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LAPLACE ’ S EQUATION AND UNIQUENESS

LAPLACE ’ S EQUATION AND UNIQUENESS. 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

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LAPLACE ’ S EQUATION AND UNIQUENESS

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  1. LAPLACE’S EQUATION AND UNIQUENESS

  2. 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

  3. 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

  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 ??? - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + +_

  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 ??? - - - - - - - - - - - - + + + + + + + + + + + + - - - - - - - - - - - -

  6. Is this a stable charge distribution for two neutral, conducting spheres? (There are no other charges around) + + - - + + - - + + - - + + - - + + - - • 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.

  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 +q Yes No ??? +q +q +q Earnshaw's Theorem +q +q

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