Question

The value of distance $r_m$ at which electric field intensity is maximum, is given by :

• A. $r_m=\frac{R}{3}$
• B. $r_m=\frac{3R}{2}$
• C. $r_m=\frac{2R}{3}$
• D. $r_m=\frac{4R}{3}$

Answer 1

The correct option is C $r_m=\frac{2R}{3}$

Using Gauss's law the field at a distance r from center of the sphere is

$E.4\pi r^2=\frac{Q_{en}}{ϵ}$

here, $Q_{en}={\int }_0^r\rho \left(4\pi r^2\right)dr=4\pi {\rho }_0{\int }_0^r[1-r/R]r^{2}dr=4\pi {\rho }_0[r^3/3-r^4/4R{]}_0^r=4\pi {\rho }_0[r^3/3-r^4/4R]$

Thus, $E.4\pi r^2=\frac{4\pi {\rho }_0[r^3/3-r^4/4R]}{ϵ}$

or $E=\frac{{\rho }_0}{ϵ}[r/3-r^2/4R]$

For max E, $\frac{dE}{dr}=0=\frac{{\rho }_0}{ϵ}[1/3-2r/4R]$

or $1/3=r/2R\Rightarrow r=r_m=\frac{2R}{3}$