Magnetism

When a magnet with total magnetic moment is divided into equal parts parallel to its length, each piece behaves as a smaller magnet with a proportionally reduced magnetic moment. Since the magnetic moment is proportional to both the pole strength and the length of the magnet, dividing the magnet into equal parts reduces the length of each piece to of the original length while keeping the pole strength the same. Therefore, the magnetic moment of each piece is: This means the total magnetic moment is conserved as the sum of the moments of all the pieces.

Ferromagnetic materials exhibit hysteresis, which is the lag between the change in magnetization of the material and the applied external magnetic field. This phenomenon occurs because the magnetic domains within the material — regions where atomic magnetic moments are aligned — do not instantly realign when the external field changes. As the magnetic field is varied, these domains gradually grow or shrink, causing the magnetization to change with some delay. When the external field is reduced to zero, the material retains some magnetization (remanence), and an opposing field (coercivity) is required to bring the magnetization back to zero. This behavior is represented graphically by a hysteresis loop, which illustrates the energy loss in the form of heat during the magnetization and demagnetization cycles.

The magnetic potential at a point located at a distance from a magnetic pole of strength is given by the expression: where is the permeability of free space. This formula is analogous to the electric potential due to a point charge, and it describes how the magnetic potential decreases inversely with distance from the pole.

The energy density , which is the energy stored per unit volume in a magnetic field of strength , is given by the formula: where is the permeability of free space. This expression shows that the energy stored in the magnetic field increases with the square of the magnetic field strength, and it quantifies how much magnetic energy is contained within a given volume.

The torque experienced by a magnetic dipole moment placed in a magnetic field is given by the vector cross product: This torque arises because the magnetic dipole tends to align itself with the external magnetic field. The magnitude of the torque is: where is the angle between the magnetic moment and the magnetic field . The direction of the torque is perpendicular to the plane formed by and , following the right-hand rule. Physically, this torque causes the magnetic dipole to rotate, reducing the angle until aligns parallel to , minimizing the potential energy of the system.

The units of magnetic field (magnetic flux density ) include:

Tesla (T),
Weber per square meter (Wb/m ),
Newton per ampere-meter (N/(A·m)).
These units are equivalent and commonly used to express the magnetic flux density. However, newton per ampere squared ( ) is not a unit of magnetic field. Instead, corresponds to the unit of magnetic permeability or related magnetic properties, but it does not represent magnetic flux density.

Step 1: The Earth's magnetic field at any location can be represented as a vector quantity, which has both horizontal and vertical components relative to the Earth's surface.
Step 2: To fully describe the magnetic field, we use magnetic elements such as:

Declination ( ): The angle between the geographic north (true north) and the horizontal component of the Earth's magnetic field. It indicates the direction of the magnetic north relative to true north.
Inclination or Dip ( ): The angle made by the total magnetic field vector with the horizontal plane. It measures how much the magnetic field dips below or rises above the horizontal.

Step 3: These magnetic elements are essential for compass navigation, geophysical surveys, and for understanding the Earth's magnetosphere. They allow navigators and scientists to accurately map the Earth's magnetic field and correct compass readings accordingly.

Magnetic flux through area is given by: where is the angle between and the area vector . If is perpendicular to , then , so:

The magnetic field at the center of a circular current-carrying coil is directed perpendicular to the plane of the coil. The direction of this magnetic field can be determined using the right-hand thumb rule: If you curl the fingers of your right hand in the direction of the current flowing through the coil, then your thumb points in the direction of the magnetic field at the center of the coil. This magnetic field is uniform near the center and its magnitude depends on the current and radius of the coil.

Step 1: Diamagnetic materials are characterized by having all their electrons paired, resulting in no permanent magnetic moment. When an external magnetic field is applied, these materials induce a magnetic field in the opposite direction due to changes in the orbital motion of electrons. This causes a very weak repulsion from the applied magnetic field.
Step 2: Paramagnetic materials possess one or more unpaired electrons, giving rise to permanent but randomly oriented magnetic moments. When subjected to an external magnetic field, these magnetic moments tend to align weakly in the direction of the field, resulting in a slight attraction towards the magnetic source. However, this alignment is usually lost once the external field is removed.
Step 3: Ferromagnetic materials contain magnetic domains, which are regions where groups of atomic magnetic moments are strongly aligned in the same direction even without an external field. Upon applying a magnetic field, these domains align more uniformly, causing a strong attraction. Ferromagnetic materials can retain this alignment, exhibiting permanent magnetism, and are widely used in the fabrication of permanent magnets and magnetic storage devices.

