Question

Two identical spheres of radius R made of the same material are kept at a distance d apart. Then the gravitational attraction between them is proportional to:
$$A. d^{-2}$$
$$B. d^2$$
$$C. d^4$$
$$D. d$$
$$E. d^{-4}$$

Answer 1

Hint: Here, we need to know about Newton's gravitational law.
It is the law of attraction between two masses. From this law, we can know the relationship between attraction force and distance between two masses.

Formula used: $$F=G\dfrac{m_1m_2}{d^2}$$

Complete step-by-step answer:
Newton’s law of gravitation states that the attraction between two objects is proportional to the product of their masses and inversely proportional to the square of their distance. Again this force acts along the line joining the two objects.

Actually this law was proposed for point masses, where it was easy to find the distance between two objects. The mathematical formula of the law is,
$$F=G\dfrac{m_1m_2}{d^2}$$
Where G is the universal gravitational constant, F is the gravitational force, d is the distance between the objects and $$m_1$$ , $$m_2$$ are the masses of the two objects.
In case of symmetrical objects, it is easy to find their distance. Luckily, here we have two identical spheres. Hence the distance can be measured from the centers of two spheres. However the distance is given to be d. So, the attraction force is proportional to $$\dfrac{1}{d^2}=d^{-2}$$
Hence, option A is the correct answer.

Additional information:
The value of the universal gravitational constant is $$G=6.674×10^{-11} Nm^2/kg^2$$ (in SI units).

Note: Apply Newton’s gravitational law when the objects are not charged. In case the objects are charged you have to use Coulomb’s law, which is almost similar.
$$F=\dfrac{1}{4π\epsilon_0}.\dfrac{q_1q_2}{d^2}$$
The q's are the charges. And $$\epsilon_0$$ is called the free space permittivity.