Question

A charge distribution with spherical symmetry has density
\[{\rho _{\text{V}}} = {\rho _0}\] when \[0 < r < R\]
\[{\rho _{\text{V}}} = 0\], when \[r > R\].
Determine \[V\] at \[r > R\]
A. \[\dfrac{{{\rho _0}{R^3}}}{{3{\varepsilon _0}r}}\]
B. \[\dfrac{{{\rho _0}{r^3}}}{{3{\varepsilon _0}R}}\]
C. \[\dfrac{{{\rho _0}{R^2}}}{{3{\varepsilon _0}}}\]
D. \[\dfrac{{{\rho _0}{r^2}}}{{3{\varepsilon _0}R}}\]

Answer 1

Hint: First of all, we will calculate the total charge on the sphere using the distribution of the charge. Then we rearrange the equation of potential by substituting the required terms followed by manipulation to obtain the result.

Complete step by step answer:
In the given problem, we are supplied the following data:
\[{\rho _{\text{V}}} = {\rho _0}\] when \[0 < r < R\]
\[{\rho _{\text{V}}} = 0\], when \[r > R\]
The electric potential at distance \[r > R\]is given by:
\[V = \dfrac{{KQ}}{r}\] …… (1)
Where,
\[{\text{Q}}\] indicates the net charge of the sphere.
\[K\] indicates a constant.
\[r\] indicates the radial distance.
Again, charge is given by the following expression which uses the charge distribution over the sphere:
\[{\text{Q}} = \dfrac{{{\rho _{\text{V}}}4}}{{3\pi {R^3}}}\]
Now, substituting the required values in the equation (1), we get:
${\text{V}} = \dfrac{{4{\rho _{\text{v}}}\pi {R^3}}}{{3r}} \times \dfrac{1}{{4\pi {\varepsilon _0}}} \\
\therefore {\text{V}} = \dfrac{{{\rho _0}{R^3}}}{{3r{\varepsilon _0}}} \\$

Hence, the required answer is \[\dfrac{{{\rho _0}{R^3}}}{{3r{\varepsilon _0}}}\], option A.

Additional Information:
Charged density: Charging density in electromagnetism is the quantity of charge per unit length, field, or volume. The volume charge density (the Greek letter \[\rho \] ) is the sum of charge per unit volume measured in coulombs per cubic meter by the SI system and at any time in volume. Surface charge density is the volume of charge per unit area measured at any time in coulombs per square meter, on a two-dimensional surface distribution. Linear charge density shall be the volume of charge, measured in coulombs per meter, per unit length at any point on a charge distribution line. The density of charge can be positive or negative as the electrical charge can be positive or negative.

Note:
We should know that electric potential is the amount of work done required to move a unit charge from a certain point to the specific point, which is done against the direction of the electric field. Higher is the strength of the electric field, higher is the potential required to do the same. It can also be said, higher is the magnitude of the charge, higher is potential required.