# Rewriting the Ampere's law in term of free currents only

• Riemann9471
In summary, the conversation discusses a problem involving finding the magnetic field inside a small air gap on a toroidal magnet with N turns of wire carrying a current I. The person is stuck and needs to use the rewritten Ampere's law, but cannot do so without first proving it. They have not been able to find the proof and are seeking help. A link to a resource is provided, and the conversation also touches on the relationship between B and H.
Riemann9471
Homework Statement
Prove that Ampere's law ( ∮ B dl = mu_0 x enclosed current ) can be rewritten in terms of free currents only like this : ∮ (1/mu_r) B dl = mu_0 x free enclosed current where mu_r is the relative permeability of the material.
Relevant Equations
Ampere's law : ∮ B dl = mu_0 x enclosed current
Ampere's law rewritten : ∮ (1/mu_r) B dl = mu_0 x free enclosed current
This is not really the assignment of my homework ( my assignment require me to find the magnetic field inside a small air gap on a toroide magnet wrapped with N turns of a wire that carry a current I ) . I'm at some point in the solution where I kind of need to use the rewritten Ampere's law to continue but I can't use it unless I prove it because it has not been proven in class. I couldn't find the proof anywhere so I was hoping to find a little help here !

You can work through this:

https://ocw.mit.edu/courses/physics/8-022-physics-ii-electricity-and-magnetism-fall-2006/lecture-notes/lecture29.pdf

Not sure what the problem is here, but

Ampere's law is actually ## \oint \vec H \cdot d \vec l = I ##
and B relates to H by ## B = \mu H = \mu_0 \mu_r H ##
so there you go. I think.

## 1. What is Ampere's law and why is it important in electromagnetism?

Ampere's law is a fundamental law in electromagnetism that describes the relationship between the magnetic field created by a current and the current itself. It states that the magnetic field around a closed loop is directly proportional to the current passing through the loop. This law is important because it helps us understand and predict the behavior of magnetic fields, which is crucial in many applications such as motors, generators, and electronic devices.

## 2. Can Ampere's law be rewritten in terms of free currents only?

Yes, Ampere's law can be rewritten in terms of free currents only. This is known as the modified Ampere's law and it takes into account the displacement current, which is a phenomenon that occurs in regions where there is a changing electric field.

## 3. How is the modified Ampere's law different from the original Ampere's law?

The modified Ampere's law takes into account the displacement current, while the original Ampere's law does not. This means that the modified law is more accurate and can be applied to a wider range of situations, such as when dealing with changing electric fields or high-frequency currents.

## 4. What are the benefits of rewriting Ampere's law in terms of free currents only?

Rewriting Ampere's law in terms of free currents only allows for a more accurate and comprehensive understanding of electromagnetic phenomena. It also simplifies calculations in certain situations, making it easier to apply in practical applications.

## 5. Are there any limitations to using the modified Ampere's law?

Like any scientific law, the modified Ampere's law has its limitations. It is most accurate in situations where the size of the objects involved is much smaller than the wavelength of the radiation being emitted. It also does not take into account quantum effects, so it may not be applicable in certain scenarios involving very small particles or high-energy phenomena.

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