Fa06Soln8 - Physics 7B Fall, 2006 Homework 8 Solutions...

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Physics 7B Fall, 2006 Homework 8 Solutions Note: The original file that these solutions came from apparently had some problems with equation editor. In some of the equations, you may see the symbol “Å”, though it’s really supposed to be an “ Å ”. 23.17 ( a ) The reference level for the potential is V = 0 at r = 8. ( c ) At points outside the sphere, the electric field is that of a point charge: E = Q /4 π Å 0 r 2 , r > r 0 . Thus the potential when r > r 0 is V = Q /4 π Å 0 r , r > r 0 . ( b ) The charge density inside the sphere is ρ = Q / 9 π r 0 3 . The electric field at a distance r < r 0 is due to the charge inside the sphere with radius r : E = ( 9 π r 3 )/4 π Å 0 r 2 = Qr /4 π Å 0 r 0 3 , r < r 0 . We find the potential by integrating along a radial line from r to r 0 : d V r r 0 = – d r r 0 =– E d r r r 0 ; Q 4 ! Å 0 r 0 V = Q 4 ! Å 0 r 0 3 r d r r r 0 = Q 8 ! Å 0 r 0 3 r 0 2 r 2 , whichgives V = ( Q /8 π Å 0 r 0 )[3 – ( r 2 / r 0 2 )], r < r 0 . 23.20 ( a ) In each region the electric field is the same as that of a point charge equal to the net charge within a spherical surface. Thus we have E = (+ Q + ! Q )/4 π Å 0 r 2 = 3 Q /8 π Å 0 r 2 , r > r 2 . E = 0, r 1 < r < r 2 (inside a conductor), which means there is a negative charge – ! Q on the inner surface; thus a positive charge 8 Q on the outer surface; E = ! Q /4 π Å 0 r 2 = Q /8 π Å 0 r 2 , 0 < r < r 1 . ( b ) The reference level for the potential is V = 0 at r = . At points outside the spherical conducting shell, the potential is that of a point charge with net charge Q + ! Q . Thus the potential when r > r 2 is V = (+ Q + ! Q )/4 π Å 0 r = 3 Q /8 π Å 0 r , r > r 2 . ( c ) The electric field within the spherical conductor is zero, so its potential must be constant and therefore equal to the potential on the surface: V = 3 Q /8 π Å 0 r 2 = 3 Q /16 π Å 0 r 1 , r 1 < r < r 2 . ( d ) The potential inside the hollow region is due to the point charge at the center with an additive constant to make the potential at the inner surface of the conductor the constant potential of the 0 r 0 V 0 r 0
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This note was uploaded on 09/09/2008 for the course PHYSICS 7B taught by Professor Packard during the Fall '08 term at University of California, Berkeley.

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Fa06Soln8 - Physics 7B Fall, 2006 Homework 8 Solutions...

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