GRAVITATIONAL METHODS 



295 



change, or gradient, at a point is the force which would act on a unit particle 

 at this point. (To find the force on other than a unit particle, the force on 

 a unit particle must be multiplied by the magnitude of the particle.) 



To denote the direction of the 

 force, it must be recalled that the 

 particle will move from higher to 

 lower potential. Thus if the gra- 

 dient of the potential, in the posi- 

 tive r direction, is positive, the 

 potential must be increasing in 

 that direction. However the force 

 is directed in the opposite direc- 

 tion, and the conclusion is that a 

 minus sign is necessary. Consider- 

 ation will show this to be true for 

 all types of conservative fields 

 (those which have a potential) , be 

 the force attractive or repulsive. 



Thus 



^*^ EQUIPOTENTIAL 



SURFACES (SPHERES) 



-Showing equipotential surfaces Ui 

 and U2 surrounding spherical mass m. Pi and 

 P2 are points on surfaces Ui and U2 respectively. 



Fig. 158. 



F= - 





(28) 



If, of course, the gradient is measured in the negative r direction, the 

 minus signs will cancel. 

 From 28, 



U= -fFdr (29) 



It is apparent that U will not be uniquely determined to the extent of a 

 constant of integration. However if the potential at infinity is arbitrarily 

 set at zero, the potential at a point may be completely defined. 



We are most concerned, however, with differences of potential. This 

 has already been defined (^U) as the negative of the work done by the 

 field in moving a particle between the points. That is, if the field has done 

 work in moving a particle, it must, by conservation of energy, be at a 

 lower potential at the final point. Its potential for doing work on a particle 

 at this new point must have decreased. 



Mathematically, in moving a particle from Pi to Pg (Figure 158) the 

 field does work 



W 



2 2 



,p =\Fdr=G\ —r- dr = Gm I ) 



12 ; ) r^ \r2 rj J 



(30) 



a negative quantity. 



