Problem 13 - Arihant AIEEE Physics

Question

Coulomb's law is applicable to: (a) point charges (b) spherical charges (c) like charges (d) all of these

Step-by-Step Solution

Answer

Coulomb's law is applicable to all of these.

Step 1: Understanding Coulomb's Law

Coulomb's law provides a mathematical model to determine the force of interaction between two charged objects. It states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Step 2: Analyzing Point Charges

Point charges are idealized charges that have no dimensions, meaning they are not extended in space, making them suitable for direct application of Coulomb's law. The law is originally defined for such point charges.

Step 3: Considering Spherical Charges

Coulomb's law can be applied to spherical charge distributions when considering the interaction between two such spheres if they can be treated as point charges. This assumes the charge is symmetrically distributed, and for analysis, each sphere can be considered as having its charge concentrated at the center.

Step 4: Evaluating Like Charges

Coulomb’s law does not impose restrictions on whether charges are like (same sign) or unlike (opposite sign). It calculates the magnitude and direction of force, which can be attractive or repulsive depending on the nature of the charges.

Step 5: Deciding Applicability

Since Coulomb's law applies to point charges, spherical charges (under certain conditions), and does not exclude like charges, the logical conclusion is that Coulomb's law is applicable to all options provided.

Key Concepts

Point Charges

In physics, point charges are idealized as having no extent in space; in other words, they are charged objects so small that they can be considered as having all their charge concentrated at a single point. This concept of point charges helps simplify the analysis when using Coulomb's Law. Coulomb's Law is primarily concerned with the electric force between two such point charges. The formula for calculating this force is given by \[ F = k \frac{|q_1 q_2|}{r^2} \] where

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

Using point charges eliminates the complexity of dealing with the charge distribution over a finite size, allowing for cleaner calculations and easier conceptual understanding. This makes them particularly useful for foundational physics experiments and theoretical explorations.

Spherical Charges

When studying spherical charges, we often treat the charge as distributed symmetrically over the surface of a sphere. Coulomb's Law still applies, providing a neat simplification for calculating electric forces. If the charge on the sphere is uniformly distributed and the sphere can be treated as a point charge, the entire charge could be imagined as residing at the sphere's center. This assumption holds due to spherical symmetry, which is an essential requirement for applying Coulomb's Law in this context. Spherical charges relate closely to concepts in electrostatics, such as

• conducting spheres where charges reside on the surface, and
• non-conducting spheres where charges may be spread through the volume.

Coulomb's Law for spherical charges offers insights into how macroscopic charged objects interact when they possess symmetrical charge distributions. Not only does applying this law make calculations straightforward, it also aids students and physicists in relating theoretical constructs to real-world phenomena.

Electric Force

The electric force is a fundamental aspect of Coulomb's Law, manifesting as either attraction or repulsion between charged objects owing to their electric charges. The nature of this force depends on the type of charges involved:

• If both charges are of the same type (e.g., both positive or both negative), they repel each other.
• If the charges are opposite (one positive and one negative), they attract each other.

Electric force, calculated through Coulomb's Law, is a vector quantity, meaning it has both magnitude and direction. The direction follows the line connecting the two charges and its magnitude is influenced by the strength of the charges and distances between them. The force is not only important in understanding electromagnetism but also crucial in fields like chemistry and biology, where it explains interactions at the atomic and molecular levels. The principles of electric force help us understand everything from how molecules bond to each other to the operations of electronic devices.