Problem 46 - Master Resource Book in JEE Main Physics

Question

The dimension of magnetic flux is (a) \(\left[\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-2} \mathrm{~A}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]\) (c) \(\left[\mathrm{ML}^{-2} \mathrm{~A}^{-2} \mathrm{~T}^{-1}\right]\) (d) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{2}\right]\)

Step-by-Step Solution

Answer

The dimension of magnetic flux is (b) \( [ML^{2}T^{-2}A^{-1}] \).

Step 1: Recall the formula for Magnetic Flux

Magnetic flux (Φ) through a surface is defined as the product of the magnetic field (B) and the area (A) through which the field lines pass, i.e., Φ = B × A.

Step 2: Determine dimensions of Magnetic Field

The dimension of the magnetic field (B) is derived from Lorentz force, which states that the magnetic force \( F = qvB \sin \theta \). Re-arranging for B gives \( B = \frac{F}{qv} \). Since \( F = qE \) (where E is the electric field) and \( q = I \cdot T \) (charge as current \( I \) times time \( T \)), it gives \( E = \frac{F}{q} = \frac{MLT^{-2}}{AT} = MLT^{-3}A^{-1} \). Thus, \( B \) becomes \( MLT^{-2}A^{-1} \).

Step 3: Determine dimensions of Area

The dimension of area (A) is \( L^{2} \) since it is represented by length times width, both having dimension \( L \).

Step 4: Combine Dimensions of Magnetic Field and Area

Magnetic flux Φ is determined by multiplying the dimensions of B and A: \( [B][A] = [MLT^{-2}A^{-1}] [L^{2}] = ML^{2}T^{-2}A^{-1} \).

Step 5: Match Derived Dimension with Options

Compare the derived dimension \( [ML^{2}T^{-2}A^{-1}] \) with the given options. It matches with option (b).

Key Concepts

Magnetic Field

A magnetic field is an invisible field that describes the magnetic influence of electric charges in relative motion. These fields exert forces on nearby moving charges and magnetic dipoles. To understand a magnetic field, consider it as lines of force through a region in which the magnetic influence is present. These fields can be visualized as a set of concentric circles around a conductor carrying electric current.

• The unit of the magnetic field is Tesla (T), representing the force on a charge.
• Magnetic fields are often visualized using magnetic field lines. The direction of the field line gives you the direction of the magnetic field and is typically represented by arrows in the field lines.
• Magnetic fields are strongest near the magnet and weaker farther away.

This concept is central to understanding phenomena such as the earth's magnetic dipole and its influence on compass needles. It also explains how electric motors work, where the magnetic field is utilized to produce motion through a conductor.

Lorentz Force

The Lorentz Force is a fundamental concept that describes the force exerted on a charged particle moving through a magnetic and electric field. The basic formula for Lorentz force is \[ F = q(E + v \times B) \]where:

• \( F \) is the force acting on the particle.
• \( q \) is the electric charge.
• \( E \) is the electric field.
• \( v \) is the velocity of the particle.
• \( B \) is the magnetic field.

This equation shows us that a charged particle experiences force due to both electric and magnetic fields, but this force is more complex than simple addition. The magnetic component of the force, \( v \times B \), is perpendicular to both the velocity of the particle and the magnetic field.
This helps us understand why a charged particle does not necessarily get faster in a magnetic field, but rather its path can be curved or circular, showing the directionality of magnetic influences.

Dimensional Analysis

Dimensional analysis is a method used to deduce or verify relationships between physical quantities by identifying their base dimensions such as mass (M), length (L), time (T), and electric current (A). This approach is essential for verifying the correctness of equations in physics.
When dealing with magnetic flux, the dimensions are determined using the formula \( \Phi = B \times A \), where \( B \)is the magnetic field, and \( A \) is the area.
The dimensions used:
Element Dimension
Magnetic field (B) \( [MLT^{-2}A^{-1}] \)Area (A) \( [L^2] \)

Multiply these to derive the dimensions for magnetic flux: \( [ML^2T^{-2}A^{-1}] \), confirming the combination of dimensions for a given physical law lies consistent with established formulas.
This method serves as a powerful tool to ensure that equations are dimensionally consistent, which is a necessary, but not sufficient, condition for a physics equation to be correct.

Area Measurement

Area measurement is a fundamental concept in physics that describes the two-dimensional space occupied by an object. In the context of magnetic flux, it’s crucial because the area determines how many magnetic field lines pass through.
- For a surface, the area is mathematically described by the unit square of length, denoted as \( L^2 \).- The measurement can vary based on the shape of the surface. For rectangles and squares, it is length times width, whereas for circles, it is \( \pi r^2 \).This quantitative measure helps in understanding how much of the magnetic field penetrates through a surface and is integral for calculating magnetic flux.
By exploring different configurations and angles, one can also consider changes in enclosed area without physical alterations, demonstrating the dynamic nature of areas in magnetic studies. The importance of precise area measurement is crucial especially in engineering and physics to ensure correct applications and interpretations of results.