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Physics 3:  Introduction to Electricity & Magnetism 

<<<  Electric Field as the Gradient of Potential  >>>

Javier Montenegro Joo


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<<<  Electric Field as the Gradient of Potential  >>>
(1) The electric potential in a space point (x,y,z), due to a small aluminum ball of radius b, located  at the origin of coordinates and which has a certain electric charge is

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


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.


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(2) The electric potential at a radial distance r from a very long straight wire with cylindrical section of radius b is given by

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

Determine the electric charge per unit length in the wire.


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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(3)The electric potential at a radial distance r from an electrically charged copper sphere of radius R is given by: 

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


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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