Magnetic field of an infinite sheet of current

[JHans]

An infinite sheet of current lying in the yz plane carries a surface current of linear density J_s. The current is in the positive z direction, and J_s represents the current per unit length measured along the y axis. Prove that the magnetic field near the sheet is parallel to the sheet and perpendicular to the current direction, with magnitude:

$$B = \frac{\mu_{0} J_{s}}{2}$$

To solve this, I know that Ampere's law should be used. It's intuitive from the right-hand rule that the direction of the magnetic field will be parallel to the sheet at points near the sheet. I'm at a loss in terms of how I can go about proving this derivation, though.

 

[collinsmark]

The first thing you need to do is to pick a closed path. It's okay to break up the path into a few sections if you need to, as long as all the sections connect together making a closed path. The goal is to pick a path such that due to symmetry, each section of the path has a constant B, parallel to the path.

Ampère's law for magneto-statics (and in a vacuum) is

$$\oint _P \vec B \cdot \vec{dl} = \mu _0 I.$$

And remember, the closed path integral can be broken up into several open path integrals, each corresponding to a section of the path, as long as all the integrals form a closed path when put together.

Let's concentrate on the left side of the equation. The goal is that for any given section, $$\vec B \cdot \vec{dl}$$ is a constant. And just to be clear, $$\vec B \cdot \vec{dl}$$ is not usually a constant -- you need to pick the pick the appropriate path that makes it a constant.

Let's take the situation where, due to symmetry, the magnitude of B is a constant along the path section, and B is perfectly parallel to dl. Then,

$$\int _P \vec B \cdot \vec{dl} = \int Bdl = B \int dl = Bl$$

Now suppose in a different section in the path B is perpendicular to dl. The dot product between perpendicular vectors is zero. So in this case,

$$\int _P \vec B \cdot \vec{dl} = \int 0 = 0$$

Given the above, here are a few hints:

• For your path, try a rectangle.
• Ampère's law will give you μ₀ times the current passing through the rectangle. So make sure the rectangle passes through the sheet/plane (otherwise there wouldn't be any current inside of the rectangle).
• Even though the plane is infinite, the rectangle doesn't have to be. Let's just assume that the rectangle has a finite length l. But the fact that the plane is infinite is very important, particular when dealing with a couple of sides of the rectangle. So keep in back of your mind that the plane/sheet is infinite in size, even though the rectangle is not.
• Since the rectangle has 4 sides, feel free to break up the closed path into 4 separate path integrals if you want.

 

[JHans]

Hmm... well, since the infinite sheet of current is along the yz-plane, with current moving in the positive z-direction, it would seem to make sense to construct a rectangular loop along the xz-plane. Then the vertical sides would have the magnetic field perpendicular to the direction of movement, while the horizontal sides would have the magnetic field parallel to the direction of movement.

Is this a wise choice to make? I want to make sure I'm not overlooking something before I go further.

 

[collinsmark]

Hmm... well, since the infinite sheet of current is along the yz-plane, with current moving in the positive z-direction, it would seem to make sense to construct a rectangular loop along the xz-plane. Then the vertical sides would have the magnetic field perpendicular to the direction of movement, while the horizontal sides would have the magnetic field parallel to the direction of movement.

Is this a wise choice to make? I want to make sure I'm not overlooking something before I go further.

You don't want the rectangular loop on the xz plane. The reason is because in that case, there wouldn't be any current going through the loop. The current would merely go across the side of the loop, but not through the loop.

The current is moving in the z-direction. So for the rectangular path, you want to pick a plane perpendicular to the z-direction (i.e. a plane that has a normal vector parallel to the z-direction). That way, current passes through the rectangle.

Ampère's law only gives you the current passing through the closed path. Any current outside the closed path does not contribute.

 

[rwisz]

I would approach this problem by first understanding the concept of a magnetic field and how it is affected by a current. I would also review the principles of Ampere's law, which states that the line integral of the magnetic field around a closed path is equal to the current passing through the surface enclosed by that path.

Next, I would consider a small section of the infinite sheet of current and apply Ampere's law to it. The path chosen for the line integral would be a rectangular loop with one side parallel to the sheet and the other side perpendicular to it. This loop would enclose a small section of the sheet, and the current passing through it would be equal to the surface current density Js multiplied by the length of the loop.

Using Ampere's law, the line integral of the magnetic field along the parallel side of the loop would be equal to the magnetic field at that point multiplied by the length of the loop. Since the magnetic field is parallel to the sheet, the line integral would be equal to B multiplied by the length of the loop. Similarly, the line integral along the perpendicular side of the loop would be equal to 0, as the magnetic field is perpendicular to that side.

Equating these two line integrals and rearranging, we get:

B = \frac{\mu_{0} J_{s}l}{l} = \frac{\mu_{0} J_{s}}{2}

Where l is the length of the loop. This proves that the magnetic field near the sheet is parallel to the sheet and perpendicular to the current direction, with magnitude B = \frac{\mu_{0} J_{s}}{2}. This result is in line with the right-hand rule, as the direction of the magnetic field is determined by the current direction and the direction of the line integral.

In conclusion, by applying Ampere's law and considering a small section of the infinite sheet of current, we can prove that the magnetic field near the sheet is parallel to the sheet and perpendicular to the current direction, with magnitude B = \frac{\mu_{0} J_{s}}{2}. This result is important in understanding the behavior of magnetic fields and their interactions with currents.