What is the effect of a wire with current on Ampère's circuital law?

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In summary, the law of magnetic field around a closed loop is that H is constant on the loop and length of a circle l = 2 Pi r.
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valjok
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Revising my old university lectures, I have encountered the famous law of magnetic field around a closed loop:
[tex]\oint_L \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \sum I[/tex]

The integral is simplified down to a product Hl when the perfectly round loop is orthogonal to the current. In this case H is constant on the loop and length of a circle l = 2 Pi r, wherefrom we can derive the [tex]\mathbf{H} = \frac{I}{2 \pi r}[/tex].
For instance, if the first current goes into the screen while another comes out of it, we can compute the field at point R, which distance is r from both currents:
attachment.php?attachmentid=17383&stc=1&d=1233518255.png


[tex]H_R = H_{1R} + H_{2R} = (I_1 + I_2)/2\pi r [/tex]As I understand the writing, it allows us to compute the field in any point by just summing H from all circles orthogonal to the current direction. Everything looks fine until a wire of current is considered:
attachment.php?attachmentid=17384&stc=1&d=1233520005.png

Here, a wire with current I surrounds the point R. The wire consits of infinitely many points and there is the current in every point, so that each point of wire contributes a finite amount of field to R, resuling in infinite H. I suspect that my treatment of shape of current is too loose in the fromula Hl = I and must be clarified.
 

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  • #2
The simple formula HL=I cannot be used for a current loop, because the loop does not have the appropriate symmetry. The law of Biot-Savart for a current loop gives
H=I/2r.
 
  • #3
Thank you, I have realized that my simplified formula describes the infinitely long stright line of current. This makes the fied round symmetric around it. Bending the wire distorts the field breaking its circle shape, the H becomes stronger in the center and the formula cannot be applied anymore. The infinite sum occurs when infinite number of stright current lines will be put tangently to the curve around the R :)
 
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1. What is Ampère's circuital law?

Ampère's circuital law is a fundamental law in electromagnetism that describes the relationship between the magnetic field and the electric current flowing through a closed loop. It states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space.

2. How does a wire with current affect Ampère's circuital law?

A wire with current passing through it creates a magnetic field around it, which can be described by Ampère's circuital law. The law states that the magnetic field strength is directly proportional to the current passing through the wire and the distance from the wire. Therefore, the presence of current in a wire affects the strength and direction of the magnetic field according to Ampère's circuital law.

3. What is the relationship between Ampère's circuital law and Faraday's law?

Ampère's circuital law and Faraday's law are two fundamental laws in electromagnetism that are closely related. Ampère's law describes the relationship between the magnetic field and electric current, while Faraday's law describes the relationship between a changing magnetic field and electric current. Together, these laws form the basis for understanding electromagnetic induction, which is the process of generating electricity using a changing magnetic field.

4. How does the current direction in a wire affect Ampère's circuital law?

The direction of the current in a wire affects the direction of the magnetic field created by the wire, as described by Ampère's circuital law. If the current flows in the same direction as the loop, the magnetic field will be stronger and in the same direction as the field created by the wire. If the current flows in the opposite direction, the magnetic field will be weaker and in the opposite direction.

5. Can Ampère's circuital law be used to calculate the magnetic field inside a wire?

No, Ampère's circuital law only applies to the magnetic field outside a wire. Inside a wire, the magnetic field is not constant and varies depending on the distance from the center of the wire. Ampère's law can only be used to calculate the magnetic field around a wire, not within it.

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