Surface Charge Density

Step 1: Understanding the concept of surface charge density.
Surface charge density ( ) is the charge per unit surface area. It is
given by:
where is the charge and is the surface area of the droplet.
Step 2: Total charge conservation.
Since all 27 droplets have the same charge, and they coalesce to form a single larger droplet, the total charge remains the same.
Step 3: Relating the radius of the droplets.
The total volume is conserved in the process. The volume of a sphere is
given by:
For 27 small droplets of radius , the total volume of the small droplets is: For the large droplet, the radius is , and the volume is: Since the total volume is conserved, we have: This gives:
Step 4: Finding the ratio of surface charge densities.
The surface area of a sphere is
given by:
Thus, the surface area of the small droplet is and the surface area of the large droplet is . Now, the surface charge densities are: Thus, the ratio of surface charge densities is: Therefore, the ratio is , not . Thus, the correct answer is
(C) 3 : 1.