MAGNETIC  EFFECT   OF  ELECTRIC  CURRENT: Short notes numericals and equations

Introduction:

Magnetism is one of the most intriguing phenomena in nature, and it has been studied by scientists for centuries. Magnetic effects are pervasive in our daily lives, from the simple magnets on our refrigerator doors to the more complex magnetic fields that are generated by the Earth's core. In this article, we will explore the magnetic effect and related topics, including the fundamental equations that describe magnetic behavior.

Magnetic Field:

A magnetic field is a region in space where magnetic forces are observed. The magnetic field is a vector field that describes the direction and magnitude of the force experienced by a magnetic dipole placed in the field. Magnetic fields are produced by moving electric charges, such as electrons. The strength of the magnetic field at a given point is given by the magnetic field strength or magnetic flux density, B. The unit of B is the tesla (T), named after the famous inventor Nikola Tesla.

Magnetic Force:

A magnetic field exerts a force on a magnetic dipole placed within it. The magnetic force is given by the following equation:

F = qv x B

Where F is the force vector, q is the charge of the particle, v is the velocity vector of the particle, and B is the magnetic field vector. The direction of the force vector is perpendicular to both the velocity vector and the magnetic field vector. The magnitude of the force is given by F = qvBsinθ, where θ is the angle between the velocity vector and the magnetic field vector.

Magnetic Dipole Moment:

A magnetic dipole is a pair of equal and opposite magnetic charges separated by a distance d. The magnetic dipole moment, m, is a measure of the strength and direction of the magnetic dipole. It is defined as the product of the magnitude of the magnetic charge and the separation between the charges, i.e., m = qd.

Magnetic Induction:

When a magnetic field is applied to a material, it induces a magnetization within the material. The induced magnetization is proportional to the applied magnetic field, and the proportionality constant is called the magnetic susceptibility, χ. The magnetic susceptibility is a dimensionless quantity and is usually expressed in units of volume per unit volume (e.g., m3/kg). The magnetic susceptibility is positive for diamagnetic materials, negative for paramagnetic materials, and very large for ferromagnetic materials.

Magnetic Materials:

Materials can be classified into three categories based on their magnetic properties: diamagnetic, paramagnetic, and ferromagnetic. Diamagnetic materials are weakly repelled by magnetic fields and have a negative magnetic susceptibility. Examples of diamagnetic materials include copper, silver, and gold. Paramagnetic materials are weakly attracted by magnetic fields and have a positive magnetic susceptibility. Examples of paramagnetic materials include aluminum, platinum, and manganese. Ferromagnetic materials are strongly attracted by magnetic fields and have a very large magnetic susceptibility. Examples of ferromagnetic materials include iron, nickel, and cobalt.

Maxwell's Equations:

Maxwell's equations describe the behavior of electric and magnetic fields in space. They are a set of four partial differential equations that relate the electric and magnetic fields to their sources. The equations are named after James Clerk Maxwell, who formulated them in the 19th century. The equations are as follows:

• Gauss's law for electric fields:

∇ ⋅ E = ρ/ε0

Where E is the electric field, ρ is the charge density, and ε0 is the electric constant.

• Gauss's law for magnetic fields:

∇ ⋅ B = 0

Where B is the magnetic field.

• Faraday's law of electromagnetic induction:

∇ × E = - ∂B/∂t

Where ∂B/∂t is the time derivative of the magnetic field, and the cross product denotes the curl of the electric field.

• Ampere's law with Maxwell's correction:

∇ × B = μ0(j + ε0∂E/∂t)

Where j is the current density, μ0 is the magnetic constant, and the curl of the magnetic field is equal to the sum of the current density and the time derivative of the electric field.

These four equations describe the behavior of electric and magnetic fields in space and their interaction with charges and currents. They are fundamental to understanding the behavior of electromagnetic waves, which are the basis of many modern technologies.

Magnetic Materials and Hysteresis:

Magnetic materials exhibit a phenomenon called hysteresis, where the magnetization of the material lags behind changes in the applied magnetic field. Hysteresis is caused by the alignment of magnetic domains within the material. These domains are regions of the material where the magnetic moments of the atoms are aligned in the same direction. When a magnetic field is applied, the domains align with the field, and the material becomes magnetized. When the field is removed, the domains retain their alignment, and the material remains magnetized. The amount of magnetization depends on the strength of the applied magnetic field.

Hysteresis can be represented graphically by a hysteresis loop. The loop shows the relationship between the magnetic field strength and the magnetization of the material. As the magnetic field strength increases, the magnetization of the material increases until it reaches saturation. When the field is reduced, the magnetization lags behind, and the material retains some magnetization even when the field is zero.

Applications of Magnetic Effect:

Magnetic effects have many applications in science and technology. Here are some of the most important applications:

• Magnetic storage: Magnetic materials are used to store data in hard drives, floppy disks, and magnetic tape. The magnetic domains within the material represent the 0s and 1s of digital data.

• Magnetic levitation: Magnetic levitation, or maglev, uses magnetic fields to suspend an object in the air. Maglev trains are a promising form of transportation that can travel at high speeds with minimal friction.

