Generalization of Ampere’s law and its applications - Application of Ampere’s law (Examples)

Ampere’s Law in Integral Form:

• States that the line integral of the magnetic field around any closed loop is equal to the permeability times the current passing through the loop.
• Mathematically, ∮ B⋅ds = μ₀I, where B is the magnetic field, ds is the differential length element, and I is the current enclosed by the loop. Applications of Ampere’s Law:

1. Solenoid:
   • A long, straight coil of wire with multiple loops closely wound together.
   • B-field inside a solenoid is proportional to the current passing through it.
   • B = μ₀nI, where n is the number of turns per unit length.

2. Toroid:
   • A hollow circular cylinder with multiple turns of wire wound around it.
   • B-field inside a toroid is uniform and parallel to the axis.
   • B = μ₀nI, where n is the number of turns per unit length.

3. Magnetic field inside and outside a current-carrying wire:
   • Inside the wire, B = μ₀I/2πr, where r is the distance from the wire.
   • Outside the wire, B = μ₀I/2πR, where R is the radius of the wire.

4. Magnetic field due to a long straight current-carrying wire:
   • B = μ₀I/2πr, where r is the distance from the wire.
   • Magnetic field lines form concentric circles around the wire.

5. Magnetic field due to a current loop:
   • At the center of the loop, B = μ₀I/2R, where R is the radius of the loop.
   • The magnetic field is perpendicular to the plane of the loop.

6. Ampere’s law for a loop carrying multiple currents:
   • If there are multiple currents passing through a closed loop, the net magnetic field is the vector sum of the magnetic fields due to each current.

7. Applications in electromagnets:
   • Electromagnets are made by winding wire in the shape of a coil around a magnetic core.
   • The magnetic field strength can be controlled by changing the number of turns or the current passing through the coil.

8. Applications in magnetic field measurement:
   • Ampere’s law provides a method to calculate the magnetic field at a specific point due to a current-carrying element or system.

9. Field due to parallel current-carrying conductors:
   • Magnetic field lines around parallel wires carrying currents in the same direction attract each other, and in opposite directions repel each other.

10. Applications in transformers:

• Transformers use the principle of electromagnetic induction to change the voltage level of an alternating current.
• Ampere’s law is used in the design and calculation of the magnetic field in the transformer coils.

Generalization of Ampere’s Law

• Ampere’s law can be generalized by considering the displacement current.
• The displacement current is a term added to Ampere’s law to account for the time-varying electric field.
• The modified form of Ampere’s law is given by ∮ B⋅ds = μ₀(I + ε₀ dΦE/dt), where ε₀ is the permittivity of free space and dΦE/dt is the rate of change of electric flux through the loop.
• This modified form is known as Maxwell’s equation.

Magnetic Field around a Current-Carrying Loop

• The magnetic field due to a current-carrying loop can be calculated using Ampere’s law.
• For a circular loop of radius R and current I, the magnetic field at a point on the axis of the loop is given by B = (μ₀IR²)/(2(R²+z²)^(3/2)), where z is the distance from the center of the loop along the axis.
• The magnetic field is maximum at the center of the loop and decreases with increasing distance.

Magnetic Field due to a Current Sheet

• Consider an infinite sheet carrying a uniform current density (J).
• The magnetic field at a point above the sheet is given by B = μ₀J.
• The magnetic field is parallel to the sheet and its magnitude is constant.
• The direction of the magnetic field can be determined using the right-hand rule.

Magnetic Field due to a Circular Current Loop

• The magnetic field at the center of a circular current loop is given by B = (μ₀nI)/2, where n is the number of turns per unit length.
• The magnetic field is perpendicular to the plane of the loop.
• The magnetic field inside the loop is uniform, while outside the loop it decreases with distance.

Magnetic Field due to a Solenoid

• A solenoid is a long, straight coil of wire with multiple turns closely wound together.
• The magnetic field inside a solenoid is uniform and parallel to the axis of the solenoid.
• The magnetic field outside the solenoid is negligible.
• The magnetic field strength inside a solenoid is given by B = μ₀nI, where n is the number of turns per unit length.

Magnetic Field due to a Toroid

• A toroid is a hollow circular cylinder with multiple turns of wire wound around it.
• The magnetic field inside a toroid is uniform and parallel to the axis.
• The magnetic field outside the toroid is negligible.
• The magnetic field strength inside a toroid is given by B = μ₀nI, where n is the number of turns per unit length.

