Current Inside long cylindrical Conductor

[Tom McCurdy]

Hi guys,

I was wondering if someone could help explain something to me. I am just starting out learning about Ampere's law. I am looking at this example in the book and I don't get it. They have a cylindrical conductor with Radius R and current I. The current is uniformly distributed over the cross sectioanl area of the conductor. They take a surface that is like an extended circle outside the cylinder which makes sense to me, but I don't get how the current would be
(r^2/R^2)*I
where r is the radius of the circle of the ampere surface outside the cylinder.

 

[Tom McCurdy]

Haha Nevermind I figured it out... I misread something sorry for posting

 

[astrokreng]

Sure, I'd be happy to help explain this example to you. Ampere's law states that the magnetic field along a closed loop is proportional to the current passing through the loop. In this case, the loop is the extended circle outside the cylindrical conductor and the current passing through it is the current I.

The key to understanding this example is to remember that the current is uniformly distributed over the cross-sectional area of the conductor. This means that at any point on the surface of the conductor, the current passing through that point is the same.

Now, let's consider a small segment of the extended circle with radius r. This segment has a certain length, let's call it dl, and it also has a certain current passing through it, which we will call dI. Since the current is uniformly distributed, we can say that dI is equal to the total current I multiplied by the ratio of the area of the segment (πr^2) to the total cross-sectional area of the conductor (πR^2). This can be written as dI = (r^2/R^2)*I.

Now, using Ampere's law, we can say that the magnetic field at any point on the extended circle is equal to the current passing through that point (dI) divided by the distance from the point to the center of the conductor (r). This can be written as B = (dI/r).

Integrating this over the entire loop gives us the total magnetic field, which is equal to the integral of (dI/r) over the extended circle. Substituting in our expression for dI, we get the integral of [(r^2/R^2)*I]/r over the extended circle. Simplifying this integral leads to the expression (r^2/R^2)*I for the magnetic field at any point on the extended circle outside the conductor.

I hope this helps to clarify the example for you. If you have any further questions, please don't hesitate to ask.