• Magnetic resonance imaging (MRI): MRI is a medical imaging technique that uses magnetic fields and radio waves to create images of the inside of the body. MRI is non-invasive and does not use ionizing radiation, making it a safer alternative to X-rays and CT scans.

• Electric motors: Electric motors use magnetic fields to convert electrical energy into mechanical energy. The interaction between the magnetic field and the current in the motor produces a force that rotates the motor.

Conclusion:

In conclusion, the magnetic effect is a fascinating phenomenon that has many important applications in science and technology. Magnetic fields, forces, and materials can be described by a set of fundamental equations known as Maxwell's equations. Understanding the magnetic effect is essential for many fields, including physics, engineering, and medicine. By exploring the properties and behavior of magnetic fields, we can develop new technologies that improve our lives and advance our understanding of the universe.

Numericals for magnetic effects

• A wire carrying a current of 2 A is placed in a magnetic field of 0.5 T. What is the force on the wire if it is perpendicular to the field?

• A coil with 100 turns is placed in a magnetic field of 0.2 T. If the coil has an area of 0.1 m2 and is perpendicular to the field, what is the magnetic flux through the coil?

• A solenoid with 500 turns has a length of 0.2 m and a radius of 0.01 m. If a current of 5 A flows through the solenoid, what is the magnetic field at the center of the solenoid?

• A charged particle with a charge of 2 C and a velocity of 5 m/s enters a magnetic field of 0.1 T at an angle of 30° to the field. What is the force on the particle?

• A transformer has 1000 turns in the primary coil and 500 turns in the secondary coil. If the voltage in the primary coil is 100 V, what is the voltage in the secondary coil?

• A wire carrying a current of 3 A is wrapped around a soft iron core with a permeability of 1000. If the coil has 100 turns and a radius of 0.05 m, what is the magnetic field inside the core?

• A wire carrying a current of 4 A is bent into a loop with a radius of 0.1 m. If the loop is placed in a magnetic field of 0.3 T, what is the torque on the loop if the field is perpendicular to the plane of the loop?

• A particle with a charge of -1 μC and a velocity of 10 m/s enters a magnetic field of 0.5 T at an angle of 45° to the field. What is the radius of the particle's path?

• A wire carrying a current of 2 A is placed in a magnetic field of 0.4 T. If the wire is perpendicular to the field and has a length of 0.2 m, what is the force on the wire?

• A transformer has 500 turns in the primary coil and 2000 turns in the secondary coil. If the current in the primary coil is 5 A, what is the current in the secondary coil?

• A solenoid with 1000 turns has a length of 0.3 m and a radius of 0.02 m. If a current of 10 A flows through the solenoid, what is the magnetic field at a point on the axis of the solenoid 0.1 m from the center?

• A charged particle with a charge of 3 C and a velocity of 2 m/s enters a magnetic field of 0.2 T at an angle of 60° to the field. What is the force on the particle?

• A wire carrying a current of 6 A is wrapped around a soft iron core with a permeability of 500. If the coil has 50 turns and a radius of 0.1 m, what is the magnetic field inside the core?

• A wire carrying a current of 5 A is bent into a loop with a radius of 0.2 m. If the loop is placed in a magnetic field of 0.2 T, what is the torque on the loop if the field is perpendicular to the plane of the loop?

• A transformer has 200 turns in the primary coil and 1000 turns in the secondary coil. If the voltage in the primary coil is 120 V, what is the voltage in the secondary coil?

• A particle with a charge of 2 μC and a velocity of 8 m/s enters a magnetic field of 0.1 T at an angle of 90° to the field. What is the force on the particle?

• A solenoid with 200 turns has a length of 0.4 m and a radius of 0.03 m. If a current of 2 A flows through the solenoid, what is the magnetic field at a point on the axis of the solenoid 0.1 m from the center?

• A wire carrying a current of 7 A is placed in a magnetic field of 0.6 T. If the wire is perpendicular to the field and has a length of 0.3 m, what is the force on the wire?

• A transformer has 500 turns in the primary coil and 100 turns in the secondary coil. If the voltage in the secondary coil is 20 V, what is the voltage in the primary coil?

• A charged particle with a charge of -3 C and a velocity of 6 m/s enters a magnetic field of 0.3 T at an angle of 30° to the field. What is the radius of the particle's path?

• A wire carrying a current of 8 A is wrapped around a soft iron core with a permeability of 200. If the coil has 20 turns and a radius of 0.2 m, what is the magnetic field inside the core?

• A wire carrying a current of 6 A is bent into a loop with a radius of 0.3 m. If the loop is placed in a magnetic field of 0.5 T, what is the torque on the loop if the field is perpendicular to the plane of the loop?

• A transformer has 2000 turns in the primary coil and 1000 turns in the secondary coil. If the current in the primary coil is 2 A, what is the current in the secondary coil?

• A solenoid with 100 turns has a length of 0.1 m and a radius of 0.01 m. If a current of 3 A flows through the solenoid, what is the magnetic field at the center of the solenoid?

• A charged particle with a charge of 4 μC and a velocity of 4 m/s enters a magnetic field of 0.2 T at an angle of 45° to the field. What is the force on the particle?

These numerical problems cover a range of concepts related to magnetic effects, including magnetic field, magnetic force, magnetic flux, solenoids, transformers, and charged particle motion in magnetic fields.