A sphere with radius R=10 cm has the potential V=10KV,in the air. Which is the load superficial density, on the sphere?

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By Gauss' Theorem, the entire charge is distributed over the surface of the sphere.

The curvature of the sphere is unoform over the entire surface.So, the load is also spread uniformly over the entire spherical surface, 4pi*r^2, where r is the radius of the sphere.

The density ,E of the charge is the charge per unit area and is given by:

E =( q/(4pi*r^2))/e_0 (1) and

q= 4pi*e_0*r*V , e_0 being permittivity of air and is nearly 1, (2).

From one and (2):

E= V/r =

Given r= 10cm = 0.01 meter and

V=10KV= 10000Volts = 10^4 volts. Substitute in (1) to get:

Superficial density of load, E = 10^4/(0.01) =10^6 volt/m

The sphere's potential is

V=Q/(4*pi*epsilon*R), so Q=4*pi*epsilon*R*V

(epsilon is electrical allowance of the environment)

The superficial load density is:

sigma=Q/S=Q/(4*pi*R^2)=(4*pi*epsilon*R*V)/(4*pi*R^2)

sigma=epsilon*V/R=0.88microC/m^2

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