Find the electric field at an arbitrary point of a sphere carrying a charge uniformly distributed over its volume.

Explanation

Let the volume density of the charge `rho=(Q)/(V)=(3Q)/(4pi R^(3))`, where R is the radius of the sphere. Choose a point M Inside the spliere al a distance `r lt R` from its centre and consider a concentric sphere through it. From the results obtained in Problem 24.13 it is evident that the spherical layer lying outside point M does not create a field at this point. The field is created solely by the charge`q=(4)/(3) pi r^(3) rho` contained inside the smaller sphere. By the result of the previous problem, the field at the surlace of this sphere, 1.e. at point M, will be
`E=(q)/(4pi epsi_(0)r^(2))=(4pi r^(3) rho)/(3xx 4pi epsi_(0)r^(2))=(rhor)/(3epsi_(0))`
We see that the field inside the sphere increases in proportion to the radius, at the centre its i tensity is zero, and on the surface it is
`E_("sur")=(rhoR)/(3epsi_(0))=(Q)/(4pi epsi_(0)R^(2))`
This dependence is plotted in Fig. 24.15. 24.16. The feld potentials on the surfaces of both spheres are `varphi=q//(4pi epsi_(0)R), varphi_(0)=q//(4pi epsi_(0), R_(1))` respectively. The potential difference is
`varphi-varphi_(1)=(q)/(4pi epsi_(0))((1)/(R)-(1)/(R_(1)))=(qR(_(1)-R))/(4pi epsi_(0)R R_(1))`
The Capacitance is `C=(q)/(varphi-varphi_(1))=(4pi epsi_(0) R R_(1))/(R_(1)-R_(2))`
If `d=R_(1)-R lr lt R`, we obtaine the approximation `C=(4pi epsi_(0)R^(2))/(d)=(epsi_(0)S)/(d)`
which is the expression for the capacitance of a plane capacitor. The error is
`delta=(C_(sp)-C_(pl))/(C_(ap))=(R_(1)-R)/(R)=(d)/(R)`