Problem 38 - Master Resource Book in JEE Main Physics

Question

The dimensions of pole strength are (a) \(\left[\mathrm{M}^{\circ} \mathrm{LT}^{0} \mathrm{~A}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{LTA}\right]\) (c) \(\left[\mathrm{M}^{\circ} \mathrm{L}^{-1} \mathrm{TA}^{-1}\right]\) (d) \(\left[\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0} \mathrm{~A}^{-1}\right]\)

Step-by-Step Solution

Answer

The correct dimensions of pole strength are \([M^0L^{-1}T^0A^{-1}]\), i.e., option (d).

Step 1: Understand Pole Strength

Pole strength is a measure of the strength of a magnetic pole to create a magnetic field. It is typically represented by the symbol \( m \) and is defined in terms of magnetic field properties.

Step 2: Identify Dimensional Formula

To identify the dimensional formula of pole strength, we start with the magnetic force equation. The magnetic force between two poles of strengths \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by the formula \( F = \frac{{ ext{k} imes m_1 imes m_2}}{{r^2}} \), where \( k \) is a constant.

Step 3: Relate Force to Magnetic Pole Strength

The dimensional formula for force \( F \) is \([MLT^{-2}]\), and for distance \( r \) is \([L]\). For the constant \( k \), its dimensions will balance the equation, leaving only the dimensions for pole strength \( m \) to be determined.

Step 4: Derive Dimensions for Pole Strength

Rewriting the force equation in terms of dimensions, we get \([MLT^{-2}] = [M^{0}L^1T^{0}A^0] \times [m]^2 / [L^2]\). Solving for the dimensions of \( [m] \), we equate \( [MLT^{-2}] \) with \([m]^2[L^{-2}]\), resulting in \([m] = [L^{-1}A^{-1}]\).

Step 5: Match With Options

Compare the calculated dimension \([L^{-1}A^{-1}]\) with the provided options. The correct option matches the result from Step 4, which is option (d) \([M^0L^{-1}T^0A^{-1}]\).

Key Concepts

Magnetic Pole Strength

Magnetic pole strength is an important concept in understanding magnetic fields and magnetic interactions. It is symbolically represented by the letter \( m \). This quantity describes the ability of a magnetic pole to produce a magnetic field around itself.
Magnetic pole strength plays a crucial role in the calculation of magnetic forces between poles. When two magnetic poles are placed in proximity, their respective strengths determine the intensity and direction of the magnetic force they exert on each other.
To easily grasp this concept, think of pole strength as the magnetic equivalent of charge in electrostatics. Just like charges create electric fields, poles create magnetic fields.

Magnetic Force Formula

The magnetic force formula provides an essential relationship between the forces experienced by two magnetic poles based on their strengths and the distance separating them. The equation is expressed as:

• \( F = \frac{k \cdot m_1 \cdot m_2}{r^2} \)

where:

• \( F \) is the magnetic force between the two poles
• \( m_1 \) and \( m_2 \) are the magnetic pole strengths
• \( r \) is the distance between the poles
• \( k \) is a constant that depends on the medium

This formula is analogous to the electrostatic force equation in Coulomb's law. It highlights the inverse square law relationship indicating that the force between poles decreases with the square of the distance between them.
Understanding the magnetic force formula helps in predicting how magnetic materials will interact under different scenarios.

Dimensional Formula

Dimensional analysis is a method used to derive or understand the dimensions of various physical quantities based on their relation to base units, such as mass \([M]\), length \([L]\), and time \([T]\). In the context of magnetic pole strength, it becomes crucial to determine the dimensional formula to understand its role in magnetic interactions.
For pole strength \( m \), the challenge is to find its dimensions using the force equation. Given the magnetic force formula \( F = \frac{k \cdot m_1 \cdot m_2}{r^2} \), we know the dimensional formula for force \( F \) is \([MLT^{-2}]\), and for distance \( r \) is \([L]\).

• Substituting into the equation, we derive dimensions \([MLT^{-2}] = [m]^2 [L^{-2}]\)
• Simplifying gives \([m] = [L^{-1}A^{-1}]\)

Thus, the dimensional formula for pole strength is \([M^0L^{-1}T^0A^{-1}]\), indicating the absence of dependency on mass or time, but a single inverse proportion with length and ampere. This highlights the independence of pole strength from mass, tying it directly to length and electric current measures.