The Curie temperature (denoted as ) is a critical temperature at which a ferromagnetic material undergoes a transition to a paramagnetic state. Below the Curie temperature, the material exhibits strong magnetic properties due to the alignment of its atomic magnetic dipoles. However, when the temperature exceeds the Curie point, thermal energy disrupts this alignment, causing the material to lose its ferromagnetic properties and become paramagnetic.
Ferromagnetic materials, such as iron, cobalt, and nickel, are characterized by their ability to retain magnetization even after the external magnetic field is removed. This property is a result of the alignment of individual magnetic moments within the material. However, as the temperature increases, the thermal energy becomes sufficient to overcome the alignment of these magnetic moments, resulting in the loss of ferromagnetic behavior.
The Curie temperature is unique to each material. For example, the Curie temperature of iron is around , which means that at temperatures above this, iron will no longer exhibit ferromagnetic behavior and will behave as a paramagnetic substance.
In summary, the Curie temperature marks the boundary between the ferromagnetic and paramagnetic states, and it plays a crucial role in the study of magnetism and materials science.

Step 1: Understanding neutral points.
Neutral points are the locations where the magnetic field produced by the Earth ( ) and the magnetic field produced by the magnet ( ) cancel each other out. At these points, the net magnetic field is zero.
Step 2: Relationship between and .
At neutral points, the magnetic field created by the magnet is exactly equal to the horizontal component of the Earth's magnetic field . This results in the cancellation of the two fields.
Step 3: Elimination.
- (A) : Incorrect, the fields cancel out, so cannot be greater.
- (B) : Incorrect, again the fields cancel out, so cannot be less.
- (C) : Correct, at neutral points, the fields are equal and cancel each other.
- (D) : Incorrect, as the magnetic field is not zero at the neutral points.
Step 4: Conclusion.
Thus, at neutral points, .

The magnetic moment ( ) is a vector quantity that represents the strength and direction of a magnetic source, such as a magnetic dipole. It is defined as the product of the current ( ) flowing through a loop and the area ( ) of the loop: Where: - is the current in amperes (A), - is the area of the loop in square meters ( ). The magnetic moment determines how a magnetic object will interact with an external magnetic field. It tends to align with the field, and the torque experienced by the magnetic moment in the field is given by: Where: - is the torque,
- is the magnetic field strength,
- is the angle between the magnetic moment and the magnetic field.
The SI unit of magnetic moment is . It is also sometimes expressed in joules per tesla (J/T), as torque (joules) per unit magnetic field strength (tesla).
In summary, the magnetic moment describes the strength and orientation of a magnetic source, and its SI unit is .

Paramagnetic substances are materials that are weakly attracted by an external magnetic field and do not retain magnetic properties once the external field is removed. The two key properties of paramagnetic substances are:
1. Weak Attraction to Magnetic Fields: Paramagnetic substances are weakly attracted to external magnetic fields. This occurs because, in these materials, the magnetic moments of individual atoms or molecules do not completely cancel out. However, the attraction is much weaker than that in ferromagnetic materials. The degree of magnetization is proportional to the applied magnetic field, following the relation , where is the magnetic susceptibility (which is positive for paramagnetic substances).
2. No Retention of Magnetization: Unlike ferromagnetic materials, paramagnetic substances do not retain magnetization once the external magnetic field is removed. In these materials, the magnetic moments tend to align with the magnetic field when applied, but random thermal motion disrupts this alignment once the field is removed. This lack of residual magnetization is a key distinguishing feature of paramagnetism.
Other characteristics of paramagnetic substances include their temperature-dependent behavior. As the temperature increases, the thermal agitation of atoms or molecules increases, leading to a decrease in the alignment of the magnetic moments and, consequently, a reduction in the magnetization.
In summary, the two key properties of paramagnetic substances are their weak attraction to magnetic fields and the absence of permanent magnetization.

Steel and soft iron are both ferromagnetic materials, but they exhibit distinct differences in their magnetic properties. The key differences are:
1. Magnetic Retention:
- Steel has a high retentivity, meaning it retains its magnetization even after the external magnetic field is removed. This property makes steel suitable for permanent magnets.
- Soft iron, on the other hand, has a low retentivity, meaning it loses its magnetization quickly when the external magnetic field is removed. This makes soft iron ideal for use in electromagnets, where quick magnetization and demagnetization are required.
2. Magnetic Susceptibility:
- Steel generally has a lower magnetic susceptibility compared to soft iron. This means that steel is not as easily magnetized as soft iron.
- Soft iron has a higher magnetic susceptibility, making it more easily magnetized and demagnetized than steel. It responds more readily to external magnetic fields.
In conclusion, steel is used for permanent magnets due to its high retentivity, while soft iron is used for electromagnets due to its ability to be easily magnetized and demagnetized.