Magnetic Properties

Magnetization is defined as the magnetic moment per unit volume . It is given by: The magnetic moment has units of , and the volume has units of . Thus, the unit of magnetization is . Magnetic Susceptibility
% Correct Answer Definition: Magnetic susceptibility is the ratio of the magnetization of a material to the applied magnetic field strength. It indicates how easily a material becomes magnetized when exposed to a magnetic field. Unit: The unit of magnetic susceptibility is dimensionless (no unit). Dimensions: The dimensions of magnetic susceptibility are (dimensionless). % Solution Solution:
Magnetic susceptibility is defined as: where is the magnetization (in ) and is the magnetic field strength (in ). Since the ratio of two quantities with the same unit is dimensionless, the magnetic susceptibility has no unit.

Step 1: Magnetic field in a toroid.
A toroid is a solenoid bent into a circular shape. The magnetic field inside the toroid can be derived using Ampère's law, which states: where is the magnetic field, is an infinitesimal length element along the path of integration, and is the total current enclosed by the path.
Step 2: Apply Ampère's Law.
For a toroid, the path of integration is a circle of radius , and the current enclosed is , where is the number of turns and is the current through each turn. Solving for , we get:
Step 3: Conclusion.
The magnetic induction inside the toroid is , where is the radius of the toroid.

Step 1: Formula for magnetic flux.
Magnetic flux is given by the formula: where is the number of turns, is the magnetic field strength, and is the area of the coil.
Step 2: Substituting the values.
Given , , and , we calculate:
Step 3: Conclusion.
The magnetic flux linked with the coil is 100 Wb, corresponding to option (3).

Step 1: Induced EMF and Current.
The energy stored in the magnetic field of an inductor can be derived using Faraday’s law of induction. The induced emf in an inductor is related to the rate of change of current by: where is the inductance and is the rate of change of current.
Step 2: Power Supplied to the Inductor.
The power supplied to the inductor is the product of the current and the induced emf:
Step 3: Energy Stored in the Magnetic Field.
The energy stored in the magnetic field is the integral of the power over time: By integrating, we get: This is the expression for the energy stored in the magnetic field in terms of the current.

Step 1: Formula for the period of oscillation.
For a bar magnet performing angular oscillations in a magnetic field, the period is related to the magnetic moment , the mass , and the magnetic field by the formula: where is the moment of inertia of the bar magnet.
Step 2: Moment of inertia.
The moment of inertia of the bar magnet about its center of mass is given by: Substitute the given values: , , .
Step 3: Solve for the magnetic field .
Given that the period , we can substitute into the period formula: Squaring both sides and solving for :
Step 4: Conclusion.
The influencing magnetic field is .

Step 1: Understanding the force on a conductor in a magnetic field.
When a current-carrying conductor is placed in a magnetic field, a force is exerted on the conductor. This force depends on several factors, such as the current, the magnetic field strength, the length of the conductor, and the angle between the magnetic field and the current direction.
Step 2: Formula for the force.
The force on a conductor of length , carrying a current , in a magnetic field of strength , is given by: where is the angle between the direction of the magnetic field and the current.
Step 3: Conclusion.
The value of the force on a closed circuit in a magnetic field is determined by the equation .

Step 1: Formula for the force between two parallel wires.
The force per unit length between two parallel wires carrying currents and is given by: where is the permeability of free space, is the distance between the wires, and and are the currents in the wires.
Step 2: Given Data.
The force per unit length is , the distance between the wires is , and the currents in the wires are equal, so .
Step 3: Solve for the current.
Substituting the known values into the formula: Simplifying: