Question
Charges $$Q, 2Q$$ and $$4Q$$ are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of radii $$\frac{R}{2},$$ $$R$$ and $$2R$$ respectively, as shown in figure. If magnitude of the electric fields at point $$P$$ at a distance $$R$$ from the centre of sphere 1, 2 and 3 are $${E_1},{E_2}$$ and $${E_3}$$ respectively, the
Charges $$Q, 2Q$$ and $$4Q$$ are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of radii $$\frac{R}{2},$$ $$R$$ and $$2R$$ respectively, as shown in figure. If magnitude of the electric fields at point $$P$$ at a distance $$R$$ from the centre of sphere 1, 2 and 3 are $${E_1},{E_2}$$ and $${E_3}$$ respectively, the
A.
$${E_1} > {E_2} > {E_3}$$
B.
$${E_3} > {E_1} > {E_2}$$
C.
$${E_2} > {E_1} > {E_3}$$
D.
$${E_3} > {E_2} > {E_1}$$
Answer :
$${E_2} > {E_1} > {E_3}$$
Solution :
$$\eqalign{ & {E_1} = \frac{1}{{4\pi { \in _0}}} \cdot \frac{Q}{{{R^2}}}; \cr & {E_2} = \frac{1}{{4\pi { \in _0}}} \cdot \frac{{2Q}}{{{R^2}}};{E_3} = \frac{1}{{4\pi { \in _0}}} \cdot \frac{{\frac{Q}{2}}}{{{R^2}}} \cr} $$
Clearly $${E_2} > {E_1} > {E_3}$$
where $${\frac{Q}{2}}$$ is the charge enclosed in a sphere of radius $$R$$ concentric with the given sphere.
$$\left[ {\frac{{4Q}}{{\frac{4}{3}\pi {{\left( {2R} \right)}^3}}} = \frac{{Q'}}{{\frac{4}{3}\pi {R^3}}}} \right]$$
$$\eqalign{ & {E_1} = \frac{1}{{4\pi { \in _0}}} \cdot \frac{Q}{{{R^2}}}; \cr & {E_2} = \frac{1}{{4\pi { \in _0}}} \cdot \frac{{2Q}}{{{R^2}}};{E_3} = \frac{1}{{4\pi { \in _0}}} \cdot \frac{{\frac{Q}{2}}}{{{R^2}}} \cr} $$
Clearly $${E_2} > {E_1} > {E_3}$$
where $${\frac{Q}{2}}$$ is the charge enclosed in a sphere of radius $$R$$ concentric with the given sphere.
$$\left[ {\frac{{4Q}}{{\frac{4}{3}\pi {{\left( {2R} \right)}^3}}} = \frac{{Q'}}{{\frac{4}{3}\pi {R^3}}}} \right]$$
