Problem 28 - Master Resource Book in JEE Main Physics

Question

A charged spherical conductor of radius \(R\) carries a charge \(q_{0} .\) A point test charge \(q_{0}\) is placed at a distance \(x\) from the surface of the conductor. The force experienced by the test charge will be proportional to (a) \((R+x)^{2}\) (b) \((R-x)^{2}\) (c) \(\frac{1}{(R-x)^{2}}\) (d) \(\frac{1}{(R+x)^{2}}\)

Step-by-Step Solution

Answer

The force is proportional to \( \frac{1}{(R+x)^2} \), so the answer is (d).

Step 1: Understanding the Setup

We have a spherical conductor of radius \( R \) with a charge \( q_0 \) on it. A test charge \( q_0 \), same as the conductor, is placed outside the surface of the sphere at a distance \( x \). The total distance from the center of the sphere to the test charge is \( R+x \).

Step 2: Applying Coulomb's Law

According to Coulomb's Law, the force \( F \) between two point charges \( q_1 \) and \( q_2 \), separated by a distance \( r \), is determined by the formula:\[F = \frac{k \cdot |q_1 q_2|}{r^2}\]For this problem, \( q_1 = q_2 = q_0 \) and \( r = R + x \).

Step 3: Substitute Values into Coulomb's Law

Plug the known quantities into the formula:\[F = \frac{k \cdot q_0^2}{(R+x)^2}\]This shows the force is proportional to \( \frac{1}{(R+x)^2} \).

Step 4: Determine the Correct Option

The expression \( \frac{1}{(R+x)^2} \) matches with option (d), which is the answer.

Key Concepts

Coulomb's Law

Coulomb's Law is pivotal in understanding how electric charges interact. It provides a mathematical relationship that describes the magnitude of the electric force between two point charges. This force depends on:

• The magnitude of each charge.
• The distance separating the charges.

Coulomb's Law is expressed by the formula:\[ F = \frac{k \cdot |q_1 q_2|}{r^2} \]where \( F \) is the magnitude of the force, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them. The force is attractive if the charges are opposite and repulsive if the charges are like. Importantly, this formula follows an inverse square law, meaning that the force decreases with the square of the distance between the charges.

Electric Force

Electric force is the result of the interaction between charges. It is a non-contact force, meaning charges do not have to physically touch to exert force on one another. Several characteristics define electric force:

• It can be either attractive or repulsive, depending on the nature of the interacting charges.
• It is a vector quantity, meaning it has both magnitude and direction.

In electrostatics, the electric force helps us understand how charged objects influence each other at rest. This force's magnitude can be calculated using Coulomb's Law, which directly ties the force to the involved charges and their separation distance. In the context of the exercise, the force on a test charge by a spherical charged conductor can be calculated if its distance from the conductor and the charge amounts are known.

Spherical Conductor

A spherical conductor uniformly distributes its charge over its surface. This is due to charge repulsion; charges move to a position where they are furthest apart, which is the surface. For a charged spherical conductor, we can assume:

• The electric field within the conductor is zero since charges reside only on the surface.
• Outside the conductor, it can be treated as if all charge is concentrated at its center, facilitating easier calculations using Coulomb's Law.

This behavior makes it similar to how a point charge operates when observed from outside. Hence, the spherical conductor simplifies the computation of electric forces on external charges. When considering the force on a test charge placed outside, the conductor can be treated as if the total charge resides at its center.

Test Charge

A test charge is a theoretical charge used to measure electric fields and forces in a given electric environment without altering them significantly. It is usually a small positive charge. Its primary purpose is to probe the effects of an electric field produced by other arrangements of charges.

• It is considered to be small enough not to disturb the field it measures.
• Its interaction with the electric field is valuable for calculating the resultant electric forces using Coulomb's Law.

In the context of the exercise, the test charge is placed at a distance from the spherical conductor. This setup allows us to apply Coulomb's Law to determine how the force on the test charge varies with distance, demonstrating its dependency on the inverse square law.

Inverse Square Law

The inverse square law is a principle seen often in physics, describing how a physical quantity decreases with increasing distance. It states that the quantity or effect is inversely proportional to the square of the distance from the source:

• Doubling the distance results in reducing the force by a factor of four.
• Tripling the distance reduces the force to one ninth.

This principle is a cornerstone in understanding electric forces via Coulomb's Law, gravitational forces, and even the intensity of light. When applying this concept to electrostatics, it's clear that as charges move further apart, the electric force they exert on each other decreases rapidly. This relationship is evident in our exercise, where the force on a test charge due to a spherical conductor follows this inverse square decline.