Magnetic Field from 2 cylinders

[bitty]

Homework Statement

Two infinitely long identical conductors carry equal current densities J in opposite directions parallel to the z-axis. They each have radius b and overlap so their centers are distance 2a apart. In the overlap a distance a from the center we have 0 current density (a hollow region).

What is the direction and magnitude of the magnetic field at the center of the hollow region?
Along the y-axis at x=0? All throughout the region?

Homework Equations

ampere's law

The Attempt at a Solution

I concluded the magnetic field everywhere inside the hollow region is the same:
B*2*pi*r=4*pi/c*J*pi*r^2 in cgs units for each cylinder
so B = 2*pi/c*J*(r2-r1)
where r2-r1 is a constant, r2 the distance from one cylinder to any pt inside the region and r1 equivalently for the other cylinder. since r2-r1 is vector addition it equals 2a, the distance between the centers of the cylinders, so B is constant for all 3. By the right hand rule, B points straight up in +y direction.

Could someone please check?

 

[ideasrule]

B*2*pi*r=4*pi/c*J*pi*r^2 in cgs units for each cylinder
so B = 2*pi/c*J*(r2-r1)
where r2-r1 is a constant, r2 the distance from one cylinder to any pt inside the region and r1 equivalently for the other cylinder. since r2-r1 is vector addition

No, r1 and r2 are both scalars, and that equation gives you only the magnitude of B. The direction of B is perpendicular to r (as given by the right hand rule).

 

[bitty]

This thread says they're vectors though...
https://www.physicsforums.com/showthread.php?t=161491

 

[bitty]

Re-solving for the magnetic field along the y-axis (at x=0)
since the two cylinders carry current in opposite direction we add the contributions from each cylinder. Each cylinder contributes a cosine component equal to a/r, where r is again the distance from one cylinder's axis to the point o interest, and 'a' is the distance from the axis of the cylinder to the y-axis.

Then B=2*pi*J*r *cos theta for each cylinder
==> B = 2*pi *J*r*a/r
so the superimposition is B= 4*pi*J*a along the axis, same answer as before.

Corrections/feedback please?

 

[ideasrule]

This thread says they're vectors though...
https://www.physicsforums.com/showthread.php?t=161491

Yes, because those equations use the cross product instead of a simple multiplication. So if you want to write everything in terms of vectors:

B = 2*pi/c*J x (r2-r1)

For this specific problem, I find it easier to think in terms of scalars, but that's just personal preferences.

 

[bitty]

these two problems though are analogous: in both we have 2 cylinders superimposed to create a hollow region. in this case though, the cylinders are identical and overlap instead of one being concentric to the other.

In this case, is my solution along the y-axis correct?

I have no idea who to solve for anywhere in the hollow region though, for the last part of the question, without claiming that the B field is the same everywhere in the hollow region. Can you toss me a bone?

 

[ideasrule]

Re-solving for the magnetic field along the y-axis (at x=0)
since the two cylinders carry current in opposite direction we add the contributions from each cylinder. Each cylinder contributes a cosine component equal to a/r, where r is again the distance from one cylinder's axis to the point o interest, and 'a' is the distance from the axis of the cylinder to the y-axis.

Then B=2*pi*J*r *cos theta for each cylinder
==> B = 2*pi *J*r*a/r
so the superimposition is B= 4*pi*J*a along the axis, same answer as before.

Corrections/feedback please?

Yes, that's correct.

 

[ideasrule]

these two problems though are analogous: in both we have 2 cylinders superimposed to create a hollow region. in this case though, the cylinders are identical and overlap instead of one being concentric to the other.

You're right, and you can apply the same equations from that thread to this problem. I just got a bit confused because in the initial post, you used * to represent a cross product.

I have no idea who to solve for anywhere in the hollow region though, for the last part of the question, without claiming that the B field is the same everywhere in the hollow region. Can you toss me a bone?

It is the same everywhere in the region, as per the other thread.

 

[bitty]

Thanks! But using the same logic, the field would not be the same everywhere outside the cylinders because we would have it be proportional to 1/r not r. Understood.

One more question: when trying to find the magnetic force between the two conductors of infinite length 'l' with currents I1, I2 and
F= 2*I1*I2 *l/(c^2*r) and so the pressure would be
F/A = 2 *I1*I2/(2*pi*l*c^2*r^2)

so P= I1*I2/(pi*r^2*c^2), and it would be pushing the cylinders away because the carry antiparallel current densities.

Is that simple derivation correct, still using superimposition? Or is there anything specially I have naively overlooked in a case where the cylinders overlap?