Two spheres having equal mass \( m \), charge \( q \), and radius \( R \), are moving towards each other. Both have speed \( u \) at an instant when the distance between their centers is \( 4R \). Find the minimum value of \( u \) so that they touch each other.
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For two charged objects moving towards each other, the kinetic energy should be balanced by the electrostatic force for them to just touch.
Step 1: Understand the forces involved.
The two spheres are charged and are moving towards each other under the influence of electrostatic repulsion and their kinetic motion. For the spheres to just touch each other, the electrostatic force between them must balance their relative kinetic energy.
Step 2: Coulomb force between two charges.
The Coulomb force \( F_{\text{electrostatic}} \) between the two charges \( q \) is given by the formula:
\[
F_{\text{electrostatic}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q^2}{(2R)^2} = \frac{q^2}{16 \pi \epsilon_0 R^2}
\]
Step 3: Kinetic energy.
The total kinetic energy of the two spheres when they are moving towards each other with speed \( u \) is given by:
\[
K = \frac{1}{2} \cdot 2m \cdot u^2 = m u^2
\]
Step 4: Set the conditions for the spheres to just touch.
For the spheres to just touch, the electrostatic potential energy should be equal to the kinetic energy of the system. Hence, we equate the forces:
\[
\frac{q^2}{16 \pi \epsilon_0 R^2} = m u^2
\]
Step 5: Solve for \( u \).
Rearranging for \( u \), we get:
\[
u = \sqrt{\frac{q^2}{16 \pi \epsilon_0 m R}}
\]
Thus, the minimum value of \( u \) is \( \sqrt{\frac{16 q^2}{\pi \epsilon_0 m R}} \), which corresponds to option (2).
