Magnetism And Matter

Step 1: Application of the Biot-Savart Law.
The magnetic field due to a segment of current-carrying wire is determined by the Biot-Savart Law, which states: where:

• is the magnetic constant.
• is the current.
• is the length vector of the wire segment.
• is the unit vector from the segment to the observation point.

Step 2: Calculation of and .
From the wire segment to the point (3 m, 4 m, 0):

Step 3: Calculation of and .
Using the right-hand rule for the cross product: Thus, points in the negative direction. The magnitude of is: Considering the direction, the field is: which matches option (C).

Understanding Ferromagnetic Materials and Ferromagnetism

Definition of Ferromagnetic Materials:
- Ferromagnetic materials are substances that exhibit strong magnetization when placed in an external magnetic field.
- These materials retain their magnetization even after the external field is removed.
- Common examples include iron (Fe), cobalt (Co), and nickel (Ni). Explanation of Ferromagnetism:
- The phenomenon of ferromagnetism arises due to the presence of magnetic domains.
- Each domain consists of a group of atomic dipoles aligned in the same direction.
- In an unmagnetized ferromagnetic material, these domains are randomly oriented, resulting in zero net magnetization.

Magnetic Domains and Alignment:
- When an external magnetic field is applied, the domains align in the direction of the field.
- The material becomes magnetized, and the alignment of the domains increases the overall magnetic moment. where:
- = Magnetic field in the material
- = Applied external field
- = Magnetization of the material
- = Permeability of free space - In fully magnetized ferromagnetic materials, most domains align in the same direction, producing a strong magnetic effect.

Key Properties of Ferromagnetic Materials:
1. High Permeability: These materials have a high ability to concentrate magnetic flux.
2. Hysteresis Effect: The magnetization does not return to zero immediately after removing the field, leading to a hysteresis loop.
3. Curie Temperature ( ): Above this temperature, the material loses its ferromagnetic properties and behaves as a paramagnet.

Conclusion:
- Ferromagnetic materials are widely used in applications like transformers, electromagnets, hard disks, and electric motors due to their ability to retain strong magnetization.

Applying Biot-Savart Law and Ampere’s Law
- The magnetic field at the center of the circular loop is given by: where , cm = 0.1 m.
- The magnetic field due to the long straight wire at distance cm is: - For net magnetic field at to be zero: Direction of Current in the Wire:
- Using the Right-Hand Rule, the circular loop produces a magnetic field into the plane at O.
- To cancel it, the wire must produce a magnetic field out of the plane at O.
- This happens when current in the wire flows in the negative x-direction.

Thus, the required current in the wire is 2 A, flowing in the negative x-direction.

Magnetic Flux Linked with Loop B
Step 1: Magnetic Field Due to Loop A
- The magnetic field at the center of a circular loop carrying a current is given by the formula: where:
- = radius of loop A,
- = permeability of free space.

Step 2: Magnetic Flux Linked with Loop B
- The magnetic flux linked with loop B is given by: where:
- is the area of loop B,
- is the radius of loop B. Thus, Simplifying: Thus, the magnetic flux linked with loop B is directly proportional to .

Step 3: Conclusion
The magnetic flux linked with loop B is proportional to , so the correct answer is:

Concept: Magnetic moment of the coil is $ B \phi \mu \mathbf{B} 90^\circ \displaystyle B=\frac{13}{18}\ \mathrm{T}\approx 0.722\ \mathrm{T}.$

A diamagnetic substance is a material that creates an opposing magnetic field when exposed to an external magnetic field. This means that diamagnetic substances are repelled by magnets. Here's why:

1. Nature of Diamagnetism: Diamagnetic materials have no unpaired electrons, and their atomic moments generally cancel out. When placed in a magnetic field, they develop an induced magnetic moment that opposes the applied field.

2. Response to Magnetic Poles: Due to their opposition to the applied magnetic field, diamagnetic substances are repelled by both the north and south poles of a magnet. This is because they tend to move from stronger to weaker parts of the magnetic field.

3. Experimental Observation: When a diamagnetic substance is brought near a bar magnet, it experiences a force that pushes it away from both poles.

Considering the behavior of diamagnetic materials, the correct answer is: "repelled by north pole as well as by south pole".

Magnetic flux and magnetic field at a point are related concepts in electromagnetism, but they are fundamentally different. Below is the concise differentiation:

In summary, magnetic flux is the total magnetic field passing through a given area, while the magnetic field at a point refers to the local magnetic influence at a specific point in space.

