Problem 110 - Master Resource Book in JEE Main Physics

Question

The electric potential at a point due to a given charge varies inversely as the square of the distance of the point from the charge. The statement is (a) true (b) false (c) neither true nor false (d) None of these Passage III The given figure shown an arrangement of four parallel, conducting plate of area \(A\) each. All the plates are equally separated by \(d .\) The plates \(A\) and \(D\) are joined together and a battery of emf \(E\) volt, is attached between the plates \(B\) and \(C\). \(A\)

Step-by-Step Solution

Answer

(b) false

Step 1: Understanding the Relationship

The electric potential (otential energy per unit charge) at a point due to a point charge is given by the formula where it is inversely proportional to the distance from the charge, specifically, \( V = \frac{k \, Q}{r} \), where \( V \) is the potential, \( Q \) is the charge, \( k \) is Coulomb's constant, and \( r \) is the distance from the charge. In this statement, it mentions 'inversely as the square of the distance', which would rather suggest the relationship \( V \propto \frac{1}{r^2} \), which is incorrect.

Step 2: Evaluating the Statement

Let's compare the relationship. The correct formula states that the potential is inversely proportional to \( r \), not \( r^2 \). Therefore, the statement that the electric potential varies inversely as \( r^2 \) is incorrect. It confuses with the electric field, which indeed varies as \( \frac{1}{r^2} \) according to Coulomb's Law for electric fields around point charges.

Step 3: Conclusion

Since the potential varies as \( \frac{1}{r} \) and not \( \frac{1}{r^2} \), the statement given is false. Thus, option (b) is correct.

Key Concepts

Understanding Point Charge

A point charge is a theoretical concept commonly used in physics to simplify the study of electric fields and potentials. It is considered to be a charge that is concentrated at a single point in space. This allows us to study how the electric field and electric potential behave in the space surrounding the charge. In reality, charges are often distributed over a region; however, approximating them as point charges can make calculations and predictions much simpler.
A real-world analogy might be how we consider stars in the night sky as point-like sources of light, even though they are massive objects.

• Point charges produce electric fields that radiate outward in all directions.
• The strength and direction of the electric field depend on the magnitude and sign of the charge.
• In calculations involving multiple point charges, the superposition principle is used to add the individual fields.

By utilizing the concept of a point charge, problems involving electrostatics can be approached more easily and accurately.

Exploring Coulomb's Constant

Coulomb's constant, often denoted as \( k \), is a fundamental constant in physics that characterizes the strength of the electric force between two charges. It appears in Coulomb's law, which describes the force between two point charges. Coulomb's constant is approximately equal to \( 8.9875 \times 10^9 \, \text{N m}^2 / \text{C}^2 \).
This constant plays a crucial role in the formulas calculating electric force, potential, and field.

• It allows for the calculation of forces and potential energies in the field of electrostatics.
• It is derived from the properties of the medium in which the charges reside, usually free space or a vacuum.

It becomes essential to consider Coulomb's constant while calculating interactions among charged particles. For instance, in the formula for electric potential due to a point charge, \( V = \frac{k \, Q}{r} \), the constant \( k \) determines how the potential decreases with distance. Knowing \( k \) aids in understanding how charges interact over large distances and helps in solving practical electrostatics problems.

The Inverse Square Law

The inverse square law is a fundamental principle that appears in various physics contexts, such as gravity, sound, and light, as well as in electrostatics with Coulomb's law. For electric fields, this law states that the intensity of the electric field produced by a point charge decreases with the square of the distance from the charge.
In the context of electrostatics, it means that as you move further away from a point charge, the influence of that charge lessens rapidly. This relationship is mathematically expressed as:\[E = \frac{k \, Q}{r^2}\]where \( E \) is the electric field, \( k \) is Coulomb's constant, \( Q \) is the point charge, and \( r \) is the distance from the charge.

• This principle is what differentiates electric fields from electric potential. While electric fields fall off with the square of the distance (\( 1/r^2 \)), electric potential decreases linearly with distance (\( 1/r \)).
• Understanding this law helps explain why forces change so dramatically even with small changes in separation distance.

Thus, the inverse square law is critical in grasping how electric forces and fields act in nature, revealing why precise positioning can heavily influence the electric interactions observed.