Introduction to magnetization

• Magnetization is the process of aligning the magnetic domains in a material to create a magnetic field.
• It can be achieved by placing the material in a strong external magnetic field or by passing an electric current through it.
• Magnetization results in the material becoming a magnet or a magnetic material.

Definition of Ampere’s law

• Ampere’s law relates the magnetic field around a closed loop to the electric current passing through the loop.
• It is derived from the principle of conservation of charge and the magnetic field generated by moving charges.
• Ampere’s law states that the line integral of the magnetic field around a closed loop is equal to the product of the permeability of free space and the total current passing through the loop.

Equation of Ampere’s law

• The mathematical form of Ampere’s law is given by: ΣB⃗ ⋅ dl⃗ = μ0Ienc.
• ΣB⃗ represents the vector sum of the magnetic field at each point on the closed loop.
• dl⃗ is the infinitesimal element of the path along the loop.
• μ0 is the permeability of free space and has a value of 4π × 10^(-7) Tm/A.
• Ienc is the net current passing through the loop enclosed by the closed path.

Explanation of the magnetic field produced by a current-carrying conductor

• When an electric current passes through a conductor, it produces a magnetic field around it.
• The magnetic field produced is circular and follows the right-hand rule: if you curl your right-hand fingers around the conductor in the direction of the current, your thumb points in the direction of the magnetic field.
• The magnetic field lines around a current-carrying conductor form concentric circles around the conductor.

Magnetic field strength and direction around a straight conductor

• The strength of the magnetic field around a straight conductor is directly proportional to the current passing through it.
• The field strength decreases with increasing distance from the conductor.
• The direction of the magnetic field can be determined using the right-hand rule.

Calculation of magnetic field using Ampere’s law

• Ampere’s law can be used to calculate the magnetic field at a given point around a current-carrying conductor.
• Choose a closed loop that encloses the conductor.
• Calculate the net current passing through the loop.
• Evaluate the line integral of the magnetic field along the loop using Ampere’s law.
• Determine the magnetic field strength and direction at the given point.

Application of Ampere’s law in solenoids

• A solenoid is a coil of wire wound in the shape of a cylinder.
• When a current passes through a solenoid, it produces a strong and uniform magnetic field inside the solenoid.
• The strength of the magnetic field inside a solenoid is directly proportional to the current passing through it and the number of turns per unit length of the solenoid.
• The magnetic field outside the solenoid is negligible.

Magnetic field inside and outside a solenoid

• Inside a solenoid, the magnetic field lines are parallel and closely spaced.
• The magnetic field inside a solenoid is uniform and can be calculated using Ampere’s law.
• Outside a solenoid, the magnetic field is negligible due to the cancellation of magnetic fields from different turns.

Explanation of the magnetic field produced by a loop of wire using Ampere’s law

• When a loop of wire carries a current, it produces a magnetic field at its center and around it.
• The magnetic field produced by a current loop is similar to that produced by a small bar magnet.
• The direction of the magnetic field can be determined using the right-hand rule.

Determination of magnetic field inside and outside a current loop using Ampere’s law

• Ampere’s law can be used to determine the magnetic field strength inside and outside a current loop.
• Choose a closed loop that encloses the current loop.
• Calculate the net current passing through the loop.
• Evaluate the line integral of the magnetic field along the loop using Ampere’s law.
• Determine the magnetic field strength and direction inside and outside the current loop.

11. Application of Ampere’s law in current loops

• Ampere’s law can be applied to calculate the magnetic field produced by current loops.
• For a circular loop, the magnetic field at the center is given by B = μ0I/2R, where I is the current and R is the radius of the loop.
• The magnetic field strength decreases as we move away from the center of the loop.
• The direction of the magnetic field is perpendicular to the plane of the loop.
• The magnetic field lines form concentric circles around the loop.

12. Calculation of magnetic field due to multiple current loops

• For multiple current loops, the magnetic field at a point due to each loop can be calculated separately using Ampere’s law.
• The magnetic field at a point due to multiple loops is found by summing up the magnetic fields from each loop using vector addition.
• The direction of the magnetic field is determined by the right-hand rule.
• Equations for the magnetic field produced by some commonly encountered current loop configurations are available, such as circular, square, and rectangular loops.

13. Magnetic field inside a toroid

• A toroid is a doughnut-shaped object formed by winding a wire around a circular ring.
• The magnetic field inside a toroid is nearly uniform and directed along the axis of the toroid.
• Ampere’s law can be used to derive an expression for the magnetic field inside a toroid: B = μ0nI, where n is the number of turns per unit length and I is the current passing through the toroid.
• The magnetic field strength inside the toroid is independent of the distance from the axis.

14. Magnetic field outside a toroid

• Outside the toroid, the magnetic field is negligible.
• The magnetic field lines inside the toroid stay confined within the toroidal structure.
• This makes toroids useful for many applications, such as in transformers and inductor coils.

15. Ampere’s law and the displacement current

• Ampere’s law was originally formulated without considering the effects of changing electric fields.
• However, it was later modified to include the displacement current, which accounts for the changing electric fields.
• The displacement current is a term in Ampere’s law that relates the change in electric flux to the change in magnetic field.
• The modified version of Ampere’s law is known as Ampere-Maxwell’s law.

16. Application of Ampere-Maxwell’s law in electromagnetism

• Ampere-Maxwell’s law is used to understand the relationship between changing electric fields and magnetic fields.
• It is essential for studying electromagnetic waves, such as light.
• This law allows us to explain various electromagnetic phenomena, including induction and electromagnetic radiation.

