Question

A solid sphere of radius $r_{1}=1 \,cm$ carries charge distributed uniformly over it with densiity $\rho_{1}=-3\, C / cm ^{3}$. It is surrounded by a concentriic spherical shell of radius $r_{2}=2 \,cm$ carrying uniform charge density $\rho_{2}=\frac{1}{2} C / cm ^{2}$. If $E_{d}$ denotes the magnitude of the electric field at distance $d$ from the common centre of the spheres, then

• A. $E_{d}=\frac{1}{3 \varepsilon_{0} d^{2}}, d \leq\, 1 \,cm$
• B. $E_{d}=\frac{1}{\varepsilon_{0} d^{2}}, d \leq\,- 1\,cm$
• C. $E_{d}=\frac{d}{3 \varepsilon_{0}}, d \,\leq \,1 \,cm$
• D. $E_{d}=\frac{d}{\varepsilon_{0}} d \,\leq \,1 \,cm$

Answer 1

Answer: (D)

Electric field at a point inside inner sphere is only due to charge enclosed.
We use Gauss' law to get,
$E=\left(\frac{q}{4 \pi \varepsilon_{0} R^{3}}\right)\, r=\frac{\rho_{R}}{3 \varepsilon_{0}}$
Here, $\rho=-3\left(\frac{ C }{ cm ^{3}}\right)$
So, $E=\frac{-d}{\varepsilon_{0}}$ or $| E |=\frac{d}{\varepsilon_{0}}$