Problem 57 - Master Resource Book in JEE Main Physics

Question

\(\left[\mathrm{ML}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\right]\) is the dimensional formula of (a) Electric resistance (b) Capacity (c) Electric potential (d) Specific resistance

Step-by-Step Solution

Answer

The dimensional formula is for specific resistance.

Step 1: Understanding the Dimensional Formula

The given dimensional formula is \([\mathrm{ML}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}]\). This implies that the physical quantity has mass dimension \(M\), length dimension \(L^3\), time dimension \(T^{-3}\), and electric current dimension \(A^{-2}\).

Step 2: Reviewing the Dimensional Formula for Options

We need to compare this dimensional formula with the known formulas for the given options. \( \text{Electric resistance (R)} = [ML^2T^{-3}A^{-2}] \). \( \text{Capacity (C)} \) has dimensions \([M^{-1}L^{-2}T^4A^2] \). \( \text{Electric potential (V)} = [ML^2T^{-3}A^{-1}] \). \( \text{Specific resistance (\(\rho\))} = [ML^3T^{-3}A^{-2}] \).

Step 3: Matching the Dimensions

Comparing the dimensional formula for specific resistance \([ML^3T^{-3}A^{-2}]\) with the given formula \([ML^3T^{-3}A^{-2}]\), they are identical. The other options do not match the given dimensional formula.

Key Concepts

Dimensional Formula

Dimensional analysis is a critical part of physics and engineering that helps describe the nature of physical quantities. The dimensional formula provides a way to express a physical quantity in terms of basic dimensions such as mass \(M\), length \(L\), time \(T\), and electric current \(A\). These dimensions help us understand the physical characteristics of a quantity by breaking it down into these basic units.

This method is used extensively in solving physical problems, checking the correctness of equations, and converting units. For instance, consider the dimensional formula \[ \text{[ML}^3 \text{T}^{-3} \text{A}^{-2]} \]. Each symbol represents a power of a fundamental dimension, indicating how many times each base quantity is involved.

• Mass (M) is at power 1, indicating the quantity is directly related to mass.
• Length (L) is at power 3, which signifies it involves volume-related aspects (since volume has the dimension of \(L^3\)).
• Time (T) is at power \(-3\), implying a relationship with the inverse of time, often associated with rates or frequency.
• Electric Current (A) is at power \(-2\), highlighting its involvement in terms of the square of current.

Dimensional formulas are essential in identifying and distinguishing between physical quantities, helping us link theoretical concepts with real-world physical phenomena.

Physical Quantities

Physical quantities are the properties of objects that we can measure. These include things like length, mass, time, and electric current, which are called fundamental quantities because they form the base of all measurements. There are seven base physical quantities in the International System of Units (SI): mass, length, time, electric current, temperature, amount of substance, and luminous intensity.

Physical quantities are divided into two broad categories:

• Base Quantities: These are independent quantities on which others are based, like mass and length.
• Derived Quantities: These are quantities derived from the base units, such as area, volume, speed, and force.

The dimensional formula helps relate these base and derived quantities by expressing the derived quantities in terms of the base units. Recognizing the dimensions allows us to ensure compatibility between different physical quantities and verify the form of equations used in scientific calculations.

Specific Resistance

Specific resistance, also known as resistivity, is a fundamental property of materials that quantifies how strongly a material opposes the flow of electric current. It is a scalar quantity and is expressed in terms of basic dimensions using the formula \([ML^3T^{-3}A^{-2}]\).

Understanding specific resistance is crucial in electrical engineering and physics as it helps in:

• Determining the efficiency of materials used in wiring and components.
• Facilitating the design of circuits that perform optimally based on available materials.

The higher the specific resistance, the more the material resists electric current, leading to energy loss in the form of heat. The dimensional formula reflects this by combining mass, length (cubed), and inverse powers of time and current, encapsulating the intrinsic resistance qualities of various materials irrespective of their dimensions. Identifying the right materials with appropriate specific resistance ensures the efficiency and safety of electrical systems.