Problem 55 - Master Resource Book in JEE Main Physics

Question

Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\), of length \(L\), of time \(T\) and of current \(I\), would be \(\quad\) [UP SEE 2007] (a) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{I}^{-1}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) [c) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1} \mathrm{l}^{-1}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}-3 \mathrm{I}^{-2}\right]\)

Step-by-Step Solution

Answer

The correct option is (d) \([ML^2T^{-3}I^{-2}]\).

Step 1: Understand the Physical Quantity

Resistance in an electrical circuit is measured in ohms (Ω), and it relates voltage (V), current (I), and resistance (R) through Ohm's Law, which is given by the formula: \( R = \frac{V}{I} \).

Step 2: Determine Dimension Figures for Voltage and Current

Voltage (V) is determined using the relation between work (energy) done per unit charge, \( V = \frac{W}{Q} \), where \( W \) is work done in joules and \( Q \) is charge in coulombs. The dimension of work \( [W] = [ML^2T^{-2}] \). Charge \( Q \) has the dimension \( [IT] \). Thus, the dimension of voltage \([V] = \left[\frac{ML^2T^{-2}}{IT}\right] = [ML^2T^{-3}I^{-1}]\).

Step 3: Compute Dimensions of Resistance

Using Ohm's Law \( R = \frac{V}{I} \), substitute the dimensions of voltage and current: \([R] = \frac{[ML^2T^{-3}I^{-1}]}{[I]} = [ML^2T^{-3}I^{-2}]\).

Step 4: Match Calculated Dimensions with Options

Check calculated dimensions \([ML^2T^{-3}I^{-2}]\) against the given options. The option that matches is (d) \([ML^2T^{-3}I^{-2}]\).

Key Concepts

Resistance Dimensions

In the world of physics education, understanding the dimensions of physical quantities is vital. When we talk about resistance in an electrical circuit, we're referring to its fundamental dimensional formula. Resistance, represented by the symbol \( R \), is calculated by considering its relationship with voltage and current. According to Ohm's Law, resistance is the ratio of voltage \( V \) to current \( I \).

To identify the resistance dimensions, we start by determining the dimensions of voltage and current separately. As per our calculations, resistance can be expressed using the dimensions of mass \( M \), length \( L \), time \( T \), and current \( I \). The final dimensional formula for resistance is \([ML^2T^{-3}I^{-2}]\). This dimension signifies how resistance relies on energy transformation and electric charge flow, coupling mechanical and electrical theories together.

Ohm's Law

Ohm's Law is fundamental in understanding electrical circuits and plays a vital role in physics education. It is a simple yet powerful equation that establishes a relationship between voltage, current, and resistance. The law states that the current \( I \) flowing through a conductor between two points is directly proportional to the voltage \( V \) across the two points, provided the temperature remains constant.

The mathematical representation of Ohm's Law is \( V = IR \). Here, \( V \) is voltage measured in volts, \( I \) is current measured in amperes, and \( R \) is resistance measured in ohms. This relationship helps in calculating one of the quantities if the other two are known. Understanding Ohm's Law is essential for solving circuit problems and analyzing how electrical components operate under various conditions.

Voltage Dimensions

Voltage, also known as electric potential difference, is an important concept in the study of electricity. It represents the energy per unit charge and is crucial in determining how electrical energy is transferred within a circuit. Voltage has physical dimensions that relate to other fundamental units of physics.

The dimensional analysis of voltage starts with its definition as the work done per unit charge. The work done is a measure of energy and has the dimension \([ML^2T^{-2}]\). Charge is represented with the dimension \([IT]\) where \( I \) is current and \( T \) is time. When work is divided by charge, we derive the voltage dimensions as \([ML^2T^{-3}I^{-1}]\). These dimensions showcase how voltage serves as a bridge between mechanical work and electric charge distribution.

Physics Education

Physics education is about building a fundamental understanding of how the universe works, from the smallest particles to complex systems. Teaching concepts like Ohm's Law and the dimensions of electrical quantities such as resistance and voltage is integral to forming a solid base in electromagnetism.

To improve physics education, it is critical to emphasize conceptual clarity and dimensional analysis. By understanding the relationships between different physical quantities, students can develop problem-solving skills that apply to both theoretical and practical scenarios.

The use of real-world examples, interactive simulations, and experiments can further enrich the educational experience, making abstract concepts more tangible. With the right guidance, students can discover the joy of learning physics and its applications in everyday life.