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2
                        the shell thickness by   = b − a, we find that K ≈ 6µ  /a when  /a 	 1.Thus
                                                                       r
                                                             3 1
                                                         κ =

                                                             2 µ r
                                                                 a
                        describes the coefficient of shielding for a highly permeable spherical enclosure, valid
                        when µ r   1 and  /a 	 1. A shell for which µ r = 10, 000 and a/b = 0.99 can reduce
                        the enclosure field to 0.15% of the applied field.





                        3.4   Static field theorems

                        3.4.1   Mean value theorem of electrostatics
                          The average value of the electrostatic potential over a sphere is equal to the potential
                        at the center of the sphere, provided that the sphere encloses no electric charge. To see
                        this, write

                                                                        ˆ


                                           1    ρ(r )      1            R    ∇  (r )






                                    (r) =            dV +        − (r )    +         · dS ;
                                         4π	   V  R        4π  S        R 2    R
                        put ρ ≡ 0 in V , and use the obvious facts that if S is a sphere centered at point r then
                                                         ˆ
                        (1) R is constant on S and (2) ˆ n =−R:

                                                   1                1

                                           (r) =     2   (r ) dS −       E(r ) · dS .
                                                 4π R  S           4π R  S
                        The last term vanishes by Gauss’s law, giving the desired result.
                        3.4.2   Earnshaw’s theorem
                          It is impossible for a charge to rest in stable equilibrium under the influence of elec-
                        trostatic forces alone. This is an easy consequence of the mean value theorem of electro-
                        statics, which precludes the existence of a point where   can assume a maximum or a
                        minimum.
                        3.4.3   Thomson’s theorem
                          Static charge on a system of perfect conductors distributes itself so that the electric
                        stored energy is a minimum. Figure 3.23 shows a system of n conducting bodies held at
                        potentials   1 ,...,  n . Suppose the potential field associated with the actual distribution
                        of charge on these bodies is  , giving

                                             W e =    E · E dV =     ∇  ·∇  dV
                                                  2  V          2  V
                        for the actual stored energy. Now assume a slightly different charge distribution, resulting

                        in a new potential   =  +δ  that satisfies the same boundary conditions (i.e., assume
                        δ  = 0 on each conducting body). The stored energy associated with this hypothetical
                        situation is


                                        W = W e + δW e =     ∇(  + δ ) ·∇(  + δ ) dV
                                          e
                                                        2  V

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