I am having trouble computing the magnetic field from a cylinder carrying a uniform circular current density j as shown below.
My approach was to use Ampere's law, with a rectangular amperian loop with the two opposite sides parallel to the cylinder axis, as shown here :
I then want to determine the magnetic field in each region :
- r < $R_i$
- $R_i$ < r < $R_e$
- r > $R_e$
For the first region, Ampere's law gives me :
$ \int \overrightarrow{B} \cdot \overrightarrow{dl} = |\overrightarrow{B}| \cdot l$
And the enclosed current is :
$I_{enc} = \mu_0 \cdot j \cdot l \cdot (R_e - R_i)$
So that the magnetic field inside is $|\overrightarrow{B}| = \mu_0 \cdot j \cdot (R_e - R_i)$.
I hope this part is correct. I am struggling with the second point :
Should I move my amperian loop so that the lower part of my loop is within the intermediate region ? Because if I do not move the loop, I find that that $\overrightarrow{B} \cdot \overrightarrow{dl} = 0$, which would mean that there is no current enclosed, and that is clearly not the case.
Without moving the loop, is there a way to get the magnetic field in this intermediate region ?
Finally I have a similar question for the outter region. My intuition is that the B field should be zero, as for an infinite solenoid, but I am not sure if I can use Ampere's law to prove this.
I have looked into another post : Contradiction using amperes law to calculate magnetic field $B$
But I am not sure how to understand what is explained there.
I hope my problem is clear, please tell me if you need more details about anything.