174 Mechanical and Electrical Character of the Heartbeat /9 : 6 



This is a special case of Ohm's law. The unique solution choosing 

 V = at infinity is 



m = I -rS 



.= i y\r- r t \ 

 This may be expanded in a series in 1/r. Expanding, one obtains 



V(r) = y r 2h + ±(lti)^+ 2^g2/i[3(vr) a - R 2 ] + ••• 



Because no net charge enters or leaves the heart, the first sum is zero. 

 The second sum is called the dipole moment, p; that is 



?«24 



ih 



A first approximation to the potential due to current sources in an 



infinite conducting medium is to replace them by an equivalent dipole p. 

 The potential at r (referred to V = at infinity) is 



V(r)J4 



' yr 3 



The preceding expression was obtained for an infinite medium. If 

 one restricts the heart to a sphere of radius R, a somewhat more com- 

 plex expression is necessary. Consider the equivalent dipole p located 

 at the center of a sphere and oriented along the 6 = axis of the sphere. 

 In this case 



— ^ — * 



p ■ r = pr cos 6 



At small values of r, the potential must approach that of a dipole in an 

 infinite medium, namely 



T/ p cos 6 



yr 2 



as r-> 



whereas at the surface, the radial current must be zero, so that 



— = at r = R 

 or 



The unique solution to this approximation is 



r 2 + R 3J 



v _p cos 6(1 



y 



which, at the surface of the sphere, reduces to 



V {R) = W (10) 