17. Inductive charging and Ampere’s law

• Ampere’s law is crucial in explaining the phenomenon of inductive charging.
• Inductive charging occurs when a magnetic field from a current-carrying coil induces an electric current in another nearby coil.
• The induced current follows Ampere’s law and generates a magnetic field that opposes the original magnetic field.
• This principle is used in wireless charging technologies and transformers.

18. Ampere’s law in circuit analysis

• Ampere’s law is often used in circuit analysis to calculate the magnetic field and magnetic flux in complex circuits.
• It helps in understanding the behavior of magnetic materials, transformers, and inductors.
• By applying Ampere’s law, we can solve circuits containing magnetic elements and determine various parameters, such as currents and voltages.

19. Magnetic field due to a straight wire using Ampere’s law

• Ampere’s law can also be applied to calculate the magnetic field due to a long, straight wire.
• Using a circular path around the wire, we can determine that the magnetic field strength is inversely proportional to the distance from the wire and directly proportional to the current passing through it.
• The direction of the magnetic field can be determined using the right-hand rule.

20. Ampere’s law and magnetic monopoles

• Ampere’s law predicts that magnetic field lines form closed loops, indicating that there are no magnetic monopoles (an isolated magnetic charge).
• Unlike electric charges, which can exist in isolation as positive or negative charges, magnetic charges have not been observed separately.
• Some theories and models propose the existence of magnetic monopoles, but they have not been experimentally confirmed yet. Sure, here are slides 21 to 30:

21. Ampere’s law and magnetic materials

• Ampere’s law applies not only to current-carrying conductors but also to magnetic materials.
• Magnetic materials, such as iron or nickel, exhibit a strong response to magnetic fields.
• When a magnetic field is applied to these materials, they become magnetized and produce their own magnetic fields.
• Ampere’s law can be used to analyze and determine the magnetic fields produced by magnetized materials.

22. Calculation of magnetic field inside a magnetic material

• When a magnetic field is applied to a magnetic material, it aligns the magnetic domains within the material.
• Ampere’s law can be applied to calculate the magnetic field strength inside the material by considering the net magnetization current passing through the loop.
• The magnetic field strength inside a magnetic material depends on its permeability, which is a measure of how easily it can be magnetized.

23. Hysteresis curve and magnetic materials

• In magnetic materials, the relationship between magnetic field strength (H) and magnetic induction (B) is not linear.
• The hysteresis curve represents this relationship and shows the magnetization behavior of the material.
• Hysteresis refers to the lagging effect observed when the magnetic field is applied, reversed, and then removed.
• The area inside the hysteresis loop represents the energy lost as heat during the magnetization process.

24. Ferromagnetic materials and Ampere’s law

• Ferromagnetic materials, such as iron, cobalt, and nickel, exhibit strong magnetic properties.
• Ampere’s law is especially useful in analyzing and understanding the behavior of ferromagnetic materials.
• The high permeability of these materials makes them valuable in various applications, including electromagnets and transformers.
• Ampere’s law can be used to calculate the magnetic field strength and magnetic flux density inside and around ferromagnetic materials.

25. Ampere’s law in electromagnets

• Electromagnets consist of a coil of wire wound around a ferromagnetic core.
• When an electric current passes through the coil, it creates a strong magnetic field due to the combined effects of Ampere’s law and the magnetic properties of the core material.
• Electromagnets are widely used in various applications, including electric motors, generators, and magnetic resonance imaging (MRI) machines.

26. Calculation of magnetic field in an electromagnet

• Ampere’s law can be applied to calculate the magnetic field strength inside and around an electromagnet.
• Consideration must be given to the number of turns in the coil, the current passing through it, and the magnetic properties of the core material.
• The magnetic field strength can be optimized by choosing an appropriate core material and adjusting the number of turns and current.

27. Practical examples of Ampere’s law

• Ampere’s law has practical applications in various fields of science, engineering, and technology.
• It is essential for understanding and designing electrical circuits and devices, such as transformers, inductors, and solenoids.
• Ampere’s law also plays a role in electromagnetism, electromagnetic radiation, and wireless charging technologies.
• It provides a theoretical basis for the behavior and manipulation of magnetic fields.

28. Ampere’s law in the study of magnetism and electromagnetic waves

• Ampere’s law is a fundamental tool in the study of magnetism and its relationship to electric fields.
• It helps in understanding the behavior of magnets, magnetized materials, and electromagnetic waves.
• The application of Ampere’s law in electromagnetism has led to significant technological advancements, including the development of medical imaging, wireless communication, and power transmission systems.

29. Limitations and extensions of Ampere’s law

• Ampere’s law is a powerful tool but has certain limitations.
• It is based on certain assumptions, such as the absence of changing electric fields and perfectly conducting materials.
• The modified version of Ampere’s law, known as Ampere-Maxwell’s law, addresses these limitations by including the displacement current term.
• Ampere-Maxwell’s law provides a more comprehensive description of the relationship between changing electric and magnetic fields.

30. Summary of Ampere’s law and its applications

• Ampere’s law is a fundamental principle in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop.
• It has various practical applications in the analysis and design of electrical circuits, devices, and magnetized materials.
• Ampere’s law was further developed into Ampere-Maxwell’s law to account for changing electric fields and is essential for understanding electromagnetic waves and phenomena.