Parameterising a cylinder and determining the flux.

[pondzo]

Hi, I am wondering how i could parameterise the following rectangular cross section of the hollow cylinder in cylindrical co-ordinates. If this surface is better parameterised via a different co-ordinate system then by all means use that system. After I have completed the parameterisation i would then like to know how to sum up the flux of all these rectangles to get the total flux through the inside of the cylinder .My aim is to determine the flux around the inside of the cylinder (if that makes any sense) between the radii a and b. The vector field in question is B= k(x/(x^2+y^2), y/(x^2+y^2),0). Where B is the magnetic field.

I know that the answer to this question is $\text{flux} = \frac{L\mu_{0}I}{2\pi}ln{\frac{b}{a}}$ and i was able to calculate (and confirm) this via a much more simplified approach to this problem. But i would now like to attempt the more mathematically rigorous way. Thank you.

 

[pasmith]

The cross section is $\{\theta = 0 , 0 \leq z \leq L, a \leq r \leq b\}$.

 

[pondzo]

The cross section is $\{\theta = 0 , 0 \leq z \leq L, a \leq r \leq b\}$.

Hi pasmith, I tried to use this parameterization but, correct me if I am wrong, the normal vector for this is (0,1,0) and the magnetic field becomes (1/r,0,0). The dot product of these two vectors is zero and hence the flux will be zero, however I know this isn't the case.

 

[vela]

It's not clear exactly what you mean by "flux around the inside of the cylinder." The magnetic field is radial, which suggests some other problems, so it's no surprise that the flux through the cross-sectional area your drew is 0.

What was the original problem? It seems like you're not taking the right approach here.

 

[pondzo]

Hi guys, the original question was this (but in this thread i am only interested in calculating the flux, and i do realize there is a much simpler way to find the flux.. but i would like to become familiar with the mathematical rigour);

"Consider a long coaxial cable made of two coaxial cylindrical conductors that carry equal currents I in opposite directions (see figure). The inner cylinder is a small solid conductor of radius a. The outer cylinder is a thin walled conductor of outer radius b, electrically insulated from the inner conductor. Calculate the self-inductance per unit length Ll of this coaxial cable. (Figure 1) ( L is the inductance of part of the cable and l is the length of that part.) Due to what is known as the "skin effect", the current I flows down the (outer) surface of the inner conducting cylinder and back along the outer surface of the outer conducting cylinder. However, you may ignore the thickness of the outer cylinder."

I realized that i incorrectly described the corresponding magnetic field for the question. I thought the direction of the magnetic field was in the radial unit vector direction, which is obviously incorrect, so i used the azimuth unit vector for the direction of the magnetic field and also the normal to the surface of the rectangle and i got the answer I desired. Thank you for your guidance and help!