Problem 54 - Master Resource Book in JEE Main Physics

Question

The dimensional formula of magnetic flux is [BVP 2007] (a) \(\left[\mathrm{ML}^{0} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{-1}\right]\) (c) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{-2}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]\)

Step-by-Step Solution

Answer

The correct answer is (d) \([ML^2 T^{-2} A^{-1}]\).

Step 1: Understanding Magnetic Flux

Magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) it penetrates through, i.e., Φ = B * A. It measures how much the magnetic field passes through a given surface.

Step 2: Dimensional Formula for Magnetic Field (B)

The dimensional formula of the magnetic field (B) can be derived from the Lorentz force law, which states that force (F) on a charge (q) moving with velocity (v) is F = qvB. Therefore, B = F/(qv). Now, force has dimensions [MLT^{-2}], charge (q) has dimensions [A^1 T^1] (from current, I=q/t), and velocity (v) has dimensions [LT^{-1}]. Thus, the dimensional formula for B becomes [M L^0 T^{-2} A^{-1}].

Step 3: Dimensional Formula for Area (A)

Area is a measure of the extent of a surface and is given by the product of two lengths. Thus, the dimensional formula for area is [L^2].

Step 4: Combining Magnetic Field and Area

Now, we multiply the dimensional formulas of magnetic field (B) and area (A) to get the dimensional formula of magnetic flux (Φ). Thus, Φ = B * A = [M L^0 T^{-2} A^{-1}] * [L^2] = [M L^2 T^{-2} A^{-1}].

Step 5: Matching with the Options

Finally, we look at the given options and match our derived dimensional formula of magnetic flux [M L^2 T^{-2} A^{-1}] to the listed choices. This corresponds to option (d).

Key Concepts

Magnetic Flux

Magnetic flux measures the total magnetic field passing through a specific area. Imagine it as the net amount of the magnetic field lines traversing through a surface.

• Mathematically, magnetic flux \( \Phi \) is given by \( \Phi = B \times A \), where \( B \) is the magnetic field strength and \( A \) is the area.
• It's crucial because it helps quantify how effective a particular magnetic field is in influencing physical systems.
• The unit of magnetic flux is the Weber (Wb) in the SI system.

Magnetic flux is vital in electromagnetics and comes into play significantly in devices like transformers, inductors, and motors.

Magnetic Field

A magnetic field is a region around a magnet where magnetic forces can be detected. It is an invisible field that exerts forces on particles that move through it.

• The magnetic field is often represented by field lines that show the field's direction and strength.
• It is denoted by \( B \) and is measured in Tesla (T).
• Both moving electric charges and magnetic dipoles (like bar magnets) create magnetic fields.

Understanding magnetic fields helps in explaining various phenomena such as the operation of electrical generators and the earth's own geomagnetic field.

Lorentz Force Law

The Lorentz force law is fundamental in the realms of electromagnetism. It describes how charged particles experience a force when moving through an electric and magnetic field.

• The law is mathematically represented as \( F = q(E + v \times B) \), where \( F \) is the force on the particle, \( q \) is the electric charge, \( E \) is the electric field, and \( v \) is the velocity of the particle.
• This law explains phenomena such as the circular trajectories of charged particles in magnetic fields.
• It also underlies the working principles of many technologies, from cathode-ray tubes to particle accelerators.

The Lorentz force law is thus central to understanding how electric and magnetic fields interact with matter.

Dimensional Formula

Dimensional formulas offer insight into the physical nature and dimensional analysis of physical quantities. They are expressions showing how fundamental units relate to derived units.

• For instance, in the problem at hand, the dimensional formula of magnetic flux is derived as \([M L^2 T^{-2} A^{-1}]\), revealing its components.
• Dimensional analysis helps verify equations and conversion between units.
• It assists in cross-checking the consistency of physical equations and can even predict relationships between different physical quantities.

Learning to derive dimensional formulas enhances one's ability to tackle a wide range of physics problems.

Physics Olympiad Problems

Physics Olympiad problems are designed to challenge students' understanding of physics concepts and their ability to apply them in complex situations. They require a deep understanding and innovative approach.

• These problems emphasize critical thinking and problem-solving skills.
• They often combine multiple topics such as electromagnetism, mechanics, or thermodynamics.
• The key to success in tackling these problems is a solid grasp of fundamental principles, calculations, and sometimes creative intuition.

Competing in Physics Olympiads can significantly improve one's ability to conceptualize and solve advanced physics problems.