube of charges, could it be in stable equilibrium?
3.5 +q +q
Yes No
??? +q +q
CORRECT ANSWER: B
USED IN: Fall 2008 (Dubson) and Spring 2008 (Pollock), Fall 2009 (Schibli)
LECTURE NUMBER: Dubson (Week 5, Lecture 13). Pollock (Lecture 13).
STUDENT RESPONSES: 73% [[27%]] 0% 0% 0% (FALL 2008)
24% [[76%]] 0% 0% 0% (FALL 2008)
61% [[35%]] 4% 0% 0% (SPRING 2008)
71% [[29%]] 0% 0% 0% (FALL 2009)
54, [[46]], 0, 0 (Sp ‘13) I talked about the “no local minimum” and revoted, but it did not help!
INSTRUCTOR NOTES: Start of class (review) question. 60% voted A, we had excellent discussion of this (and how the "no local minimum of V in a charge free region" PROVES the answer is no. ) The "Earnshaw's theorem" text should appear *after* the concept test is done. It's very iinteresting to think about how crystals are stable, given this theorem! (quantum mechanics matters)
In ‘13, I pointed out that if you think of a small region AROUND the center, where there are no charges, (except the test charge), you can apply the principles of the previous slide. But this did not help. Apparently this is a subtle/difficult concept.
Some students realized they could use Gauss’ law (if there was stable equilibrium at the middle, E must point “in” towards the middle point from all directions, to move a little +test charge back to the center - but that’s in violation of Gauss’ law!)
-SJP
Dubson: Great discussion, lots of questions. A winner.
Modifed by SVC after Fall 08 to make charges excplicitly positive.
WRITTEN BY: Steven Pollock (CU-Boulder) +q +q
Earnshaw's Theorem +q +q