If N drops of same size each having the same charge, coalesce to form a bigger drop. How will the following vary with respect to single small drop? (i) Total charge on bigger drop (ii) Potential on the bigger drop (iii) Capacitance

When N identical droplets, each carrying the same charge, merge to form a larger droplet, several physical properties change in relation to the original smaller droplets. Let’s break down how the total charge, potential, and capacitance of the larger droplet compare to those of a single small droplet.

Total Charge on the Bigger Drop

The total charge on the larger droplet is straightforward to determine. Since each small droplet has the same charge, when N droplets coalesce, the total charge on the larger droplet is simply the sum of the charges of the individual droplets.

• If the charge on a single small droplet is Q, then the total charge (Q_total) on the larger droplet becomes:
• Q_total = N × Q

This means that the total charge on the bigger drop increases linearly with the number of smaller droplets that combine.

Potential on the Bigger Drop

The electric potential (V) of a charged sphere is given by the formula:

• V = k × (Q / R)

where k is Coulomb's constant, Q is the charge, and R is the radius of the sphere. When the smaller droplets combine, the radius of the larger droplet increases. The volume of a sphere is given by:

• V = (4/3)πR³

Since the volume of the larger droplet is equal to the sum of the volumes of the smaller droplets, we can express the radius of the larger droplet in terms of the radius of the smaller droplets (r):

• R = r × N^(1/3)

Substituting this into the potential formula, we find:

• V = k × (N × Q) / (r × N^(1/3))
• V = (k × Q / r) × N^(2/3)

This indicates that the potential of the larger droplet increases with the cube root of the number of smaller droplets, specifically by a factor of N^(2/3).

Capacitance of the Bigger Drop

Capacitance (C) is defined as the ratio of charge (Q) to potential (V):

• C = Q / V

For the larger droplet, substituting the expressions we derived for Q and V gives:

• C = (N × Q) / ((k × Q / r) × N^(2/3))
• C = (N^(1/3) × r) / k

This shows that the capacitance of the larger droplet increases with the cube root of the number of smaller droplets, indicating that:

• C increases as N^(1/3)

Summary of Variations

To summarize the variations when N small droplets coalesce into a larger droplet:

• Total Charge: Increases linearly as Q_total = N × Q.
• Potential: Increases by a factor of N^(2/3).
• Capacitance: Increases by a factor of N^(1/3).

Understanding these relationships helps in grasping the principles of electrostatics and the behavior of charged bodies in different configurations. Each of these properties plays a crucial role in applications ranging from electronics to fluid dynamics.