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Find the electric field at an arbitrary point of a sphere carrying a charge uniformly distributed over its volume. |
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Answer» Solution : 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)` 
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