More Applications Of Ampere’S Law

Slide 1: More Applications of Ampere’s Law - Introduction

Ampere’s law is a powerful tool for calculating magnetic fields.
In the previous lecture, we discussed how to use Ampere’s law to calculate the magnetic field generated by a current-carrying wire.
In this lecture, we will explore more applications of Ampere’s law in various scenarios.
Ampere’s law relates the magnetic field around a closed loop to the electric current passing through the loop.
Let’s dive into some interesting examples of Ampere’s law in action.

Slide 2: Magnetic Field Inside a Solenoid

A solenoid is a long coil of wire with multiple turns.
When a current flows through a solenoid, it creates a magnetic field inside it.
The magnetic field inside a solenoid is approximately uniform.
The magnitude of the magnetic field inside a solenoid can be calculated using Ampere’s law.
The equation for the magnetic field inside a solenoid is given by: B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.

Slide 3: Magnetic Field Due to a Toroid

A toroid is a donut-shaped object with multiple turns of wire wrapped around it.
When a current flows through a toroid, it creates a magnetic field inside and outside the toroid.
The magnetic field inside a toroid is stronger compared to the magnetic field outside.
The magnitude of the magnetic field inside a toroid can also be calculated using Ampere’s law.
The equation for the magnetic field inside a toroid is given by: B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.

Slide 4: Magnetic Field Inside a Cylindrical Conductor

When a current flows through a cylindrical conductor, it creates a magnetic field around it.
The magnetic field inside a cylindrical conductor is inversely proportional to the distance from the axis of the conductor.
The magnitude of the magnetic field inside a cylindrical conductor can be calculated using Ampere’s law.
The equation for the magnetic field inside a cylindrical conductor is given by: B = (μ₀IR²) / (2πL²), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, R is the radius of the cylindrical conductor, and L is the distance from the axis.

Slide 5: Magnetic Field Due to a Long Straight Wire

When a current flows through a long straight wire, it creates a magnetic field around it.
The magnetic field due to a long straight wire decreases with the distance from the wire.
The magnitude of the magnetic field due to a long straight wire can be calculated using Ampere’s law.
The equation for the magnetic field due to a long straight wire is given by: B = (μ₀I) / (2πr), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, and r is the distance from the wire.

Slide 6: Magnetic Field Between Two Parallel Wires

When currents flow in two parallel wires, they create magnetic fields around them.
The magnetic field between two parallel wires is dependent on the direction and magnitude of the currents.
The magnitude of the magnetic field between two parallel wires can be calculated using Ampere’s law.
The equation for the magnetic field between two parallel wires is given by: B = (μ₀I₁I₂) / (2πd), where B is the magnetic field, μ₀ is the permeability of free space, I₁ and I₂ are the currents in the two wires, and d is the distance between the wires.

Slide 7: Magnetic Field Due to a Circular Wire Loop

When a current flows through a circular wire loop, it creates a magnetic field inside and outside the loop.
The magnetic field inside the loop is concentrated at the center and follows the shape of a magnetic dipole.
The magnitude of the magnetic field due to a circular loop can be calculated using Ampere’s law.
The equation for the magnetic field due to a circular loop is given by: B = (μ₀IR²) / (2(R² + x²)^3/2), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, R is the radius of the loop, and x is the distance from the center of the loop along its axis.

Slide 8: Magnetic Field Due to a Current-Carrying Arc

When a current flows through an arc of a circle, it creates a magnetic field around it.
The magnetic field due to a current-carrying arc is dependent on the shape of the arc and the distance from the arc.
The magnitude of the magnetic field due to a current-carrying arc can be calculated using Ampere’s law.
The equation for the magnetic field due to a current-carrying arc is more complex than other scenarios and involves integration.
In practical scenarios, the magnetic field due to a current-carrying arc is often approximated using simplified formulas based on symmetry.

Slide 9: Ampere’s Law and Magnetic Flux

Ampere’s law can also be used to relate the magnetic field and magnetic flux.
Magnetic flux represents the total magnetic field passing through a certain area.
The equation for the magnetic flux through a closed loop can be derived from Ampere’s law.
The equation for the magnetic flux through a closed loop is given by: Φ = B⋅A, where Φ is the magnetic flux, B is the magnetic field, and A is the area of the closed loop.
This relationship is known as the magnetic flux formula.

