Problem 1956 - JEE Physics

Question

Magnetic intensity for an axial point due to a short bar magnet of magnetic moment \(\mathrm{M}\) is given by (a) \(\left(\mu_{0} / 4 \pi\right)\left(\mathrm{M} / \mathrm{d}^{3}\right)\) (b) \(\left(\mu_{0} / 4 \pi\right)\left(\mathrm{M} / \mathrm{d}^{2}\right)\) (c) \(\left(\mu_{0} / 2 \pi\right)\left(\mathrm{M} / \mathrm{d}^{3}\right)\) (d) \(\left(\mu_{0} / 2 \pi\right)\left(\mathrm{M} / \mathrm{d}^{2}\right)\)

Step-by-Step Solution

Answer

The magnetic intensity for an axial point due to a short bar magnet of magnetic moment \(M\) is given by (a) \(\left(\frac{\mu_0}{4\pi}\right)\left(\frac{M}{d^3}\right)\).

Step 1: Recall the formula for Magnetic Intensity

The formula for magnetic intensity \(H\) at a point located on the axis of a short bar magnet is given by: \[H = \left(\frac{\mu_0}{4\pi}\right)\left(\frac{2M}{d^3}\right)\] Where \(\mu_0\) is the permeability of free space, \(M\) is the magnetic moment of the bar magnet, and \(d\) is the distance from the point to the center of the magnet.

Step 2: Compare the given options with the correct formula

Now, let's compare the correct formula for magnetic intensity with the given options: 1. Option (a): \(\left(\frac{\mu_0}{4\pi}\right)\left(\frac{M}{d^3}\right)\) 2. Option (b): \(\left(\frac{\mu_0}{4\pi}\right)\left(\frac{M}{d^2}\right)\) 3. Option (c): \(\left(\frac{\mu_0}{2\pi}\right)\left(\frac{M}{d^3}\right)\) 4. Option (d): \(\left(\frac{\mu_0}{2\pi}\right)\left(\frac{M}{d^2}\right)\) The correct formula for magnetic intensity is \(H = \left(\frac{\mu_0}{4\pi}\right)\left(\frac{2M}{d^3}\right)\). Comparing this formula, we see that it matches Option (a).

Step 3: Conclusion

Based on our comparison, we can conclude that the magnetic intensity for an axial point due to a short bar magnet of magnetic moment \(M\) is given by: (a) \(\left(\frac{\mu_0}{4\pi}\right)\left(\frac{M}{d^3}\right)\)

Key Concepts

Magnetic Moment

Magnetic moment is a crucial concept in understanding magnetism. It quantifies the ability of a magnet to exert torque in a magnetic field. Essentially, it describes how strong and in which direction a magnet will interact with an external magnetic field.
Here are some important points about magnetic moment:

• A magnetic moment has both a magnitude and a direction, much like a vector.
• The unit of magnetic moment in the International System (SI) is Ampere-square meter (A·m²).
• For a bar magnet, the magnetic moment is determined by the pole strength and the distance between the poles, mathematically expressed as \( M = p \times l \), where \( p \) is the pole strength and \( l \) is the separation between the poles.

Understanding the magnetic moment is vital for solving problems involving magnetic fields and forces, as it directly influences the intensity and direction of the magnetic field produced by the magnet.

Axial Point

When we talk about an axial point in relation to a bar magnet, we're referring to a specific location along the line that extends from the magnetic poles through the center of the magnet. This line is called the magnetic axis.
Points on this line experience the magnetic field produced by the magnet differently compared to points off this axis.

• The magnetic field intensity at an axial point is stronger than at any other point at the same distance from a bar magnet.
• This is because the axial line is directly aligned with the magnetic moment.
• The axial magnetic field formula is crucial for calculating magnetic intensity and is represented by \( H = \left( \frac{\mu_0}{4\pi} \right) \left( \frac{2M}{d^3} \right) \), where \( d \) is the distance from the axial point to the center of the magnet.

Understanding axial points is key to analyzing the strength and direction of a magnet's field in various situations.

Bar Magnet

A bar magnet is one of the simplest forms of magnets and most common examples of permanent magnets. It is a rectangular object made of ferromagnetic material, which generates a magnetic field around itself.
You might find it interesting that:

• Bar magnets have two distinct poles: the North (N) pole and the South (S) pole.
• The magnetic field lines emanate from the North pole and curve back into the South pole, creating a closed loop outside the magnet.
• The strength of a bar magnet is concentrated at its poles.
• Bar magnets are used in numerous applications, including compasses, scientific experiments, and as a teaching tool in physics education.

Their magnetic moment, which determines the strength and orientation of the magnetic field, is along the axis that runs from the North to the South pole.

Permeability of Free Space

Permeability of free space, denoted as \( \mu_0 \), is a fundamental physical constant used in electromagnetism. It is also known as the magnetic constant.
This constant plays a significant role in defining the strength of the magnetic field in a vacuum:

• The value of \( \mu_0 \) is approximately \( 4\pi \times 10^{-7} \) H/m (Henries per meter).
• It is part of the fundamental equation for the magnetic field strength in free space: \( B = \mu_0 H \), where \( B \) is the magnetic flux density and \( H \) is the magnetic field intensity.
• \( \mu_0 \) helps describe how a magnetic field propagates in a vacuum, without the influence of any material medium.

In exercises involving magnetic fields, including those of bar magnets, \( \mu_0 \) is crucial for accurately calculating the magnetic field characteristics and analyzing real-world magnetic interactions.