Question

A uniform metal sphere of radius $a$ and mass $M$ is surrounded by a thin uniform spherical shell of equal mass and radius $4a$. The centre of the shell lies on the surface of the inner sphere as shown in the figure below.Find the gravitational field intensity at point ${\text{P}}_1$ lying along the same vertical line joining the two centers.

• A. $\frac{21GM}{900a^2}$
  
• B. $\frac{61GM}{450a^2}$
• C. $\frac{61GM}{900a^2}$
• D. $\frac{61GM}{1800a^2}$

Answer 1

The correct option is C $\frac{61GM}{900a^2}$


The point is outside solid sphere at a distance $6a$ from its centre and also outside the spherical shell at a distance $5a$ from its center.

So, the net gravitational field at point ${\text{P}}_1$ will be due the sum of the fields due to solid sphere and shell.

$E_{net}=E_{sphere}+E_{shell}$

Substituting the values,

$\Rightarrow E_{net}=\frac{GM}{(6a{)}^2}+\frac{GM}{(5a{)}^2}$

$\Rightarrow E_{net}=\frac{GM}{36a^2}+\frac{GM}{25a^2}$

$\Rightarrow E_{net}=\frac{61GM}{900a^2}$

Hence, option (c) is the correct answer.

Key Concept: Field due to solid sphere: Outside: $\frac{Gm}{r^2}$ Inside: $\frac{Gm(R-r)}{r^3}$ Field due to a hollow sphere: Outside: $\frac{Gm}{r^2}$ Inside: 0 Why this question: It is absolutely important to be well versed with the field due to a solid sphere and a hollow sphere both inside and outside. This concept is extremely important from JEE standpoint.