Problem 66 - Master Resource Book in JEE Main Physics

Question

Let \(\left[\varepsilon_{0}\right.\) ] denote the dimensional formula of the perimitivity of vacuum. If \(M=\) mass, \(L=\) length, \(T\) = Time and \(A=\) electric current, then [JEE Main 2013] (a) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{2} \mathrm{~A}\right]\) (b) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{4} \mathrm{~A}^{2}\right]\) (c) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{2} \mathrm{l}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{-2}\right]\) (d) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{-} \mathrm{l}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{2}\right]\)

Step-by-Step Solution

Answer

Option (b): \([M^{-1}L^{-3}T^4A^2]\).

Step 1: Understand the Problem

We need to determine the dimensional formula for the permittivity of vacuum, denoted as \( \varepsilon_0 \). This involves deriving its expression in terms of the fundamental dimensions: mass \((M)\), length \((L)\), time \((T)\), and electric current \((A)\).

Step 2: Recall the Relevant Formula

One relevant equation that involves \( \varepsilon_0 \) is given by the force equation for two point charges in a vacuum: \[ F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \] where \( F \) is force, \( q \) is charge, and \( r \) is distance.

Step 3: Analyze Dimensional Formula of Force

The dimensional formula for force \( F \) is known to be \( [MLT^{-2}] \).

Step 4: Analyze Dimensional Formula of Charge

The dimensional formula for charge \( q \) is \([AT]\), because charge is the product of current and time.

Step 5: Set up the Dimensional Equation

From the force equation, we have: \[ [F] = [\varepsilon_0]^{-1} \cdot [q]^2 \cdot [r]^{-2} \] which becomes: \[ [MLT^{-2}] = [\varepsilon_0]^{-1} \cdot [AT]^2 \cdot [L]^{-2} \]

Step 6: Solve the Dimensional Equation

Substitute the dimensions: \[ [MLT^{-2}] = [\varepsilon_0]^{-1} \cdot [A^2T^2] \cdot [L^{-2}] \] Simplifying, we get: \[ [\varepsilon_0] = [M^{-1}L^{-3}T^4A^2] \]

Step 7: Verify and Select the Correct Answer

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

Key Concepts

Vacuum Permittivity

Vacuum permittivity, denoted by the symbol \( \varepsilon_0 \), is a fundamental characteristic of the vacuum that describes how electric fields interact with the vacuum environment. It's also known as the "electric constant." Understanding vacuum permittivity is crucial in the field of electrostatics, as it appears in equations governing the behavior of electric fields and forces between charges.

• Symbol and Units: Vacuum permittivity is denoted by \( \varepsilon_0 \) and has a value of approximately \( 8.85 \times 10^{-12} \ \, \text{F/m} \).
• Role in Equations: It is integral to Coulomb's Law, which describes the force between two point charges.
• Physical Interpretation: It reflects the capacity of the vacuum to permit electric field lines, which essentially shows how much resistance the vacuum offers against the electric field formation.

This property helps define the scale at which electric forces operate in free space, differentiating it from other types of media. Understanding \( \varepsilon_0 \) not only helps in solving theoretical problems but also has practical implications in areas such as telecommunications and electronics.

Dimensional Formula

Dimensional formulas are expressions that denote the dependence of a physical quantity on the fundamental dimensions such as mass \((M)\), length \((L)\), time \((T)\), and electric current \((A)\). They are vital in analyzing and solving problems involving various physical quantities without getting entangled in the units themselves.

• Purpose: It helps in verifying the equations' dimensional consistency and aids in deriving relationships between different physical quantities.
• Structure: A dimensional formula is typically represented as \([M^aL^bT^cA^d]\), where \(a, b, c,\) and \(d\) are the respective powers denoting the involvement of each dimension.
• Application in Exercises: By deriving the dimensional formula, as done for vacuum permittivity, students can understand how to express new and unfamiliar physical quantities in terms of known dimensions.

Effective use of dimensional formulas ensures that equations are balanced and reveals potential errors in calculations by highlighting discrepancies in dimensions.

Electrostatics

Electrostatics is the study of electric charges at rest. It includes understanding the forces exerted by charges, the electric fields they create, and potentials involved.

• Core Concepts: Electrostatics involves charge interactions, electric field strength, and potential energy. The fundamental law governing electrostatics is Coulomb's Law, which calculates the force between two stationary charges in vacuum.
• Equations: One essential equation in electrostatics is \[ F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \], where \(F\) is the force, \(q\) are the values of the charges, and \(r\) is the distance between them. This illustrates how vacuum permittivity \(\varepsilon_0\) plays a critical role.
• Applications: The principles of electrostatics are vital in designing electronic components, such as capacitors and various sensing devices, as well as in understanding natural phenomena like lightning.

Electrostatics helps students understand and predict the behavior of charged particles and is foundational for more advanced topics in electromagnetism.

Fundamental Dimensions

Fundamental dimensions, such as mass \((M)\), length \((L)\), time \((T)\), and electric current \((A)\), form the building blocks of all physical measurements. In physics, these dimensions are used to describe the properties of physical quantities.

• Significance: Recognizing these dimensions helps in expressing and manipulating complex relationships in physics. They are universal, making them a common language across different scientific disciplines.
• Applications: Fundamental dimensions simplify unit conversion and allow for the dimensional analysis, a tool to check the correctness of equations and derive relations between units of different physical quantities.
• Link to Exercises: Previously derived dimensional formulas, such as that for \(\varepsilon_0\), leverage understanding the roles these dimensions play, showcasing interdependence in equations related to electrostatics, and other fields.

By mastering fundamental dimensions, students can effectively engage with physics problems, ensuring they can convey complex ideas in a standardized physical language.