Magnetic field within cylinder using Ampere's Law

[Silversonic]

Homework Statement

Using Ampere's law, show that the magnetic field strength in a region within a cylinder, which has a constant current density j (flowing in the direction parallel to its axis), is equal to

B = (mu-nought)*j*r/2

The Attempt at a Solution

It doesn't say specifically, but this is the field in the theta-hat direction - i.e. in the direction of the cylinder's axis of rotation. I can prove this easily and the actual question isn't the problem. I'm assuming that this question means I have to show that the magnetic field in the r-hat direction (radially outwards) and the z-hat direction (in the direction of the current flow) are both zero.

I can prove there is no component in the r-hat direction by taking an imaginary cylinder, placing it within and using the fact that the flux through the cylinders surface is always equal to zero.

However, how do I prove that there is no component in the z-hat direction? Any help/hints appreciated.

 

[Inna]

I can prove there is no component in the r-hat direction by taking an imaginary cylinder, placing it within and using the fact that the flux through the cylinders surface is always equal to zero.

However, how do I prove that there is no component in the z-hat direction? Any help/hints appreciated.

Did you try using the same reasoning? That is, finding a surface that would allow to calculate the circulation of z - component of the field?

 

[Silversonic]

Did you try using the same reasoning? That is, finding a surface that would allow to calculate the circulation of z - component of the field?

What surface would I use? If I use a cylinder, that only tells me that the flux in the z-hat direction through one end is equal and opposite to the flux in the z-hat direction flowing through the other end - but it doesn't tell me that the magnetic field for that component is zero, only that the sum of the two fluxes flowing through the ends sums to zero.

 

[Inna]

Wait - it seems like you are using a Gauss law instead of Ampere's law. You need a circulation of B-field around the boundary of your surface. It will be proportional to the current going through the surface.

 

[Silversonic]

Wait - it seems like you are using a Gauss law instead of Ampere's law. You need a circulation of B-field around the boundary of your surface. It will be proportional to the current going through the surface.

I used Ampere's law to find the direction of the field in the theta-hat direction, I haven't touched Gauss' law at all.

I used the fact that $$\int$$B.dS = 0, using a closed cylinder, to show that it is zero in the r-hat direction, but can't prove it for the z-hat direction.

 

[Inna]

Try a rectangle with one side placed inside the cylinder, parallel to the z-axis, and the opposite side outside of the cylinder.