APPENDIX II 

 A FISH IN A UNIFORM ELECTRIC FIELD 



We wish to analyze here the electrical problem of a fish located in a uniform electric field with 

 the long axis of the fish parallel to the field. The problem is to determine the head-to-tail potential 

 in the fish and the current through the fish. 



In order to reduce this problem to mathematical analysis we must approximate the shape of the 

 fish by some suitable geometrical nnodel. A long, thin ellipsoid of revolution would approximate the 

 shape of a fish very well, but would still leave the mathematical solution rather difficult. A simple 

 model from a mathematical point of view is a sphere, and even though this is a rather poor model of 

 a fish, the solution of this problem will at least give a qualitative answer to the original one. 



We can assume that the uniform electric field is produced by two large, plane, parallel 

 electrodes placed in the water with a wide separation as compared with the length of the fish. Let the 

 strength of this uniform field be Eq. Let the conductivity of the water be cr^ and that of the fish 

 (sphere) cf. 



axis 



We shall choose the coordinate system so that the origin is at the center of the sphere of radius a 

 and the Z-axis is in the direction of the field Eq. Any point inside or outside the sphere can then be 

 designated by the polar coordinates r and 9. A third coordinate is not required since there is 

 symmetry about the Z-axis. 



The potential relative to the origin at any point inside or outside the sphere is found by the 

 solution of Laplace's Equation 



-^<^^-«T7-'^^<^^-^' = °- 



(1) 



subject to the following boundary conditions: 



a. The potential at the origin shall be taken as zero. Thus V = at r = 0. 



b. The field at large distances from the sphere must be equal to Eq. This requires that the 

 potential at large distances be equal to -E^z, where z is the coordinate along the Z-axis and z = rcosO. 

 Thus V = -E^rcosQ for large r. 



c. The component of the current density normal to the surface of the sphere must be the same 

 on both sides of the sphere. If E£ and E^,, are the fields inside (in the fish) and outside the sphere (in 

 the water) respectively, then (Ef)i. and {E^)^ are the radial and hence normal components of these 

 fields. This boundary condition then requires that jr = <rf(Ef)i. = (r^(E^)r at r = a, jj. being the normal 

 component of the current density. 



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