Problem 54 - Master Resource Book in JEE Main Physics

Question

The work done by a battery is \(W=\varepsilon \Delta q\), where \(\Delta q\) charge transferred by battery, \(\varepsilon=\) emf of the battery. What are dimensions of emf of battery? (a) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{-2} \mathrm{~A}^{-2}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\right]\) (c) \(\left[\mathrm{M}^{2} \mathrm{~L}^{0} \mathrm{~T}^{-3} \mathrm{~A}^{0}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\right]\)

Step-by-Step Solution

Answer

(d) \([ML^{2} T^{-3} A^{-1}]\)

Step 1: Understanding EMF in terms of Work and Charge

In the formula \( W = \varepsilon \Delta q \), \( W \) is the work done, \( \varepsilon \) is the electromotive force (EMF) of the battery, and \( \Delta q \) is the charge transferred. We need to find the dimension of \( \varepsilon \) in terms of fundamental dimensions.

Step 2: Determine Dimensions for Work (W)

Work, \( W \), is expressed as force times distance. The dimension of force is \([M L T^{-2}]\), so the dimension of work, which is energy, will be \([M L^{2} T^{-2}]\).

Step 3: Determine Dimensions for Charge (Δq)

Electric charge, \( \Delta q \), has the dimension \([A T]\), where \( A \) stands for ampere, the unit of current, and \( T \) is time.

Step 4: Find Dimensions for EMF (ε)

Using the expression \( W = \varepsilon \Delta q \), we rearrange to find \( \varepsilon = \frac{W}{\Delta q} \). Therefore, the dimension of \( \varepsilon \) is \( \frac{[M L^{2} T^{-2}]}{[A T]} \).

Step 5: Calculate the Dimensions of EMF

Simplifying \( \frac{[M L^{2} T^{-2}]}{[A T]} \) results in \([M L^{2} T^{-3} A^{-1}]\). This indicates that the dimension of \( \varepsilon \) is \( \left[ML^{2} T^{-3} A^{-1}\right] \).

Key Concepts

Electromotive Force (EMF)

Electromotive Force, often abbreviated as EMF, is a fundamental concept in physics, particularly in the fields of electricity and magnetism. It represents the energy provided by a battery or a cell per unit charge. This might sound complicated, but it is basically the voltage across the terminals of a battery when it is not supplying any current.
\[ \text{EMF} = \varepsilon = \frac{W}{\Delta q} \]
Here, \( W \) is the work done or energy supplied, and \( \Delta q \) is the amount of charge. EMF is usually measured in volts, which encompasses dimensions of several fundamental quantities, making it essential to understand its dimensional analysis to comprehend its nature fully.

• EMF is often a source of electrical energy, providing potential energy that pushes electrons through a circuit.
• It is not dependent on the resistance of the circuit, unlike the terminal voltage which varies with the current.
• Recognizing its role in circuits helps in understanding the flow and supply of electrical energy.

Work and Energy Dimensions

Understanding the dimensions of work and energy is crucial, as these concepts are frequently used in physics. Work and energy have the same dimensions because they are related physical quantities. Work is done when a force moves an object over a distance, and energy is the capacity to do work.
In terms of dimensions, force is measured as mass times acceleration. Acceleration has the dimensions of distance over time squared. Thus, the dimensions of force can be written as \([M L T^{-2}]\).
When work is done (force applied over a distance), its dimensions are:
\[ [M L T^{-2}] \times [L] = [M L^2 T^{-2}] \]
This indicates that the dimensions of work, and therefore energy, are \([M L^2 T^{-2}]\).

• This understanding is helpful when calculating energy transformations, evaluating system performances, and resolving physical problems involving forces and motions.
• It provides a solid foundation when converting between different forms of energy, such as potential and kinetic energy, in physics problems.

Charge and Current Dimensions

The concepts of charge and current are fundamental in understanding electromagnetism and electrical circuits. Despite being so common, students often overlook their dimensional aspects. Let's break it down simply.
Electric charge, often denoted by \( \Delta q \), has the dimensions \([A T]\). Here, \( A \) represents amperes, the unit of electric current, and \( T \) stands for time.
Current (\( I \)) is essentially the rate of flow of electric charge. Its dimensional formula can be represented as:
\[ I = \frac{\Delta q}{T} \quad \Rightarrow \quad [A] = \frac{[A T]}{[T]} = [A] \]

• Understanding these dimensions helps us comprehend how circuits function and partake in energy exchanges.
• It's vital for designing electrical systems where accuracy in measurements of current and charge impacts the system's performance and safety.

Recognizing these dimensions ensures a deeper understanding of how charges move and interact in an electrical field and aids in solving more complex circuit problems.