VDU_Physics_3_002.htm 
Solved UniversityPhysics ProblemsPhysics 3: Introduction to Electricity & Magnetism <<< Electric Charge Density >>>Javier Montenegro Joo 

<<< Electric Charge Density >>> 
(1) An spherical shell of radiuses ‘a’ and ‘b’ , ( a < b), made of plastic material has a total charge Qo. Compute the charge existing in the shell from radiuses ‘a’ to ‘r’, (a < r < b). Solution

(2) (a) Determine the total electric charge in an spherical shell of exterior radius R and width b, when its electric charge density is D(r) = D_{o} r, where D_{o} is a constant and r is the radial distance from the center of the sphere. (b) From the obtained result, calculate the total charge if the sphere were not hollowed but compact. Solution. Note that the charge density D is variable because it depends on r. Then in order to calculate the total charge in the shell it is necessary to add (integrate) differentials of charge dQ in differential shells, to do so, consider a differential shell of radius r and width dr:

The VirtualDynamics Intelligencer
Multidisciplinary Open Access Journal. Call for papers http://www.virtualdynamicssoft.com/VD_Intelligencer/VDI_Main.htm English / Spanish / Portuguese 
(3)
A cylindrical tube
of radiuses ‘a’ and ‘b’, (with a < b), is made of rubber and has a total
electric charge Qo per unit length. Determine the charge per unit
length in the cylinder as a function of its radius. Solution.

(4) Two metallic and compact objects, a ball of radius R and a cube of side L, have electric charges Q1 and Q2, respectively. Both objects are placed in contact with each other during a time t_{o} and then they are separated. Determine: (a) The initial charge density in each object (b) The charge in each object after separation. Solution. Since both objects are metallic, their electric charges Q1 and Q2 distribute on their surfaces. When the objects are placed in contact with each other the total charge Q1 + Q2 redistributes on their surfaces. Once the objects are separated the sum of the redistributed charges must be the total charge Q1 + Q2. The time t_{o }is there just to bother the students, because the process is instantaneous.
