Magnetic field of rotating cylinder

In summary, the problem is to find the magnetic field within a rotating cylinder (infinitely long) that has on its surface a given surface charge density p. I made a picture of the problem to illustrate this. The only hint given: "the magnetic field outside the cylinder is zero.
  • #1
Kruger
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0

Homework Statement



The problem is to find the magnetic field within a rotating cylinder (infinitely long) that has on its surface a given surface charge density p. I made a picture of the problem to illustrate this. The only hint given: "the magnetic field outside the cylinder is zero.

Homework Equations



Sorry, I cannot use Latex, but:

integral(B,ds) = u0*integral(J,da) where

B is the magnetic field, ds the vector line element, J the current density, da the vector area element.

The Attempt at a Solution



Well, to be cleary, I can solve this problem, but I do not understand a very basic thing.
First I argued by symmetry, that the magnetic field can only be in direction of the angular frequency vector w. Then I took a curve c, illustrated in my picture and now I want to apply the integral law I have written above to this curve c.

I know the left hand side of the integral, which is simply (given that the curve has lengt L):

B*L,

Now I want to calculate the right hand side. Doing so, I first need J. logically:

|J|=(p/(2*pi*R))*(w*R) where (w*R) = v is simply the velocity.

Up to here all is quite easy, but now I have some questions:
i) Is it correct, that the J vector is the tangential vector to the outer surface of the cylinder perpendicular to the angular frequency vector w?
ii) The integral law states that I have to integrate J over the area enclosed by the curve c. But this doesn't seems to be correct. I mean, the area within the cylinder and enclosed by the curve c would then be (R-r1)*L and outside undefined. So over which area do I have to integrate and WHY?
 

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  • #2
Is it a hollow cylinder? Also, do you have a surface current flowing only on the outside of the cylinder, or is it a volume current?
 
  • #3
Well, I think we can assume the cylinder to be a perfect conductor carrying only a surface charge density at its outside. So inside there will not be any current.
 
  • #4
So, you'll only need to calculate the net current enclosed by the curve C you drew. If [tex]\sigma[/tex] is the surface charge density, then the surface current will be [tex]K=\sigma v = \sigma \omega R[/tex]. From this, you can find the current enclosed by the loop and the magnetic field inside.
 
  • #5
Ah, so you mean I have to integrate the surface current density K (unit A/m) over the line my curve intersects the cylinder? So I get K*length(curve) = K*L.
 
  • #6
Kruger said:
Ah, so you mean I have to integrate the surface current density K (unit A/m) over the line my curve intersects the cylinder? So I get K*length(curve) = K*L.

Yeah, looks right.
 
  • #7
Ah, ok, then I understand it now. I think the first thing I were trying to do was integrating an object, namely the current density J, which is simply everywhere zero except at the boundaries of the cylinder over an area which intersects the current density vector J perpendicular. This cannot really work, because then J would be of zero measure for the integral of J over the area which is enclosed by the curve I have drawn (because with respect to the integral J*d(area), J is just defined on a line, namely the intersection of my rectangular area with the cylinder and zero elsewhere).

<--- This seems to be written in a strange manner, but well, this was just what confused me.

Thank you for help.
 

1. How does the rotation of a cylinder affect its magnetic field?

The rotation of a cylinder creates a changing magnetic field due to the movement of charged particles within the cylinder. This changing magnetic field is known as an induced magnetic field and adds to the existing magnetic field of the cylinder.

2. What is the direction of the magnetic field in a rotating cylinder?

The direction of the magnetic field in a rotating cylinder is determined by the right-hand rule. This rule states that if you point your thumb in the direction of the current, your fingers will wrap around in the direction of the magnetic field.

3. How does the speed of rotation affect the magnetic field of a rotating cylinder?

The speed of rotation has a direct impact on the strength of the magnetic field in a rotating cylinder. The faster the cylinder rotates, the stronger the induced magnetic field will be.

4. Can the direction of rotation be changed to alter the magnetic field of a cylinder?

Yes, the direction of rotation can be changed to alter the magnetic field of a cylinder. Reversing the direction of rotation will also change the direction of the induced magnetic field, resulting in a change in the overall magnetic field of the cylinder.

5. How is the magnetic field of a rotating cylinder measured?

The magnetic field of a rotating cylinder can be measured using a magnetometer, a device that detects and measures the strength and direction of magnetic fields. The cylinder can also be placed in a magnetic field and its movement can be observed to determine the strength and direction of its magnetic field.

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