Problem 1977 - JEE Physics

Question

Two iclentical short bar magnets, each having magnetic moment \(\mathrm{M}\) are placed a distance of \(2 \mathrm{~d}\) apart with axes perpendicular to each other in a horizontal plane. The magnetic induction at a point midway between them is. (a) \(\sqrt{2}\left(\mu_{0} / 4 \pi\right)\left(\mathrm{M} / \mathrm{d}^{3}\right)\) (b) \(\sqrt{3}\left(\mu_{0} / 4 \pi\right)\left(\mathrm{M} / \mathrm{d}^{3}\right)\) (c) \(\sqrt{4}\left(\mu_{0} / 4 \pi\right)\left(\mathrm{M} / \mathrm{d}^{3}\right)\) (d) \(\sqrt{5}\left(\mu_{0} / 4 \pi\right)\left(\mathrm{M} / \mathrm{d}^{3}\right)\)

Step-by-Step Solution

Answer

The short answer is: The magnetic induction at a point midway between the two identical short bar magnets is \(\sqrt{2}\left(\frac{\mu_{0}}{4\pi}\right)\left(\frac{M}{d^{3}}\right)\).

Step 1: Find the magnetic field due to one bar magnet

To find the magnetic field at the midpoint between the two bar magnets, we first need to find the magnetic field due to one bar magnet at that point. Let's call this bar magnet A. The magnetic moment of bar magnet A is given as M. Since the bar magnets are placed in a horizontal plane with a distance of 2d between them, the distance from the midpoint to bar magnet A is d. The formula for the magnetic field due to a magnetic dipole (bar magnet) at a distance r from it is given by: \[B = \frac{\mu_0}{4\pi}\frac{M}{r^3}\] Substituting the given values for M and r (d) in the formula, we get: \[B_A = \frac{\mu_0}{4\pi}\frac{M}{d^3}\]

Step 2: Find the magnetic field due to the second bar magnet

Similarly, let's find the magnetic field due to the second bar magnet, B, at the midpoint. The magnetic moment of bar magnet B is also given as M and the distance from the midpoint to bar magnet B is also d. Using the formula for magnetic field due to magnetic dipole, we get: \[B_B = \frac{\mu_0}{4\pi}\frac{M}{d^3}\]

Step 3: Combine the magnetic fields of the two bar magnets

Now, let's find the total magnetic induction at the midpoint by combining the magnetic fields due to both bar magnets A and B. Since the bar magnets are placed perpendicular to each other and the axes are perpendicular, their magnetic fields at the midpoint are at right angles to each other. We can use the Pythagorean theorem to combine these fields: \[B_{total} = \sqrt{B_A^2 + B_B^2}\] Substituting the values for B_A and B_B, we get: \[B_{total} = \sqrt{\left(\frac{\mu_0}{4\pi}\frac{M}{d^3}\right)^2 + \left(\frac{\mu_0}{4\pi}\frac{M}{d^3}\right)^2}\]

Step 4: Simplify the expression

Now, we simplify the expression for B_total: \[B_{total} = \sqrt{2\left(\frac{\mu_0}{4\pi}\frac{M}{d^3}\right)^2}\] \[B_{total} = \sqrt{2}\left(\frac{\mu_0}{4\pi}\right)\left(\frac{M}{d^3}\right)\] Comparing the expression of B_total with the given options, we find that the correct answer is: (a) \(\sqrt{2}\left(\frac{\mu_{0}}{4\pi}\right)\left(\frac{M}{d^{3}}\right)\)

Key Concepts

Bar Magnets

Bar magnets are simple forms of magnets that have two distinct poles: a north and a south pole. These magnets are considered as magnetic dipoles, which means that they have a positive and a negative side. Bar magnets create a magnetic field around them, influencing any magnetic material within its range.

• Due to their dipole nature, the field lines emerge from the north pole and curve around to enter the south pole.
• When placed in proximity to another bar magnet, they will either attract or repel depending on their orientation.

This characteristic makes bar magnets useful in various applications, ranging from simple compass needles to complex magnetic sensors. They are also commonly used in educational contexts to demonstrate basic magnetic principles.

Magnetic Fields

Magnetic fields are invisible forces that surround magnetic materials and are represented by field lines. The direction of these lines is conventionally from the north pole to the south pole of a bar magnet.

• Magnetic fields are vector fields, which means they have both magnitude and direction. This dual nature is crucial in understanding how they interact with other fields or charged particles.
• The strength of a magnetic field depends on how close the field lines are to one another; the closer they are, the stronger the field.

Understanding magnetic fields is integral when studying topics such as electromagnetism or when working with electronic devices, as these fields can induce electrical currents in conductive materials.

Magnetic Dipole Moment

At the heart of understanding magnets is the concept of a magnetic dipole moment, which is a measure of the magnetic strength and orientation of a magnet. For a bar magnet, this represents the product of its pole strength and the distance between its poles.

• The magnetic dipole moment is crucial in determining the torque a magnet will experience in an external magnetic field. A larger magnetic dipole moment implies a stronger magnetic field it can produce.
• It also signifies how a bar magnet will align itself when placed in an external magnetic field.

In practical terms, the magnetic dipole moment helps predict how a magnet will behave when subjected to external fields, making it essential in designing magnetic materials and devices.

Superposition of Magnetic Fields

The principle of superposition is a key concept in understanding how multiple magnetic fields interact. When different magnetic fields overlap, the net effect is the vector sum of all the individual fields.

• This means if two magnetic fields are acting perpendicular to each other, as in the exercise, their combined effect can be determined by calculating the root of the sum of the squares, following the Pythagorean theorem.
• This superposition allows one to analyze complex magnetic environments by breaking them down into simpler individual components.

Knowing how to apply superposition helps in predicting the resultant magnetic fields in intricate settings like those found in motors or MRI machines. It simplifies the otherwise complex interactions between multiple magnetic sources.