If the electric field is known everywhere, Gauss's law makes it quite easy, in principle, to find the distribution of electric charge: The charge in any given region can be deduced by integrating the electric field to find the flux.However, much more often, it is the reverse problem that needs to be solved: The electric charge distribution is known, and the electric field needs to be computed. This is much more difficult, since if you know the total flux through a given surface, that gives almost no information about the electric field, which (for all you know) could go in and out of the surface in arbitrarily complicated patterns.An exception is if there is some symmetry in the situation, which mandates that the electric field passes through the surface in a uniform way.
For a sphere of charge, the electric field outside the sphere, with
a total charge Q and uniform charge density rho, is the same as the field due to a point charge Q at the center
of the sphere. In other words, to an observer outside the sphere of uniform charge density (or a charge density
that depends only on r, the distance from the center) the sphere appears to be a point charge at the center.
For a sphere of charge, with uniform charge density rho, the
electric field inside the sphere, at a distance r from the center, is (rho) * r / (3*epsilon_0).
Gausss law may be used to find the electric field inside a
spherical cavity with a sphere of charge. It may also be used to find the E-field both inside and outside a
sphere of charge with a charge density that varies with r, the distance from the center.
Application of Gauss's Law, Part 1
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