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Electric Potential Due to a Continuous Distribution of Charges

Suppose we have volume charge density (ρ) and its position vector is r´ then to calculate the electric potential at point P due to the continuous distribution of charges, entire charge distribution is integrated. 

Where we have:
ρ = Volume charge density
dT ׳ = Small volume element
r = position vector at point P
r׳ = position vector at dT ׳
 
Uniformly charged spherical shell:
Point lying outside the spherical shell:
Here point is lying outside the uniformly charged spherical shell and the radius is greater then R, hence electric potential is

Electric field outside a uniformly charged spherical shell is equal to the electric field generated by the shell as the s hell is concentrated at the center.

Point lying inside the spherical shell:
Here point is lying inside the spherical shell and hence the radius is smaller then R, and the electric potential is


Like in the electric field intensity we have three different formulas for each type of charge distribution.

Line Charge Distribution


, where ρ is the line charge density.

Surface Charge Distribution


, where ρ is the surface charge density.

Volume Charge Distribution


, where ρ is the volume charge density.

Calculation:
Example-1: A point lying inside a hollow charged sphere has ………….. electric potential.
a)  Directly proportional to the distance from the centre of the sphere

b)  Constant
c)  Inversely proportional to the distance from the centre of the sphere
d)  None of above

Answer: A point lying inside a hollow charged sphere has constant electric potential.


Example-2: Unit of electrical potential is …………………..
a)  joule / coulomb
b)  joule/ coulomb2
c)  joule/volt
d)  volt/coulomb

Answer = joule/coulomb


Example-3: Potential energy acquired by the charged particle in a particular electric field is associated with the ……………. ……of the particle itself.
a)  Velocity
b)  Density

c)  Kinetic energy
d)  None of above

Answer: Potential energy acquired by the charged particle in a particular electric field is associated with the kinetic energy of the particle itself.

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