Question

A solid sphere of radius R has a charge Q distributed in its volume with a charge density \[\rho = k{r^a}\], where k and a are constants and r is the distance from its center. If the electric field at \[r = \dfrac{R}{2}\]is\[\dfrac{1}{8}\] times that at r=R, the value of a is
A. 3
B. 5
C. 2
D. 7

Answer 1

Hint: Electric field in physics is defined as the property associated with an electric charge due to which it can interact with other charged particles or can influence them. It is also defined as the force applied on any unit charge in a particular area.

Formula Used:
Gauss law is used to calculate the electric field in a closed surface. According to this law, the electric flux linked with a closed surface is \[\dfrac{1}{{{ \in _0}}}\] times the charge in that closed surface. Mathematically, Gauss law can be written as
\[\int {\overrightarrow E .\overrightarrow {ds} = \dfrac{1}{{{ \in _0}}}q} \]
where, \[{ \in _0}\] is the permittivity constant and q is the charge.

Complete step by step solution:
Using equation of Gauss law written above and substituting the values given, we get
\[\int {\overrightarrow E .\overrightarrow {dA} = \dfrac{1}{{{ \in _0}}}\int {\rho dv} } \]
\[\Rightarrow \int {\overrightarrow E .\overrightarrow {dA} = \dfrac{1}{{{ \in _0}}}\int {k{r^a} \times 4\pi {r^2}dr} } \]
On integrating, the above equation can also be written as
\[{E_1} \times 4\pi {R^2} = (\dfrac{{4\pi k}}{{{ \in _0}}})\dfrac{{{R^{a + 3}}}}{{a + 3}}\]
\[\Rightarrow {E_1} = \dfrac{{k{R^{a + 1}}}}{{{ \in _0}(a + 3)}}\]

The electric field at \[r = \dfrac{R}{2}\] is \[\dfrac{1}{8}\], then we will get,
\[{E_2} = \dfrac{{k{{(\dfrac{R}{2})}^{a + 1}}}}{{{ \in _0}(a + 3)}}\]
\[\Rightarrow \dfrac{{{E_1}}}{8} = \dfrac{{k{{(\dfrac{R}{2})}^{a + 1}}}}{{a + 3}}\]
\[\Rightarrow \dfrac{{k{R^{a + 1}}}}{{8{ \in _0}(a + 3)}} = \dfrac{{k{{(\dfrac{R}{2})}^{a + 1}}}}{{{ \in _0}(a + 3)}}\]
\[\Rightarrow \dfrac{1}{8} = {(\dfrac{1}{2})^{\dfrac{1}{{a + 1}}}}\]
\[\Rightarrow \dfrac{1}{{{{(2)}^3}}} = {(\dfrac{1}{2})^{\dfrac{1}{{a + 1}}}}\]
$\Rightarrow a+1=3$
$\Rightarrow a=3-1$
$\therefore a=2$
The value of a is 2.

Therefore, Option C is the correct answer.

Note: It is important to note that Gauss law is used to find out the amount of electric field which is there due to an electric charge in any closed surface. Therefore, if the closed surface does not have any electric charge in it, the total electric flux for that surface will be zero. Gauss law is similar to Coulomb’s law, but it is to be used for closed surfaces. It can also be used to find electric fields for an isolated charge that can attract or repel other charges.