GRAVITATIONAL METHODS 297 



Potential difference between two points in a field has been logically and physically 

 defined as the negative of the work done by the field in moving a particle between the 

 points. A field doing negative work in moving a particle implies that work has been 

 done against the field by outside forces. In such a situation the potential difference is 

 positive (the negative of negative work) and the final point is at a higher potential 

 than the initial. This is physically sound, since work has been done against the field 

 by outside forces, and its potential for doing work on a particle at the final point has 

 been increased. This situation corresponds to moving a mass away from the earth. 

 Outside forces must do work (lifting) : the potential of the final point is greater, since 

 the earth's gravitational field will return the work in moving the particle back to the 

 earth's surface. 



The difference in potential may thus be equivalently considered as the positive of 

 the work done by outside forces against the field in moving the particle between the 

 points. Mathematically, the equivalence can be seen as follows : 



The outside force (Fo) necessary is equal but opposite to the gravitational field 



(Fo=: + (j-^), thus making one sign change from our original definition. But A {/ 



is the positive of the work done by these outside forces, thus making the second sign 

 change. The mathematical formulae remain identical. Care must be exercised to use 

 one or the other definition consistently. 



Finally it has been stated that the potential is such that F =. — ^-^ ; gravitational 



force per unit mass along r equals the negative space derivative of the potential. 



U = -G^ and ^-^=Gt^ 



F-- '^Jl = — G — 



consistent with our original definition, m denotes the mass creating the field. 



Figure 159 illustrates a short section taken 



between two equipotential surfaces over an anti- ^ 



cline, showing their convergence. -f — — ^'__ ^ 



The gravity potential on the surface AB = ;^_ ~~~--l^B 



Us, and on the surface CD below it is Ui. The [^ / "^\ 



distance separating the surfaces from A to C = ^' ~~~~'~~^'^ /^z 



hi, and between B and D = hi. The mean gravity ^^^ / 



force along AC is gi, and the mean gravity force D^\ 



along BD is gt. Such a pair of equipotential sur- ^\ 



faces might pass through the upper and the lower Fig. 159. — A short section taken between 



weights of a torsion balance. It is assumed that ^wo equipotential surfaces over an anti- 



vvx-igiiij vji o, uv^ ^ v^ L^ . . cline, showing their convergence, gi :s 



g vanes along an equipotential surface such that the mean gravity force along AC, and g2 



g, and Oo are not equal. However the variation is the mean gravity force along BD. hi 



r 1 r TT irri • r ^^'^ "'' ^""^ "'^ distances separating sur- 



of g between surfaces Uj and U2 along a given h face U2 and lower equipotential surface 



(plumb line direction) is slight and the mean Ui at p'oints A and B respectively, 

 value practically constant. 



If we move a unit mass from C to A against gi the work done = gi • hi. If we 



move a unit mass from D to B, the work done = gi • ht. The difference in potential 



between AC and BD ■=. Ut-Ui and is a constant. Then : 



gJu = gjii = At/ (34) 



From this it follows that as hi is less than hi, gi must be greater than gt. 



This is an important concept in torsion balance work and indicates that while the 

 gravity potential is equal at every point on an equipotential surface, the gravity force 

 will be different at different points. This further shows that the gravity force on an 