Slide 10: Summary

Ampere’s law is a powerful tool for calculating magnetic fields in various scenarios.
It allows us to determine the magnetic field inside a solenoid, toroid, cylindrical conductor, long straight wire, between two parallel wires, circular wire loop, and current-carrying arc.
The magnetic field calculations are based on the application of Ampere’s law and involve factors such as current, permeability of free space, radius, distance, and number of turns per unit length.
Ampere’s law can also be used to relate the magnetic field and magnetic flux through a closed loop.
Understanding and applying Ampere’s law will help us analyze and predict magnetic field behaviors in different situations.

Magnetic Field Due to a Current-Carrying Coil

A current-carrying coil, also known as a circular loop, creates a magnetic field around it.
The magnetic field due to a current-carrying coil depends on the number of turns in the coil and the current flowing through it.
The magnitude of the magnetic field at the center of the coil can be calculated using Ampere’s law.
The equation for the magnetic field at the center of a current-carrying coil is given by: B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.
Example: If a coil with 100 turns per meter carries a current of 2 Amperes, calculate the magnetic field at its center.

Ampere’s Law and Magnetic Field of a Straight Conductor

Ampere’s law can also be applied to calculate the magnetic field around a straight conductor.
When a current flows through a straight conductor, it creates a magnetic field that forms concentric circles around the conductor.
The magnitude of the magnetic field around a straight conductor can be determined using Ampere’s law.
The equation for the magnetic field around a straight conductor is given by: B = (μ₀I) / (2πr), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, and r is the distance from the conductor.
Example: Calculate the magnetic field at a distance of 5 cm from a straight conductor carrying a current of 3 Amperes.

Magnetic Field Inside and Outside a Toroidal Solenoid

A toroidal solenoid is a coil wound in the shape of a torus.
The magnetic field inside and outside a toroidal solenoid can be calculated using Ampere’s law.
Inside the solenoid, the magnetic field depends on the current and the number of turns per unit length.
The equation for the magnetic field inside a toroidal solenoid is given by: B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.
Outside the solenoid, the magnetic field is nearly zero.
Example: Find the magnetic field inside and outside a toroidal solenoid with 500 turns and a current of 4 Amperes.

Ampere’s Law and Magnetic Field of a Coaxial Cable

Ampere’s law can also be used to calculate the magnetic field in a coaxial cable.
A coaxial cable consists of two concentric conductors with opposite currents.
The magnetic field between the conductors can be determined using Ampere’s law.
The equation for the magnetic field in a coaxial cable is given by: B = (μ₀I₁) / (2πr₁) - (μ₀I₂) / (2πr₂), where B is the magnetic field, μ₀ is the permeability of free space, I₁ and I₂ are the currents in the two conductors, r₁ and r₂ are the distances from the respective conductors.
Example: Calculate the magnetic field at a distance of 2 cm from the inner conductor of a coaxial cable carrying currents of 5 Amperes and 3 Amperes respectively.

Ampere’s Law and Magnetic Field Inside a Cylindrical Conductor with Current

A cylindrical conductor with a current flowing through it creates a magnetic field inside and outside the conductor.
The magnetic field inside the conductor can be calculated using Ampere’s law.
The equation for the magnetic field inside a cylindrical conductor is given by: B = (μ₀IR²) / (2πL²), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, R is the radius of the conductor, and L is the distance from the axis.
Example: Determine the magnetic field at a distance of 10 cm from the axis of a cylindrical conductor with a radius of 2 cm carrying a current of 8 Amperes.

Ampere’s Law and Magnetic Field Inside and Outside a Current Sheet

A current sheet refers to a region with a uniform distribution of current.
The magnetic field inside and outside a current sheet can be calculated using Ampere’s law.
Inside the current sheet, the magnetic field is constant and depends on the current density.
The equation for the magnetic field inside a current sheet is given by: B = μ₀j, where B is the magnetic field, μ₀ is the permeability of free space, and j is the current density.
Outside the current sheet, the magnetic field is zero.
Example: A current sheet with a current density of 4 Amperes/m² is present. Calculate the magnetic field inside the sheet.

Magnetic Field Due to a Current Loop

A current loop, also known as a circular coil, creates a magnetic field around it.
The magnetic field due to a current loop depends on its radius and the distance from the loop.
The magnitude of the magnetic field at a point on the axis of a current loop can be calculated using Ampere’s law.
The equation for the magnetic field at a point on the axis of a current loop is given by: B = (μ₀IR²) / (2(R² + x²)^3/2), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, R is the radius of the loop, and x is the distance from the center of the loop along its axis.
Example: A current loop of radius 5 cm carries a current of 6 Amperes. Calculate the magnetic field at a distance of 8 cm from the center along its axis.

