Question

N identical drops of mercury are charged simultaneously to 10 volt. when combined to form one large drop, the potential is found to be 40 volt, the value of \(\mathrm{N}\) is \(\ldots \ldots\) (A) 4 (B) 6 (C) 8 (D) 10

Short answer

The value of N is approximately 6. Answer: (B) 6.

Step-by-step solution

Write the capacitance and charge equations for the small drops

Since there are N identical small drops, each with a potential of 10 volts, we can express their capacitance as: \(C_{small} = k \cdot R_{small}\) and their charge as: \(Q_{small} = N \cdot q = C_{small} \cdot V_{small} = k \cdot R_{small} \cdot 10\)

Write the capacitance and charge equations for the large drop

When the N small drops combine to form a large drop, we can express the capacitance of the large drop as: \(C_{large} = k \cdot R_{large}\) and its charge as: \(Q_{large} = q' = C_{large} \cdot V_{large} = k \cdot R_{large} \cdot 40\)

Relate the charges of the small and large drops

Since the charges of the small drops are summed up to form the charge of the large drop, we can write: \(Q_{large} = N \cdot Q_{small}\)

Substitute the charge equations and express the radii in terms of volume

Using the expressions for charges, we can write: \(k \cdot R_{large} \cdot 40 = N \cdot k \cdot R_{small} \cdot 10\) Note that the total volume of small drops is equal to the volume of the large drop: \(N \cdot \frac{4}{3} \pi R_{small}^3 = \frac{4}{3} \pi R_{large}^3\) From this, we can express the radii in terms of their volumes: \(R_{large} = \sqrt[3]{N} \cdot R_{small}\)

Substitute the radii relation and find the value of N

Substituting the radii relation into our previous equation, we get: \(k \cdot \sqrt[3]{N} \cdot R_{small} \cdot 40 = N \cdot k \cdot R_{small} \cdot 10\) Now, we can solve for N: \(4 \cdot \sqrt[3]{N} = N\) Since the problem provides options to choose from, we can test each option: - (A) N = 4: \(4 \cdot \sqrt[3]{4} = 4 \cdot 1.587 = 6.348 \neq 4\) - (B) N = 6: \(4 \cdot \sqrt[3]{6} = 4 \cdot 1.817 = 7.268 \approx 6 \) Therefore, the value of N is approximately: Answer: (B) 6

Key concepts

Capacitance

Capacitance is a crucial property of a capacitor, reflecting its ability to store charge. For a simple capacitor, capacitance is defined as the ratio of the charge stored on one plate of the capacitor to the potential difference between the plates. The formula is given by:

• \[ C = \frac{Q}{V} \]

where \( C \) is the capacitance, \( Q \) is the charge, and \( V \) is the potential difference.
This helps us understand how much charge a capacitor can store for a given potential difference.
The unit of capacitance is the farad (F), and it often appears in microfarads (\( \mu \text{F} \)) or picofarads (\( \text{pF} \)).
In electrostatics exercises involving charged drops merging to form a larger drop, understanding the change in capacitance is essential for calculating new potentials and charges.
The capacitance of a spherical object like a drop relates directly to its radius by:

• \[ C = k \cdot R \]

where \( R \) is the sphere's radius, and \( k \) is a proportionality constant.
Thus, when smaller charged bodies merge into a larger one, capacitance changes depending on the new radius.

Charge relation

Charge relation in the context of electrostatics focuses on how charge is conserved and distributed between objects. When analyzing problems like merging charged drops, keeping the total charge constant is key.
For multiple charged bodies combining into a single body, the charge conservation principle states that the total charge before and after the combination remains the same:

• \[ Q_{\text{total, initial}} = Q_{\text{total, final}} \]

Thus, if we have \( N \) identical charged drops each with charge \( Q_{small} \), their combination into a single large drop results in:

• \[ Q_{large} = N \cdot Q_{small} \]

This principle ensures that the charge accounted in multiple bodies is equal to the single larger body they form.
This conservation allows us to relate changes in potential, capacitance, and geometry upon these physical transformations.

Potential difference

Potential difference, often simply called voltage, indicates the work done per unit charge to move a charge between two points. It's a central concept in electrostatics, portraying the readiness to drive charge from one place to another. The basic formula connecting potential difference with charge and capacitance is:

• \[ V = \frac{Q}{C} \]

where \( V \) refers to the potential difference, \( Q \) is the charge, and \( C \) is the capacitance.
When small charged drops are considered, each has its own potential. If we combine \( N \) identical drops (each at the same potential) to form a larger drop, the potential of this larger drop is calculated by taking into account changes in geometry and capacitance:

• The relation of radii: \( R_{large} = \sqrt[3]{N} \cdot R_{small} \)
• The new potential: \( V_{large} = 40 \text{ volts} \)

In this scenario, understanding the transition of potential from individual items to a collective unit allows us to deduce the new overall potential and the necessary adjustments in charge magnitude.
This showcases how electrostatic potential difference adapts with changes in an object's configuration.