Ampere's Law and Magnetic Fields

[datran]

Homework Statement

I have a coaxial cable with current density Jo in the center, with radius a, going in -z_hat direction. This generates a magnetic field. The outside of the cable, radius c, also carries a current density Jout going in the +z_hat direction. This generates its own magnetic field.

Find the value of the outside current density to make the magnetic field 0 for r > c

Homework Equations

I used Ampere's equation.

The Attempt at a Solution

I do not use Ampere's equation explicitly (starting from dot product and such), but conclude that H = 0 if the net current = 0. and then find out Jout in terms of Jo.

Is this correct thinking? I mean, if I drew a loop that was outside of the coaxial cable, the Inet would be 0, but there is still an H-field being contributed by both the current densities.

How else would I go solving this problem then?

Thanks!

 

[rude man]

We need to know the cross-sectional area of the outide conductor.

You are right in assuming the net current = 0 for H to be 0. If the inside conductor cross-sectional area = a then you know the inside current is J_oπa² but we also need to know the outside area & it can't be πc² obviously.

 

[datran]

Ok thank you! This was more of a conceptual question than anything. So if the two current densities are in opposite direction but equal to each other in magnitude, this will cancel out the magnetic field entirely?

I'm lost in the understanding because if we take an amperian loop outside of the coaxial cable and assume that the currents were different (meaning they each generate their own magnetic field, which does not cancel each other out), the enclosed current will be 0, but that does not necessarily guarantee the magnetic field is 0 right?

 

[rude man]

Ok thank you! This was more of a conceptual question than anything. So if the two current densities are in opposite direction but equal to each other in magnitude, this will cancel out the magnetic field entirely?

I'm lost in the understanding because if we take an amperian loop outside of the coaxial cable and assume that the currents were different (meaning they each generate their own magnetic field, which does not cancel each other out), the enclosed current will be 0, but that does not necessarily guarantee the magnetic field is 0 right?

It's not the current densities that cancel each other to attain H = 0, it's the currents. That's why you have to know the cross-sectional area of the outer conductor. Then (area of inner conductor) * Jo = (area of outer conductor) * Jout to give ∫Hds = I = net current = 0 so by symmetry H = 0 everywhere along any closed loop outside the outer conductor.

 

[paranormal]

Your approach is correct. Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the current enclosed by the loop. In this case, since the loop is outside the coaxial cable, the current enclosed is zero. Therefore, the line integral of the magnetic field around the loop must also be zero. This means that the magnetic field outside the cable must be zero.

To find the value of the outside current density, you can use the fact that the magnetic field outside the cable is due to the current density Jout. So, you can set the magnetic field outside the cable equal to zero and solve for Jout in terms of Jo. This will give you the value of Jout that will make the magnetic field outside the cable equal to zero.

Alternatively, you can also use the fact that the total current must be conserved, so the current flowing in the -z_hat direction in the center of the cable must be equal to the current flowing in the +z_hat direction on the outside of the cable. This will also give you the value of Jout in terms of Jo.

Overall, your approach is correct and you have correctly applied Ampere's Law to solve this problem. Good job!