Magnetic Field due to a Current-Carrying Wire

• Inside a current-carrying wire, the magnetic field is given by B = (μ₀I)/(2πr), where r is the distance from the wire.
• The magnetic field forms concentric circles around the wire.
• Outside the wire, the magnetic field decreases with increasing distance and is given by B = (μ₀I)/(2πR), where R is the radius of the wire.

Forces on Current-Carrying Conductors

• When two parallel conductors carrying currents in the same direction, they attract each other.
• When two parallel conductors carrying currents in opposite directions, they repel each other.
• The force per unit length (F) between two parallel conductors is given by F = (μ₀I₁I₂L)/(2πd), where I₁ and I₂ are the currents, L is the length of the conductors, and d is the distance between them.

Magnetic Field Inside a Straight Current-Carrying Conductor

• According to Ampere’s law, the magnetic field inside a straight current-carrying conductor is circular and perpendicular to the direction of current flow.
• The magnetic field strength at a distance r from the conductor is given by B = (μ₀I)/(2πr), where I is the current and r is the distance.

Magnetic Field due to Multiple Current-Carrying Wires

• When a closed loop encloses multiple current-carrying conductors, the net magnetic field at a point is the vector sum of the magnetic fields due to each current.
• The direction of the magnetic field can be determined using the right-hand rule.
• The strength of the magnetic field at a point can be calculated using the superposition principle.

21. Magnetic Field due to a Coaxial Cable

• A coaxial cable consists of an inner conductor, an insulating layer, a metallic shield, and an outer conductor.
• The magnetic field inside a coaxial cable is given by B = (μ₀I)/(2πρ), where I is the current and ρ is the distance from the axis of the cable.
• The magnetic field strength decreases as we move away from the axis.

22. Magnetic Field due to a Circular Coil

• The magnetic field at the center of a circular coil carrying current I is given by B = (μ₀nI)/2, where n is the number of turns per unit length.
• The magnetic field is perpendicular to the plane of the coil.
• The magnetic field at a point on the axis of the coil is given by B = (μ₀nIR²)/(2(R²+z²)^(3/2)), where R is the radius of the coil and z is the distance from the center of the coil along the axis.

23. Magnetic Field due to a Helical Coil

• A helical coil is a coil with a wire wound in a spiral shape.
• The magnetic field at the center of a helical coil carrying current I is given by B = (μ₀nI)/(2√(R² + h²)), where n is the number of turns per unit length, R is the radius of the coil, and h is the pitch (vertical distance between consecutive turns).
• The magnetic field is parallel to the axis of the coil.

24. Magnetic Field due to a Current Loop in an External Magnetic Field

• When a current loop is placed in an external magnetic field, it experiences a torque.
• The torque experienced by the loop is given by τ = μ₀IA sinθ, where I is the current, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop.

25. Ampere’s Law for a Closed Surface

• Ampere’s law can be applied to closed surfaces as well.
• The integral form of Ampere’s law for a closed surface is ∮ B⋅dA = μ₀Φ, where B is the magnetic field, dA is the differential area element, and Φ is the magnetic flux through the closed surface.

26. Calculation of Magnetic Field using Ampere’s Law

• Ampere’s law can be used to calculate the magnetic field of symmetric current distributions.
• By choosing an appropriate closed loop, we can simplify the integration and calculate the magnetic field easily.

27. Magnetic Field due to a Long Straight Wire using Ampere’s Law

• Ampere’s law can be used to calculate the magnetic field due to a long straight wire.
• By choosing a circular closed loop centered on the wire, we can calculate the magnetic field using the equation B = (μ₀I)/(2πr), where r is the distance from the wire.

28. Magnetic Field due to a Current-Carrying Loop using Ampere’s Law

• Ampere’s law can also be used to calculate the magnetic field due to a current-carrying loop.
• By choosing a circular closed loop around the loop, we can calculate the magnetic field using the equation B = (μ₀nI)/2, where n is the number of turns per unit length.

29. Magnetic Field due to a Solenoid using Ampere’s Law

• Ampere’s law is particularly useful in calculating the magnetic field inside a solenoid.
• By choosing a rectangular closed loop that contains the solenoid, we can calculate the magnetic field using the equation B = μ₀nI, where n is the number of turns per unit length.

30. Application of Ampere’s Law in Electromagnetic Induction

• Ampere’s law is one of the fundamental principles used in understanding electromagnetic induction.
• By analyzing the change in magnetic field through a closed loop, we can determine the induced electromotive force (emf) in a circuit.
• This is the basic principle behind the working of generators and transformers.