Problem 9 - Master Resource Book in JEE Main Physics

Question

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

Step-by-Step Solution

Answer

The dimensional formula is option (a): \([\mathrm{M}^{0} \mathrm{L}^{0} \mathrm{T}^{0} \mathrm{A}^{0}]\).

Step 1: Identify Dimensional Formulas

First, let's identify the dimensional formulas for each physical quantity involved in the expression \(\frac{1}{\varepsilon_{0}} \frac{e^{2}}{h c}\).- The permittivity of free space, \(\varepsilon_0\), has the dimensional formula \([\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{T}^4 \mathrm{A}^2]\).- The elementary charge, \(e\), has the dimensional formula \([\mathrm{A} \mathrm{T}][\mathrm{L}^{0} \mathrm{M}^{0}]\).- Planck's constant, \(h\), has the dimensional formula \([\mathrm{M}^1 \mathrm{L}^2 \mathrm{T}^{-1}][\mathrm{A}^{0}]\).- The speed of light, \(c\), has the dimensional formula \([\mathrm{L}^{1} \mathrm{T}^{-1}].\)"

Step 2: Calculate Dimensional Formula of Each Component

Now, calculate the dimensional formula of \( e^2 \), which is:\[[e]^2 = ([\mathrm{A} \mathrm{T}])^2 = [\mathrm{A}^2 \mathrm{T}^2]\]Next, identify the dimension formula of the expression:\[\frac{e^2}{h c} = \frac{[\mathrm{A}^2 \mathrm{T}^2]}{[\mathrm{M}^1 \mathrm{L}^2 \mathrm{T}^{-1}][\mathrm{L}^{1} \mathrm{T}^{-1}]} = [\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{T}^4 \mathrm{A}^2]\]

Step 3: Calculate Dimensional Formula of the Entire Expression

Combine results to find the dimensional formula of the entire expression \(\frac{1}{\varepsilon_{0}} \frac{e^{2}}{h c}\):\[\frac{1}{\varepsilon_0} \cdot \frac{e^2}{hc} = [\mathrm{M}^1 \mathrm{L}^{3} \mathrm{T}^{-4} \mathrm{A}^{-2}] \times [\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{T}^{4} \mathrm{A}^{-2}] = [\mathrm{M}^{0} \mathrm{L}^{0} \mathrm{T}^{0} \mathrm{A}^{0}]\]

Step 4: Choose Correct Option

From our dimensional analysis, the correct formula of the entire expression is \([\mathrm{M}^{0} \mathrm{L}^{0} \mathrm{T}^{0} \mathrm{A}^{0}]\). This corresponds to option (a).

Key Concepts

Physical Quantities

Physical quantities are the properties or characteristics of a phenomenon that can be measured or quantified. They form the foundation for which scientific concepts and equations are developed, allowing us to understand and predict natural occurrences. In physics, physical quantities are typically divided into two broad categories: base quantities and derived quantities. Base quantities include fundamental measurements such as length, mass, time, electric current, and more. Derived quantities, on the other hand, are those formed by combining base quantities, like speed (length/time) or force (mass × acceleration). Each physical quantity has dimensions that describe its physical reality in terms of these base units. This description—its dimensional formula—allows us to keep equations dimensionally consistent. When dealing with complex expressions, analyzing the dimensions of physical quantities involved can provide insights into their relationships and properties.

Permittivity of Free Space

The permittivity of free space is a critical physical constant, denoted by \( \varepsilon_0 \). It describes the ability of a vacuum to permit electric field lines and impacts how electric forces interact over a distance. This constant primarily features in equations like Coulomb's law and the equations of electromagnetism.The dimensional formula for the permittivity of free space is \([\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{T}^4 \mathrm{A}^2]\). This dimensionality indicates how permittivity relates mass, length, time, and electric current. It essentially connects how electric fields arise and propagate in a vacuum.Understanding the role of \( \varepsilon_0 \) helps students dive deeper into electromagnetism and electrical engineering fields. Recognizing its presence in calculations not only aids theoretical comprehension but also practical applications.

Planck's Constant

Planck's constant is one of the most fundamental constants in quantum mechanics. It is denoted by \(h\) and relates the energy of a photon to the frequency of its electromagnetic wave. This can be seen in the famous equation \(E = hu\), where \(E\) is energy and \(u\) is frequency.The dimensional formula for Planck's constant is \([\mathrm{M}^1 \mathrm{L}^2 \mathrm{T}^{-1}]\). This formula essentially shows it as incorporating mass, length squared, and an inverse of time, linking it directly to the quantum realm where matter and energy are interconnected.Planck’s constant signifies the discreetness of physical properties like energy. It underpins quantum theory, providing the scale at which quantum effects become significant. For students, grasping how Planck's constant integrates into both theoretical calculations and experimental physics is crucial.

Speed of Light

The speed of light in a vacuum, denoted as \(c\), is a fundamental constant in physics. It represents the maximum speed at which information or matter can travel through space. The speed of light is essential in the realms of relativity and electromagnetism.The dimensional formula is \([\mathrm{L}^{1} \mathrm{T}^{-1}]\), meaning it consists of length per unit of time. It brings clarity to determining how distances and times relate in a vacuum.With a constant value approximately equal to \(299,792,458 \) meters per second, the speed of light establishes a definitive speed limit in the universe. Understanding \(c\) allows students to bridge concepts between classical mechanics and the more complex frameworks of modern physics, such as Einstein's theories of relativity.