Two identical spheres are placed in contact with each other. The force of gravitation between the spheres will be proportional to (R = radius of each sphere)

The correct option is R⁴.

Let's analyze the gravitational force between two identical spheres placed in contact step-by-step.

Step 1: Express the mass of each sphere in terms of its radius
For a sphere of radius $R$ and uniform density $\rho$, the mass $M$ is given by the formula:
$M=\text{Volume}\times \text{density}$
$M=\frac{4}{3}\pi R^3\rho$

From this equation, we can see that the mass of each sphere is proportional to the cube of its radius:
$M\propto R^3$

Step 2: Determine the distance between the centers of the two spheres
Since the two identical spheres are placed in contact with each other, the distance $d$ between their centers is equal to the sum of their radii:
$d=R+R=2R$

Step 3: Apply Newton's Law of Gravitation
According to Newton's law of gravitation, the force of attraction $F$ between two masses $M$ separated by a distance $d$ between their centers is:
$F=\frac{G{M}^2}{d^2}$

Step 4: Substitute the values of mass and distance into the force equation
Substituting $M=\frac{4}{3}\pi R^3\rho$ and $d=2R$ into the gravitational force equation gives:
$F=\frac{G{\left(\frac{4}{3}\pi R^3\rho \right)}^2}{{\left(2R\right)}^2}$

Simplifying the expression:
$F=\frac{G·\frac{16}{9}{\pi }^2{R}^6{\rho }^2}{4R^2}$
$F=\left(\frac{4}{9}G{\pi }^2{\rho }^2\right)R^4$

Conclusion
Since $G$, $\pi$, and $\rho$ are constants, the gravitational force $F$ is directly proportional to $R^4$:
$F\propto R^4$