203, 204] DIELECTRICS AND MAGNETIZABLE BODIES. 395 



204. Dissipation of energy in Static Hysteresis. Since 

 we have seen that p is not uniquely determined by the value of F, 

 so it must be for the energy of the field. Accordingly the forces 

 acting on polarized bodies cannot be derived from a single-valued 

 potential, but must be non-conservative. In taking a body through 

 a cycle of magnetization, accordingly, a certain portion of the work 

 done upon it fails to be stored up as energy, and is therefore 

 dissipated into heat. We may easily find an expression for the 

 value of this dissipated energy. The potential energy of a 

 polarized body in a field whose potential is V is, by 126 (2), 

 equal to 



or in terms of the field 



W=- \l\(AX + BY+CZ)dT. 



If we consider an element of volume dr, and suppose it moved to 

 a point where the field is 



X + dX, Y+dY, Z+dZ, 



the work dW done upon the particle during the motion is accord- 

 ingly equal to the increase in the value of the energy, 



(i) dW = - dr(AdX + BdY + CdZ). 



In the second position the values of A, B, C have changed to the 

 values 



A+dA, B + dB, C + dC, 



but the change made by using these values in the expression for the 

 work would be of the second order and may be neglected.* If instead 

 of moving the particle we change the strength of the field the work 

 done will be the same. Inserting the values of A, B, G in terms 

 of the induction and force we obtain 

 j 



(2) 



If now we vary X, Y, Z through a cycle of values, coming back to 

 the value from which we started, the integral 



(3) 



