# Understanding Gauss's Law Regarding Magnetic Field in Current Sheets

In summary: NurizadeIn summary, Yusif Nurizade is trying to understand the Gauses's Law and how it applies to a current sheet with a width. He is getting help from the community on the forums, but is still having some trouble understanding it. He is hoping someone can help him out with a derivation of the formula.

Hello all,

I've gotten some solid advice on these forums before and I was hoping someone could help me out again.

I'm learning about Gauses's Law and am having some trouble understanding how it pertains to the magnetic field in a current sheet with a width. The online examples I've found with regard to a current sheet talk about infinite sheets that you treat as N number of wires carrying current and boggle down to the following formula:

$B = (\mu IN)/(2)$

My question is whether or not the formula can be modified by substituting the unknown N with a known W (width) or if N can be scrapped with no replacement since there is only one piece carrying the current? Or would the resulting formula be more complicated making me very far off base?

Any help would be appreciated. I am trying to grasp the concept as it applies to multiple conductors not just the generic examples given.

Thanks,

I would hazard a guess that this formula is the result of an integration of the Biot-Savart law for all the wires in a parallel sheet. If this is correct I think the case of a thick sheet would give a different result, because you will start getting contributions from out-of-plane wires (ie. sqrt(x^2 + y^2) distances rather than just x).

Does your source for this formula contain a derivation? That would probably be very helpful in deriving a "thick conductor" version.

MikeyW,

I'm either unfamiliar or can't recall the Biot-Savart Law although the quick glance I took after reading your message has examples of single wires.

The scenario I am considering involves no given thickness but a known width and infinite length. Later elements of the example include an identical parallel sheet with current running in the opposite direction to derive inductance but so far I am just considering the magnetic field.

A couple of sites document the derivation that got me to the original formula I posted. One of the best documented was on Wikipedia:

http://en.wikipedia.org/wiki/Current_sheet

The example I am studying differs in that it has a known width so I'm trying to figure out how the formula would be adjusted.

Thanks,
Yusif

## 1. What is Gauss's Law regarding magnetic field in current sheets?

Gauss's Law states that the total magnetic flux through a closed surface surrounding a current sheet is equal to the total current passing through the sheet. In other words, the magnetic field around a current sheet is directly proportional to the amount of current passing through it.

## 2. How is Gauss's Law applied to current sheets?

Gauss's Law can be applied to current sheets by using the concept of Ampere's Law. This law states that the magnetic field around a current-carrying conductor can be calculated by multiplying the current passing through the conductor by the permeability of free space and dividing by the distance from the conductor.

## 3. What is the significance of understanding Gauss's Law for current sheets?

Understanding Gauss's Law for current sheets is important for various applications in physics and engineering, such as designing electromagnetic devices and calculating the magnetic field strength around current-carrying wires. It also helps in understanding the behavior of magnetic fields in different scenarios.

## 4. Are there any limitations to Gauss's Law in relation to current sheets?

Yes, there are some limitations to Gauss's Law regarding current sheets. It assumes that the current is evenly distributed across the sheet, and the sheet is infinitely thin. In reality, this may not always be the case, and the results may not be accurate in such situations.

## 5. Can Gauss's Law be used to calculate the magnetic field inside a current sheet?

No, Gauss's Law cannot be used to calculate the magnetic field inside a current sheet. This is because the law only applies to closed surfaces, and the field inside a current sheet is not enclosed by any surface. To calculate the field inside a current sheet, other methods such as the Biot-Savart Law or the Ampere's Law can be used.