Magnetic Field Due to a Solenoid with Iron Core

When a current flows through a solenoid with an iron core, the magnetic field increases significantly.
The presence of the iron core enhances the magnetic field due to the phenomenon of magnetization.
The magnitude of the magnetic field inside a solenoid with an iron core can be calculated using Ampere’s law, taking into account the permeability of the iron.
Example: A solenoid with 200 turns per meter carries a current of 3 Amperes. The iron core has a relative permeability of 500. Calculate the magnetic field inside the solenoid.

Magnetic Field Due to a Straight Wire Bent into a Loop

A straight wire bent into a loop creates a magnetic field inside and outside the loop.
The magnetic field inside the loop is nearly uniform, while outside the loop, it follows a pattern similar to a magnetic dipole.
The magnitude of the magnetic field due to a straight wire loop can be calculated using Ampere’s law.
Example: Consider a circular loop formed by bending a straight wire. The loop has a radius of 4 cm and carries a current of 5 Amperes. Calculate the magnetic field at the center of the loop.

Magnetic Field Due to a Current-Carrying Helical Coil

A helical coil, also known as a solenoid, creates a magnetic field inside and outside the coil.
The magnetic field inside a helical coil is uniform and depends on the number of turns, current, and pitch length.
The magnitude of the magnetic field inside a helical coil can be calculated using Ampere’s law.
Example: A helical coil with a pitch length of 3 cm, having 100 turns per meter, carries a current of 4 Amperes. Calculate the magnetic field inside the coil.

Magnetic Field Due to a Cylindrical Capacitor

A cylindrical capacitor consists of two concentric cylinders carrying opposite charges.
When a current flows through a cylindrical capacitor, it creates a magnetic field around it.
The magnetic field due to a cylindrical capacitor can be calculated using Ampere’s law.
The equation for the magnetic field due to a cylindrical capacitor is given by: B = (μ₀Q) / (2πr), where B is the magnetic field, μ₀ is the permeability of free space, Q is the total charge, and r is the distance from the axis.
Example: Calculate the magnetic field at a distance of 2 cm from the axis of a cylindrical capacitor with a total charge of 6 μC.

Magnetic Field Due to a Current-Carrying Disk

A disk-shaped conductor with a current flowing through it creates a magnetic field around it.
The magnetic field due to a current-carrying disk depends on its radius and the distance from the disk.
The magnitude of the magnetic field at a point on the axis of a current-carrying disk can be calculated using Ampere’s law.
The equation for the magnetic field at a point on the axis of a current-carrying disk is given by: B = (μ₀I) / (2R²), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, and R is the radius of the disk.
Example: A disk with a radius of 8 cm carries a current of 10 Amperes. Calculate the magnetic field at a distance of 12 cm from the center along its axis.

Magnetic Field Due to a Current-Carrying Coil with Iron Core

A current-carrying coil with an iron core creates a magnetic field around it.
The presence of the iron core significantly increases the magnetic field due to its high permeability.
The magnitude of the magnetic field at the center of a coil with an iron core can be calculated using Ampere’s law, taking into account the permeability of the iron.
Example: A coil with 500 turns carries a current of 3 Amperes. The iron core has a relative permeability of 5000. Calculate the magnetic field at the center of the coil.

Magnetic Field Due to a Current-Carrying Conductor - Right Hand Rule

The right-hand rule is a technique used to determine the direction of the magnetic field around a current-carrying conductor.
To apply the right-hand rule, point your right thumb in the direction of the current flow.
The direction of the curled fingers represents the direction of the magnetic field.
Example: If the current flow is from top to bottom in a vertical wire, use the right-hand rule to determine the direction of the magnetic field around the wire.

Ampere’s Law and Magnetic Field of a Rectangular Loop

Ampere’s law can also be used to calculate the magnetic field around a rectangular loop.
A rectangular loop carrying a current creates a magnetic field inside and outside the loop.
The magnitude of the magnetic field at a point along the axis of a rectangular loop can be calculated using Ampere’s law.
The equation for the magnetic field at a point on the axis of a rectangular loop is given by: B = (μ₀Iw) / (4πd), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, w is the width of the loop, and d is the distance from the center of the loop along its axis.
Example: A rectangular loop with a width of 10 cm carries a current of