Generalization Of Ampere’S Law And Its Applications - Derivation Of Ampere’S Law For Arbitary Current Carrying Loop

Slide 1

Topic: Generalization of Ampere’s law and its applications
Subtopic: Derivation of Ampere’s law for arbitrary current carrying loop

Slide 2

Ampere’s law allows us to calculate the magnetic field around a current-carrying wire or a straight conductor.
It relates the magnetic field (B) to the current (I) and the distance from the wire (r).
However, Ampere’s law is limited to situations with symmetry and only works for wires with a simple shape.

Slide 3

In certain cases, we need to calculate the magnetic field for a more complex current carrying loop.
For this purpose, we can generalize Ampere’s law to include arbitrary loops by introducing the concept of line integration.
- Line integration allows us to integrate the magnetic field over the closed path of the loop.

Slide 4

The general form of Ampere’s law for an arbitrary current carrying loop is given by:
- ∮ B · dl = μ₀Iᵢₙ,
- where ∮ represents line integration around the loop, B is the magnetic field, dl is an infinitesimal element of length along the loop, μ₀ is the magnetic constant (4π × 10⁻⁷ Tm/A), and Iᵢₙ is the total current enclosed by the loop.

Slide 5

Let’s derive this formula for an arbitrary current carrying loop.
Consider a loop with a small element dl located at a position vector r along the loop.
The magnetic field due to this element at a distant point P can be expressed as d𝐵 = (μ₀/4π) (𝐈 d𝐥 × 𝟏/r²), where d𝐥 is the small element of current and r is the distance from dl to P.

Slide 6

By using the right-hand rule, we find that the magnetic field d𝐵 is perpendicular to both dl and r.
Therefore, we can write the equation as d𝐵 = (μ₀/4π) (𝐈 d𝐥 sinθ/r²), where θ is the angle between dl and r.

Slide 7

We need to determine the net magnetic field B at point P due to all the elements dl around the loop.
To find the net field, we integrate d𝐵 along the entire loop using line integration represented as ∮ d𝐵.
By definition, line integration involves integrating a vector quantity along a closed path.

Slide 8

By integrating ∮ d𝐵, we get ∮ 𝐵 · dl = ∮ (μ₀/4π) (𝐈 d𝐥 sinθ/r²) · dl, where we substitute d𝐵 by its expression.
However, ∮ d𝐥 sinθ is equal to the circumference of the loop, so the equation becomes: ∮ 𝐵 · dl = μ₀𝐼 (where ∮ 𝐿 stands for closed loop line integral).

Slide 9

This result states that the net magnetic field B due to a current carrying loop of any shape is directly proportional to the current I enclosed by the loop.
The proportionality constant μ₀ is known as the permeability of free space or vacuum permeability.
In SI units, μ₀ = 4π × 10⁻⁷ Tm/A.

Slide 10

Ampere’s law for an arbitrary current carrying loop is a powerful tool for calculating the magnetic field around complex current configurations.
It allows us to determine the magnetic field at any point outside or inside the loop, provided we know the current enclosed by the loop.
This law finds its applications in various areas of physics, including electromagnetism and magnetic field analysis.

Slide 11

Ampere’s law for an arbitrary current carrying loop allows us to calculate the magnetic field strength at any point due to the current enclosed by the loop.
This law is applicable for both open and closed loops.
It is based on the principle of electromagnetic induction and the relationship between magnetic field and electric current.
Ampere’s law is a fundamental concept in electromagnetism and has wide-ranging applications.
Let’s look at a few examples and equations to understand its practical use.

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Example 1: Consider a circular loop of radius R carrying a current I. Find the magnetic field at a point P on its axis.
Solution: Since our loop is symmetric, we can choose a circular path that encompasses the entire loop. Thus, we can use Ampere’s law.
Applying Ampere’s law, we have ∮ B · dl = μ₀I, where ∮ represents line integration along the closed path enclosing the loop and B is the magnetic field along dl.
Since the magnetic field is radial and its magnitude is constant on the chosen loop, the equation simplifies to B(2πR) = μ₀I.
Solving for B, we get B = (μ₀I) / (2πR).
This equation gives us the magnetic field at point P on the axis of the circular loop.

Slide 13

Example 2: Let’s consider a straight current-carrying wire with infinite length. Find the magnetic field at a distance r from the wire.
Solution: We can take a circular loop centered on the wire.
Applying Ampere’s law, ∮ B · dl = μ₀I, we find that B(2πr) = μ₀I.
Solving for B, we get B = (μ₀I) / (2πr).
This equation gives us the magnetic field around the wire at a distance r.

Slide 14

Example 3: Consider two parallel wires carrying currents I₁ and I₂ in opposite directions. Find the magnetic field at a point between the wires.
Solution: We can choose a rectangular loop that encloses one of the wires.
Applying Ampere’s law, ∮ B · dl = μ₀I, we find that B(2L) = μ₀I₁, where L is the distance between the wires.
Solving for B, we get B = (μ₀I₁) / (2L).
Similarly, for the other wire, we get B = -(μ₀I₂) / (2L).
The net magnetic field at the point between the wires is the sum of the fields due to individual wires.

