Determine:  (a) The electric field at the point (x,y,z)   (b) The total electric charge in the ball

Solution.-

Calculating the gradient in Cartesian coordinates is a little bit laborious, because it includes three derivatives, in this particular case it is easier to express (x,y,z) as r.

                        V(r)= -Ao ln( r/ b),           Ao: Constant

Determine the electric charge per unit length in the wire.

Solution.-

The electric field is calculated as the negative gradient of the electric potential. In order to calculate the electric charge in the wire, the Gauss theorem must be applied to the lateral surface of a cylinder of radius r around the wire. Since the wire is very long the electric field at the extreme basis of the cylinder are not considered.

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Determine the charge in the sphere: (a) surface    (b) interior     (c) Total

Solution

Since in a metal body all the charge is on its surface, it doesn’t matter whether the sphere is compact or a shell, all the charge distributes necessarily on its surface. This means that there must be a surface density of charge. The electric field at a radial distance r may be obtained as the negative gradient of the potential:

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