42 Electrostatics Field of Force [OH. n 



This is known as Poisson's Equation; clearly if we know the value of the 

 potential at every point, it enables us to find the charges by which this 

 potential is produced. 



50. In free space, where there are no electric charges, the equation 

 assumes the form 



and this is known as Laplace's Equation. We shall denote the operator 



.* . 



8^ dy* a* 2 



by V 2 , so that Laplace's equation may be written in the abbreviated form 



V 2 F=0 .................................... (20). 



Equations (18) and (20) express the same fact as Gauss' Theorem, but 

 express it in the form of a differential equation. Equation (20) shews that 

 in a region in which no charges exist, the potential satisfies a differential 

 equation which is independent of the charges outside this region by which 

 the potential is produced. It will easily be verified by direct differentiation 

 that the value of V given in equation (10) is a solution of equation (20). 



We can obtain an idea of the physical meaning of this differential 

 equation as follows. 



Let us take any point and construct a sphere of radius r about this 

 point. The mean value of F averaged over the surface of the sphere is 



terr*)) 



VdS 



~ 



where r, 9, < are polar coordinates, having as origin. If we change the 

 radius of this sphere from r to r + dr, the rate of change of F is 



8F 1 /Y8F . 



- smOdOdd) 



dr 47T./J dr 



*8F , 



= 0, by Gauss' Theorem, 



shewing that JF is independent of the radius r of the sphere. Taking r 0, 

 the value of F is seen to be equal to the potential at the origin 0. 

 This gives the following interpretation of the differential equation : 



F varies from point to point in such a way that the average value of F 

 taken over any sphere surrounding any point is equal to the value of V at 0. 



