Problem 12 - Master Resource Book in JEE Main Physics

Question

The dimensional formula of coefficient of permittivity for free space \(\left(\varepsilon_{0}\right)\) is (a) \(\left[\mathrm{ML}^{3} \mathrm{~A}^{-2} \mathrm{~T}^{-4}\right]\) (b) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{4} \mathrm{~A}^{2}\right]\) (c) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~A}^{-2} \mathrm{~T}^{-4}\right]\) (d) \(\left[M L^{3} A^{2} T^{-4}\right]\)

Step-by-Step Solution

Answer

The dimensional formula of \(\varepsilon_{0}\) is \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{4} \mathrm{~A}^{2}\right]\), matching option (b).

Step 1: Understand the Given Problem

We need to find the dimensional formula for the coefficient of permittivity of free space, denoted as \( \varepsilon_0 \). This will involve understanding the related physical laws and dimensions involved.

Step 2: Recall the Relevant Physical Formula

The coefficient of permittivity \( \varepsilon_0 \) appears in the formula for Coulomb's law of electrostatics for force between two point charges: \[ F = \dfrac{1}{4\pi\varepsilon_0} \dfrac{q_1q_2}{r^2}\]In this formula, \( F \) is the force, \( q_1 \) and \( q_2 \) are charges with the dimension \([\mathrm{I}\mathrm{T}]\), and \( r \) is the distance between charges.

Step 3: Find the Dimensional Formula for Force

Using Newton's law, the dimensional formula for force \( F \) is: \[ F = [\mathrm{MLT}^{-2}]\]

Step 4: Determine the Units of \( q_1q_2 \) and \( r^2 \)

The charge \( q \) has the dimension: \[ [\mathrm{I}\mathrm{T}] \]Thus, the product \( q_1 q_2 \) has the dimension: \[ [\mathrm{I}^2 \mathrm{T}^2] \]Distance \( r \) has the dimension: \[ [\mathrm{L}] \]Thus, \( r^2 \) has the dimension: \[ [\mathrm{L}^2] \]

Step 5: Compute the Dimensional Formula for \( \varepsilon_0 \)

Rewriting Coulomb's law:\[ \varepsilon_0 = \dfrac{q_1 q_2}{4\pi r^2 F} \]Substitute dimensions:\[ [\varepsilon_0] = \dfrac{[\mathrm{I}^2 \mathrm{T}^2]}{[\mathrm{L}^2][\mathrm{MLT}^{-2}]} \]Simplification results in:\[ [\varepsilon_0] = [\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{T}^{4} \mathrm{A}^{2}] \]

Step 6: Match the Result with the Given Options

Compare our derived dimensional formula \([\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{T}^{4} \mathrm{A}^{2}]\) with the options. It matches option (b).

Key Concepts

Coefficient of Permittivity

The coefficient of permittivity is a fundamental concept in electromagnetism. It quantifies how much electric field (or force) is allowed to permeate a certain medium. Particularly, in a vacuum or free space, it's denoted by the symbol \( \varepsilon_0 \). This is also known as the permittivity of free space.
The coefficient plays a crucial role in determining the strength of electric forces between charges. It appears prominently in Coulomb's Law as a factor governing interactions in free space. Understanding this coefficient is key to solving many problems in electrostatics and understanding the behavior of electric fields in various media.

Coulomb's Law

Coulomb's Law is a pivotal theory in electrostatics that explains how forces act between electrically charged particles. Named after the French physicist Charles-Augustin de Coulomb, this law forms the foundation of understanding interactions in electric fields.
The law is mathematically represented as: \[ F = \frac{1}{4\pi\varepsilon_0} \frac{q_1q_2}{r^2} \] Here, \(F\) denotes the force between charges \(q_1\) and \(q_2\), which are separated by a distance \(r\). Within this formula, \(\varepsilon_0\) represents the permittivity of free space, acting as a moderating factor for the interaction strength.
Coulomb's Law helps us understand how charged particles attract or repel each other, directly influencing real-world phenomena like lightning, the operation of electronic devices, and even atomic structure.

Dimensional Formula Calculation

Dimensional analysis is a powerful tool to verify equations or deduce new ones. It's the method used to find the dimensional formula for physical quantities, allowing us to understand how different units relate.
To find the dimensional formula of the coefficient of permittivity (\(\varepsilon_0\)), one must consider the dimensions of each element in Coulomb's Law.
- Start with the force \( F \), which has dimensions \([MLT^{-2}]\). - The charge \(q\) has dimensions \([IT]\), leading to \(q_1q_2\) being \([I^2T^2]\). - The distance \(r\) is simply \([L]\), making \(r^2\) equal to \([L^2]\).
Using the relationship \(\varepsilon_0 = \frac{q_1q_2}{4\pi r^2 F}\), we substitute these dimensions, resulting in \([M^{-1}L^{-3}T^4A^2]\). This is the dimensional formula for \(\varepsilon_0\), helping verify the consistency of physics laws.

Electrostatics

Electrostatics is the branch of physics that studies stationary electric charges. Unlike moving charges that generate magnetic fields, electrostatics focuses on the forces and fields resulting from charges at rest. This study provides the basis for understanding electric forces in various materials.
Main concepts in electrostatics include:

• Electric Charge: The intrinsic property of matter responsible for electric forces; can be positive or negative.
• Electric Field: A field around a charged particle where forces are exerted on other charges.
• Electric Potential: The potential energy per unit charge at a point in a field.

Electrostatics plays critical roles in numerous technologies, from capacitors storing energy in circuits to understanding forces at molecular levels in chemistry and biology. It helps in designing circuits, predicting molecule behaviors, and even in new medical technologies.