Question

The relation between \([E]\) and \([B]\) is (A) \([E]=[B][L][T]\) (B) \([E]=[B][L]^{-1}[T]\) (C) \([E]=[B][L][T]^{-1}\) (D) \([E]=[B][L]^{-1}[T]^{-1}\)

Short answer

The correct relation between [E] and [B] is (C) [E]=[B][L][T]^{-1}.

Step-by-step solution

Identify the Physical Quantities

Identify the physical quantities represented by the symbols [E] and [B]. Here, [E] likely stands for electromotive force or electric field, and [B] likely stands for magnetic field. The symbols [L] and [T] represent the dimensions of length and time respectively.

Analyze Dimensional Consistency

To find the relation between [E] and [B], we need to check the dimensional consistency for each option by comparing the dimensions of both sides of the equation. The dimensions of electromotive force, or electric field (in case [E] stands for that), are given by \(MLT^{-2}Q^{-1}\), where M is mass, L is length, T is time, and Q is electric charge.

Match the Dimensions

The dimensions of the magnetic field [B] are \(MT^{-2}Q^{-1}\). We must now determine which combination of [B], [L], and [T] gives the correct dimensions for [E]. This step involves substituting the dimensions of [B], [L], and [T] into each option and comparing the result to the dimensions of [E].

Check Each Option

Option (A) gives \(MT^{-2}Q^{-1}L^{1}T^{1}\), which simplifies to \(MLT^{-1}Q^{-1}\).Option (B) gives \(MT^{-2}Q^{-1}L^{-1}T^{1}\), which simplifies to \(MT^{-1}Q^{-1}\).Option (C) gives \(MT^{-2}Q^{-1}L^{1}T^{-1}\), which simplifies to \(MLT^{-2}Q^{-1}\).Option (D) gives \(MT^{-2}Q^{-1}L^{-1}T^{-1}\), which simplifies to \(MT^{-3}Q^{-1}\).

Select the Correct Option

Compare the dimensional expressions obtained from the options with the dimensions of [E] given earlier (\(MLT^{-2}Q^{-1}\)). Only option (C) has the dimensions that match \(MLT^{-2}Q^{-1}\), which are the dimensions of electromotive force or electric field.

Key concepts

Electromotive Force

Electromotive force, commonly referred to as EMF, is a foundational concept in physics, particularly in the area of electromagnetism. It refers to the amount of energy provided by a source per coulomb of charge that passes through the circuit. Imagine it as the 'pressure' that pushes the charges through a conducting loop, overcoming the internal resistance of the battery and the external resistance of the circuit. It's because of EMF that currents are able to flow through circuits, powering everything from flashlights to smartphones.

When we talk about EMF in dimensional analysis, it is represented as \( MLT^{-2}Q^{-1} \), which encompasses mass (M), length (L), time (T), and electric charge (Q). In terms of units, EMF is usually expressed in volts. A fundamental understanding of EMF is necessary for students tackling advanced physics problems, such as those in the JEE Advanced exam. Grasping this concept involves being familiar with other physical quantities like electric field, magnetic field, resistance, and current.

Magnetic Field

A magnetic field is a vector field that permeates space and can exert magnetic forces on moving charges and magnetic materials like iron. It is represented by the symbol \( \vec{B} \), and its strength and direction can determine how charges and currents behave within its influence. This is a central concept when discussing electromagnetic phenomena, including electromotive force.

When we describe the magnetic field using dimensions, we use the expression \( MT^{-2}Q^{-1} \). Understanding the properties of magnetic fields, including how they're generated and how they interact with other magnetic fields and charges, is integral for solving complex physics problems. In the context of JEE Advanced physics, comprehending the interplay between electric and magnetic fields is crucial, especially in the development of equations governing electromagnetic induction and waves.

Physical Quantities

Physical quantities are the building blocks of physics, helping us describe and quantify the properties of objects and phenomena. They include notions like length, time, mass, and charge, which can be measured and expressed through units. Dimensional analysis is a powerful tool used to compare the relationships between different physical quantities and to check the consistency and validity of physics equations.

In the context of the given problem, we used dimensional analysis to deduce the connection between electromotive force and magnetic field. Such analysis is essential not only for theoretical understanding but also for practical problem-solving in contexts like examinations or real-world applications. To master JEE Advanced physics, students must be adept at manipulating and understanding the dimensions of various physical quantities.

JEE Advanced Physics

The Joint Entrance Examination (JEE) Advanced is a highly competitive exam that serves as a gateway for admission into some of India's most esteemed engineering institutes, including the IITs. JEE Advanced physics comprises a rigorous curriculum, involving a deep dive into concepts like electromagnetism, mechanics, thermodynamics, and quantum physics.

To excel in JEE Advanced physics, students must have a profound understanding of fundamental concepts and be capable of applying them to solve complex problems, often involving multiple steps and concepts. Dimensional analysis, as demonstrated in the exercise, is one such skill that can help to check the validity of physical equations and deduce relationships between diverse physical quantities. By strengthening these skills, students set themselves up for success in both JEE Advanced and their future engineering endeavors.