The potential energy of a magnetic dipole is minimum when the magnetic moment is aligned with the magnetic field, i.e., when . At this position, the cosine term is maximum (equal to 1), and thus the potential energy is at its minimum: Therefore, the potential energy of the magnet is minimum when the magnetic moment is aligned with the magnetic field.

Magnetic Moment and Potential Energy

Given:

• Magnetic moment,
• Magnetic field,

The potential energy of a magnetic dipole in a uniform magnetic field is given by the formula:

(i) Potential Energy of the Bar Magnet:

Since the magnetic moment is initially aligned with the magnetic field, , and thus:

Substitute the values:

Thus, the potential energy of the bar magnet is -2 J.

(ii) Work Done in Turning the Magnet by 180°:

When the magnet is turned by 180°, the angle between the magnetic moment and the magnetic field becomes . The new potential energy is:

Substitute the values:

The work done in turning the magnet is the change in potential energy:

Thus, the work done in turning the magnet by 180° is 4 J.

1. Magnetic Moment of a Current-Carrying Coil:

The magnetic moment ( ) of a current-carrying coil is a vector quantity that represents the strength and orientation of the coil's magnetic field. It is defined as the product of the current flowing through the coil and the area of the coil, along with the direction normal to the plane of the coil (perpendicular to the coil's surface).

The formula for the magnetic moment of a coil is given by:

Where:

• is the current flowing through the coil (in amperes, A).
• is the area of the coil (in square meters, m²).

2. Direction of Magnetic Moment:

The direction of the magnetic moment vector is determined by the right-hand rule. If the fingers of the right hand curl in the direction of the current, then the thumb points in the direction of the magnetic moment.

3. SI Unit of Magnetic Moment:

The SI unit of magnetic moment is the ampere-square meter (A·m²), which is derived from the current in amperes and the area in square meters.

4. Conclusion:

• The magnetic moment of a current-carrying coil is .
• The SI unit of magnetic moment is ampere-square meter (A·m²).

The magnetic dipole moment of a current-carrying coil is given by: Where: - is the current in the coil, - is the area of the coil, - is the unit vector perpendicular to the plane of the coil, indicating the direction of the dipole moment. The direction of is given by the right-hand rule. If the fingers of the right hand curl in the direction of the current, the thumb points in the direction of the magnetic dipole moment.

Number of Holes Created in P-type Silicon Due to Doping

Given:

• The average number of dopant atoms per silicon atoms
• The number density of silicon atoms in the specimen is atoms/m
• We need to calculate the number of holes created per cubic centimetre due to doping.

Solution:

Step 1: Number of dopant atoms per silicon atom

The ratio of dopant atoms to silicon atoms is given as:

Step 2: Number of dopant atoms per cubic metre of silicon

The number density of silicon atoms is atoms/m . The number of dopant atoms per cubic metre is:

Step 3: Number of holes created per cubic metre

Since each dopant atom creates one hole in the semiconductor, the number of holes created per cubic metre is equal to the number of dopant atoms:

Step 4: Convert the number of holes to per cubic centimetre

Since , the number of holes per cubic centimetre is:

Example of a Dopant in P-type Silicon:

One example of a dopant used in p-type silicon is **boron (B)**. Boron atoms have one fewer valence electron than silicon, creating a hole in the crystal lattice when substituted for a silicon atom.

Final Answer:

The number of holes created per cubic centimetre in the specimen due to doping is .

To solve this assertion-reason type question, we need to evaluate the truth of each statement and their relationship to each other.

Assertion (A): The deflection in a galvanometer is directly proportional to the current passing through it.
The assertion states a fundamental principle of how galvanometers work. When a current passes through the coil of a galvanometer, it experiences a torque due to the magnetic field, causing a deflection which is generally proportional to the current.
Therefore, the assertion is true.

Reason (R): The coil of a galvanometer is suspended in a uniform radial magnetic field.
The reason describes the specific construction of a galvanometer where the coil is placed in a uniform radial magnetic field. This design ensures that the torque experienced by the coil is proportional to the current, regardless of its position, as the angle between the plane of the coil and magnetic field is always constant.
Hence, the reason is also true.

While the reason correctly describes an aspect of the galvanometer's design, it is not the direct explanation for why the deflection is proportional to the current. The deflection being proportional to the current is a result of the linear relationship between current, magnetic field, and torque, not merely due to the coil being in a radial field.

Therefore, the correct conclusion is: Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).