Slide 15

Ampere’s law can also be used to find the magnetic field inside a solenoid, which is a long coil of wire wound in a helix.
For an ideal solenoid with closely spaced turns and negligible resistance, the magnetic field is nearly uniform inside.
Applying Ampere’s law to a rectangular loop inside the solenoid gives ∮ B · dl = μ₀I, where I is the current passing through each turn and B is the magnetic field.
Since B is constant inside the solenoid and the length of line integration is the same as the length of the solenoid, we have B(L) = μ₀nI, where n is the number of turns per unit length.

Slide 16

Ampere’s law provides a powerful tool for calculating the magnetic field due to current in various configurations.
It is applicable to simple and complex loops and allows us to determine the magnetic field at any point outside or inside the loop.
Ampere’s law is based on the principle of symmetry and line integration.
When using Ampere’s law, it is important to choose a closed path that encloses the current-carrying region of interest.
The law is valid for steady currents and assumes the absence of time-varying electric fields.

Slide 17

Ampere’s law has important applications in many areas of physics and engineering, including:
- Design and analysis of electromagnets.
- Calculation of the magnetic field around wires, coils, and solenoids.
- Study of magnetic forces and field interactions.
- Understanding the behavior of magnetic materials.
- Application in practical devices such as transformers and electric motors.

Slide 18

The generalization of Ampere’s law for arbitrary current carrying loops allows us to solve complex problems in electromagnetism.
By using line integration, we can calculate the magnetic field at any point due to the current enclosed by the loop.
Ampere’s law provides a mathematical approach to analyze magnetic fields and their interactions with currents.
It serves as an important tool in the study and application of electromagnetic phenomena.

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In summary, Ampere’s law for arbitrary current carrying loops extends the application of Ampere’s law beyond simple wire scenarios.
It allows us to calculate the magnetic field at any point due to the current enclosed by the loop.
Ampere’s law is a fundamental concept in electromagnetism and finds diverse applications in various fields.
Understanding and applying Ampere’s law is essential for analyzing and solving complex electromagnetic problems.
Practice and application of this law will enhance your understanding of magnetic fields and their behavior.

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Thank you for your attention!

Slide 21

Consider a current carrying loop with a non-uniform magnetic field.
To determine the magnetic field at a point due to this loop, we divide the loop into small segments.
Each segment produces a magnetic field that can be calculated using the Biot-Savart law.
The total magnetic field at the point is the vector sum of the magnetic fields produced by all the segments.

Slide 22

The Biot-Savart law describes the magnetic field produced by a current carrying segment of a wire.
Its formula is given by dB = (μ₀/4π) (𝐈 d𝐥 × 𝟏/r²), where dB is the magnetic field produced by the segment, d𝐥 is the differential length of the segment, r is the distance from the segment to the point of interest, 𝐈 is the current in the segment, and μ₀ is the magnetic constant.

Slide 23

To calculate the magnetic field due to the entire loop, we integrate the magnetic field produced by each segment over the entire loop.
This can be expressed as B = ∫ (μ₀/4π) (𝐈 d𝐥 × 𝟏/r²), where the integral is taken over the entire loop.

Slide 24

In cases where the loop carries a steady current and exhibits symmetry, Ampere’s law can simplify the calculation of the magnetic field.
Ampere’s law states that the line integral of the magnetic field around a closed loop is equal to μ₀ times the total current enclosed by the loop.
This law is a consequence of the laws of electromagnetism and is valid in vacuum and in the presence of non-magnetic materials.

Slide 25

Ampere’s law can be written as ∮ B · dl = μ₀I, where ∮ represents line integration around the loop, B is the magnetic field, dl is an infinitesimal element of length along the loop, μ₀ is the magnetic constant, and I is the total current enclosed by the loop.

Slide 26

Ampere’s law is a powerful tool for calculating the magnetic field due to current in symmetric systems.
It simplifies calculations by allowing us to find the magnetic field directly using the symmetry of the system.
Some common examples where Ampere’s law is applied include infinite straight wire, solenoids, and toroidal coils.

Slide 27

For an infinite straight wire carrying current I, the magnetic field at a distance r from the wire can be calculated using Ampere’s law.
By choosing a circular loop of radius r centered on the wire, we can simplify the line integral to ∮ B · dl = B(2πr) = μ₀I.
Solving for B, we get B = (μ₀I) / (2πr).

Slide 28

A solenoid is a tightly wrapped coil of wire that generates a uniform magnetic field along its axis.
By choosing a rectangular loop inside the solenoid, we can apply Ampere’s law to determine the magnetic field inside the solenoid.
For an ideal solenoid with closely spaced turns and negligible resistance, the magnetic field is nearly uniform inside.
Applying Ampere’s law to the rectangular loop gives ∮ B · dl = B(2L) = μ₀NI, where L is the length of the loop, N is the number of turns, and I is the current passing through each turn.
Solving for B, we get B = (μ₀NI) / (2L).

Slide 29

In more complex scenarios, Ampere’s law can be applied to calculate the magnetic field due to current in systems with symmetry.
The choice of the loop and the symmetry of the system determine the simplifications that can be made.
It is important to consider the direction of the magnetic field and assign appropriate signs to the currents and loop integrals.
The use of Ampere’s law requires a good understanding of the system’s geometry and the symmetries present.

Slide 30

In conclusion, Ampere’s law allows us to calculate the magnetic field due to current in symmetric systems.
By choosing an appropriate loop and utilizing the symmetry of the system, the calculations can be simplified.
Ampere’s law is based on the principles of electromagnetism and is valid in vacuum and non-magnetic materials.
Understanding and applying Ampere’s law is an essential skill in the study of electromagnetism and its applications.