Problem 53 - Master Resource Book in JEE Main Physics

Question

Electric displacement is given by \(D=\varepsilon E\), Here, \(\varepsilon=\) electric permittivity \(E=\) electric field strength The dimensions of electric displacement are (a) \(\left[\mathrm{ML}^{-2} \mathrm{TA}\right]\) (b) \(\left[\mathrm{L}^{-2 \mathrm{~T}}{\underline{\phantom{xx}}}^{-1} \mathrm{~A}\right]\) (c) \(\left[\mathrm{L}^{-2} \mathrm{TA}\right]\) (d) None of these

Step-by-Step Solution

Answer

The dimensions of electric displacement are \([\mathrm{L}^{-2}\mathrm{TA}]\), option (c).

Step 1: Identify dimensions of given quantities

First, identify the dimensions of electric permittivity (\(\varepsilon\)) and electric field strength (\(E\)). \(\varepsilon\) has the dimensions of charge squared per force times length, which can be expressed as \([\mathrm{M}^{-1}\mathrm{L}^{-3}\mathrm{T}^4\mathrm{A}^2]\). The electric field strength \(E\) has dimensions of force per charge, which can be represented as \([\mathrm{M}\mathrm{L}\mathrm{T}^{-3}\mathrm{A}^{-1}]\).

Step 2: Calculate dimensions of D

Use the relation \(D = \varepsilon E\) to find the dimensions of electric displacement. Multiply the dimensions of \(\varepsilon\) and \(E\):\[[\varepsilon] \times [E] = [\mathrm{M}^{-1}\mathrm{L}^{-3}\mathrm{T}^4\mathrm{A}^2] \times [\mathrm{M}\mathrm{L}\mathrm{T}^{-3}\mathrm{A}^{-1}] = [\mathrm{L}^{-2}\mathrm{TA}]\]

Step 3: Match with given options

Compare the calculated dimensions \([\mathrm{L}^{-2}\mathrm{TA}]\) with the options provided. The dimensions match option (c).\(\left[\mathrm{L}^{-2}\mathrm{TA}\right]\). Thus, option (c) is the correct answer.

Key Concepts

Electric Permittivity

Electric permittivity, often symbolized as \(\varepsilon\), is a fundamental property of materials that affects how they respond to electric fields. It essentially measures a material's ability to permit the passage of the electric field through it. This property is crucial in defining how much resistance a given medium offers when an electric field is applied. The electric permittivity is used in the relation \(D = \varepsilon E\), where \(D\) is the electric displacement and \(E\) is the electric field strength.

• In vacuum, the permittivity is denoted as \(\varepsilon_0\), also known as the permittivity of free space.
• This constant has a value of approximately \(8.85 \times 10^{-12}\) F/m (Farads per meter).
• In other materials, the permittivity will be relative to this value and is denoted \(\varepsilon_r\).

Knowing the electric permittivity helps in calculating how electric fields interact with different materials, which is essential for designing capacitors and understanding the behavior of insulators and dielectrics.

Electric Field Strength

Electric field strength, represented by \(E\), is a crucial concept in electromagnetism. It describes the force per unit charge experienced by a small positive test charge placed in the field. This means that the electric field strength indicates how strong the electric force will be on a charge within that field. The strength depends on both the source charge that creates the field and the distance from the source charge.

• Mathematically, it is defined as \(E = F/Q\), where \(F\) is the force experienced by the charge \(Q\).
• The unit of electric field strength is volts per meter (V/m) or newtons per coulomb (N/C).
• The dimensional formula of electric field strength is \([\mathrm{M}\mathrm{L}\mathrm{T}^{-3}\mathrm{A}^{-1}]\).

Understanding electric field strength is essential in the study of electric forces, capacitors, and overall electric laws governing forces and interactions between charges.

Dimensional Analysis

Dimensional analysis is a mathematical technique used to convert units from one system to another and to check the consistency of equations by analyzing the dimensional units. It involves breaking down each unit into its basic dimensions such as mass (M), length (L), time (T), and electric current (A), to ensure that physical quantities have consistent units across an equation.

• It helps in verifying the correctness of physical relations and equations involving physical quantities.
• In the context of electric displacement \(D\), dimensional analysis was used to determine its dimensions by multiplying the dimensions of electric permittivity \(\varepsilon\) and electric field strength \(E\).
• It is a powerful tool across various scientific and engineering disciplines as it confirms that equations are dimensionally homogeneous.

This technique ensures that the equations used in physics are reliable and can be universally applied regardless of the units in question.