Problem 73 - Master Resource Book in JEE Main Physics

Question

A soap bubble is charged to a potential of \(16 \mathrm{~V}\). Its radius is, then doubled. The potential of the bubble now \begin{tabular}{l|l} will be & [BVP Engs- 2007] \end{tabular} [a) \(16 \mathrm{~V}\) (b) \(8 \mathrm{~V}\) (c) \(4 \mathrm{~V}\) (d) \(2 \mathrm{~V}\)

Step-by-Step Solution

Answer

The potential of the bubble is now 8 V (Option b).

Step 1: Understanding the Problem

We need to determine the change in the potential of a soap bubble when its radius is doubled while it is charged to an initial potential of 16 V.

Step 2: Formula for Potential of a Sphere

The potential (V) of a charged sphere is given by the formula: \[ V = \frac{kQ}{R} \]where Q is the charge, R is the radius, and k is the Coulomb's constant.

Step 3: Initial Conditions

Initially, the potential is 16 V, and we denote the initial radius as R. So, \[ 16 = \frac{kQ}{R}. \]

Step 4: Doubling the Radius

When the radius is doubled, the new radius R' becomes 2R. The expression for the potential becomes: \[ V' = \frac{kQ}{2R}. \]

Step 5: Expressing New Potential

Substituting the initial condition into the new potential formula: \[ V' = \frac{kQ}{2R} = \frac{1}{2} \cdot \frac{kQ}{R}. \]This simplifies to \( V' = \frac{1}{2} \times 16 = 8 \mathrm{~V}. \)

Step 6: Conclusion

After doubling the radius, the potential of the soap bubble becomes 8 V.

Key Concepts

Potential of a Charged Sphere

The potential of a charged sphere is an important concept in electrostatics. A sphere that carries an electric charge will have an electric potential depending on its charge and size. We express this potential as \( V = \frac{kQ}{R} \), where \( V \) denotes the potential, \( k \) is Coulomb's constant, \( Q \) is the charge on the sphere, and \( R \) is the radius of the sphere.

• Potential (\( V \)): Measured in volts, it reflects the potential energy per unit charge at the surface of the sphere.
• Charge (\( Q \)): This is the total electrical charge on the sphere in coulombs.
• Radius (\( R \)): The radius is the distance from the center of the sphere to its surface.

When the radius of the sphere changes, as seen in the question about the soap bubble, the potential changes as well. Specifically, if the radius is doubled and the charge remains constant, the potential is halved. This relationship directly results from the inverse relationship between potential and radius in the equation \( V = \frac{kQ}{R} \).

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics used to calculate the electric force between two point charges. It helps us understand how charged objects influence one another across a distance. The law is given by the formula:\[F = k \frac{|q_1 \cdot q_2|}{r^2}\]where:

• \( F \) is the magnitude of the force between the charges, in newtons.
• \( q_1 \) and \( q_2 \) are the amounts of the two charges, in coulombs.
• \( r \) is the distance between the centers of the two charges, in meters.
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \text{ Nm}^2/ ext{C}^2 \).

Coulomb's Law illustrates that the electric force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Even though it describes the force, understanding this principle is essential in predicting how an increase or decrease in sphere radius (affecting distance for point charges) might influence potential changes.

Electric Potential

Electric potential is a concept that represents the work needed to move a unit of electric charge from a reference point to a specific point inside the field, without any acceleration. Imagine it as the electric "height" that indicates potential energy available to charges in an electric field. The formula is:\[V = \frac{W}{Q}\]where:

• \( V \) is the electric potential, measured in volts.
• \( W \) is the work done or energy required in joules.
• \( Q \) is the charge, measured in coulombs.

In the context of a charged sphere, electric potential helps us understand how charge and energy are stored and influenced by changes in the sphere's dimensions. For instance, if a soap bubble's radius doubles, you can use this basic understanding of electric potential to predict how its potential changes, just like solving the exercise problem by substituting into the formula used for a spheres potential. This concept lays the foundation for comprehending larger, more complex arrangements of charges and fields.