Problem 44 - Master Resource Book in JEE Main Physics

Question

The dimensions of emf in MKS is (a) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2} \mathrm{Q}^{-2}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{Q}^{-2}\right]\) (c) \(\left[\mathrm{MLT}^{-2} \mathrm{Q}^{-1}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{Q}^{-1}\right]\)

Step-by-Step Solution

Answer

The dimensions of EMF in MKS is \([ML^2T^{-2}Q^{-1}]\), which is option (d).

Step 1: Understanding the Concept of EMF

Electromotive force (EMF) is the energy provided by a source to move a charge through an electric circuit. The unit of EMF in the MKS system is the volt (V), and 1 Volt is equivalent to Joules per Coulomb (J/C). Therefore, we need to derive the dimensions of 1 Joule and 1 Coulomb first.

Step 2: Dimensions of Joule

A Joule (J) is the unit of energy or work in the MKS system. Joules are defined as the work done when a force of 1 Newton is applied over a distance of 1 meter. The dimension of energy (or work) is \([ML^2T^{-2}]\), where \(M\) is mass, \(L\) is length, and \(T\) is time.

Step 3: Dimensions of Coulomb

A Coulomb (C) is the unit of electric charge, and it is defined by the equation for current: \(I = \frac{Q}{T}\), where \(I\) is current. The unit of current is the Ampere (A), which can be expressed as Coulombs per second \(C/s\). Therefore, the dimension of charge \(Q\) is \([Q]\).

Step 4: Calculating Dimensions of EMF

Since EMF is defined in Volts, which can be expressed as Energy (Joules) per Charge (Coulombs), we need to divide the dimensions of Energy by the dimensions of Charge. EMF will have the dimensions: \([ML^2T^{-2}]/[Q]\). This can be simplified to \([ML^2T^{-2}Q^{-1}]\).

Step 5: Identifying the Correct Answer

From our calculation, the dimensions of EMF in the MKS system are \([ML^2T^{-2}Q^{-1}]\). From the given options, this matches option (d).

Key Concepts

MKS System

The MKS system, or Meter-Kilogram-Second system, is one of the key systems of measurement used in physics. This system provides the standard units for various physical quantities, making it easier to compare and compute experimental data across different settings and disciplines. In the MKS system, the unit for length is the meter (m), mass is measured in kilograms (kg), and time is measured in seconds (s).
The simplicity and universality of the MKS system have made it a fundamental part of the International System of Units (SI), which is widely adopted around the globe. It aids in unifying the ways we discuss physical concepts, such as length, area, force, and energy.

Electromotive Force

Electromotive force (EMF) is a physical quantity that reflects the potential to drive electric current through a circuit. It is not a force in the traditional sense, but rather an energy difference per charge that causes free charges to move, generating an electric current. When we talk about the EMF supplied by a battery or a generator, we refer to how much energy that source provides per charge to move it through the circuit.
The unit of electromotive force in the MKS system is the volt (V). A single volt represents one joule of energy supplied per coulomb of charge. Understanding this concept is fundamental to comprehending how electric circuits function and how various electrical devices are powered.

Unit of Energy in Physics

Energy in physics is a measure of the capacity to perform work or produce change. In the MKS system, the unit of energy is the joule (J). A joule is defined as the work done when a force of one newton is applied to move an object one meter. This provides a convenient and consistent way of quantifying energy, making calculations more standardized.
Energy can take many forms, such as kinetic, potential, thermal, electrical, or mechanical energy, depending on the context. Understanding the units of energy helps to better grasp these concepts as well as the principle of energy conservation, which is a cornerstone of physics.

MKS Units Conversion

Conversion between different units in the MKS system is essential for solving a wide variety of physics problems. This often involves converting quantities like energy, force, or time into different units without altering their inherent dimensions. For instance, converting joules to kilojoules simply involves the relationship that 1 kilojoule equals 1000 joules.
Accurate unit conversion is vital in ensuring that calculations remain correct and meaningful. It also helps bridge gaps in understanding between various measurement systems, like converting between MKS units and those of the CGS (Centimeter-Gram-Second) system, often used in more precise scientific contexts. This skill is invaluable for students tackling complex scientific problems.