d. The inside and outside potentials must be equal on the surface of the sphere. Thus Vf(r = a) 

 = Vw(r = a). 



In order to satisfy the boundary conditions, the solution of Laplace's Equation must be of the 

 form 



Vw = A^rcose + ^^^°/^ (2) 



r ^ 



outside the sphere, and so that boundary condition (a) is satisfied, 



Vf = AfrcosQ (3) 



inside the sphere. A^, B^, and A-f are constants to be determined by the boundary conditions. 



At large distances from the sphere, eq. (2) becomes V^v = A^rcos9. According to boundary 

 condition (b), at large distances V = -EQrcosS. Thus we see that A^^, = -Eq. 



The radial electric field is found by Ej. = . Hence (E^)j. = ^ = -EqCosS ^^^ 



(Ef)r = --37^= Afcose. 



According to boundary condition (c), these multiplied by (7^ and (Tf respectively must be equal at r = a. 

 This gives 



^,A, = - a E - '"W^.fw ,4) 



f f w o a-5 



Equating the inside and outside potentials on the surface of the sphere as required by boundary 

 condition (c) we obtain 



Afa = -E^a +-|^ . (5) 



Multiplying eq, (4) by a and eq. (5) by -(r£, and adding, we obtain 



= Eoa((rf - a^) ^2~ ("^f ■•" ^'^w)- 



Solving for B^: 



.3, °-f - "• 

 <^f 



Substitution of this value of B^ into eq. (4) or (5) yields 



Bw = Eo^^( :;;27 >. f^) 



The solution for the potential is thus 



Af = -Eo( ^/;7^ ). (7) 



-EqZ (1 - ^^ -/W ^_) (8) 



3 



Vf = -Eoz ( ':- ). (9) 



Head-to-tail Potential 



The head-to-tail potential on the fish corresponds here to the potential difference between the 

 points z = a and z = -a. If the length of the fish is L = 2a, and the head-to-tail voltage is. Vj^, then 



Vl = Vf (-a) - Vf (+a) = E„L ( _l£^). (10) 



In other words, the head-to-tail voltage on a fish in a uniform electric field Eq ia not merely its 

 length times the field strength, but there is an additional factor depending on the relative conductivi- 

 ties of the fish and the surrounding water. In the special case where they are the same, erf = cr^, then 

 the factor reduces to 1. When the conductivity of the fish is greater than that of the water as could be 



22 



