VirtualDynamics UniversitySolved UniversityPhysics ProblemsPhysics 3: Introduction to Electricity & Magnetism <<< Electric Field as the Gradient of Potential >>>Javier Montenegro Joo jmj@VirtualDynamics.OrgReturn to Electricity and Magnetism Index page Return to VirtualDynamics University home page

<<< 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 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.

(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. 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.

Imagery 37: Digital Image Processing Virtual Lab. https://www.researchgate.net/publication/280576589_Imagery_37_Digital_Image_Processing_Virtual_Lab

(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 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:

