Hairy Electro-Magnetics problem.

[frankR]

A surface current equal to J_s is flowing on the surface of a perfect conductor in the x-z-plane traveling in the positive x direction. At a distance y = L along the y-axis lies the central axis of a cylindrical conductor with radius “a” and having a volumetric current distribution J_v= J_o*r*e_x traveling in the positive x-direction, where L > a. Find the condition for J_o where the H field is equal to zero in the regions, r <= a and r > a.


What I know:

Without posting a ton of equations I’ll tell you where I’m stuck. I used Biot-Savart laws for a surface current and a volumetric current and combined them to find the condition where the H-fields cancel.

I’m not sure how to handle the integrals for the Biot-Savart laws. I get infinities in the limits and everything explodes.Do you have make a one axis simplification to find where the fields cancel? I’m thinking you do because the math gets hairy. Or am I going in the wrong direction, do you use something other than Biot-Savart’s law?

Thanks

 

[M98Ranger]

for your help!



It seems like you are on the right track by using Biot-Savart's law to solve this problem. However, you may need to use some simplifications or approximations in order to avoid getting infinities in your integral limits. One approach could be to make a one-axis simplification, as you mentioned, by considering only the x-component of the magnetic field. This would essentially reduce the problem to a simpler 1-dimensional case, where you can then use the Biot-Savart law to find the condition for Jo where the H-field is equal to zero.

Another approach could be to use symmetry arguments to simplify the problem. Since the surface current is flowing in the x-z-plane and the cylindrical conductor is also aligned along the x-axis, the problem has certain symmetries that could be exploited. For example, you could consider the problem in terms of cylindrical coordinates (r, θ, z) and use the symmetry of the problem to simplify the integrals.

It's also worth noting that the infinities you are encountering could be due to the fact that you are using a perfect conductor. Perfect conductors have infinite conductivity, so they do not allow for any magnetic fields to penetrate through them. This could be causing your calculations to explode, as the Biot-Savart law may not be valid in this case. You may need to consider the problem in terms of a finite conductivity material instead.

In summary, you are on the right track by using Biot-Savart's law to solve this problem, but you may need to use some simplifications or approximations to avoid getting infinities in your calculations. Additionally, considering the symmetries of the problem or using a different material model may also help in finding the condition for Jo where the H-field is equal to zero.

 

[M98Ranger]

for any help!



Hi there,

Thank you for sharing your thoughts and where you are currently stuck in solving this Hairy Electro-Magnetics problem. It seems like you have a solid understanding of the problem and have made some progress by using Biot-Savart's law to combine the surface and volumetric currents. However, as you mentioned, the math can get complicated and the limits may lead to infinities.

In situations like this, it is always helpful to simplify the problem as much as possible. This can involve making assumptions or approximations to make the math more manageable. In this case, it may be helpful to consider a one-axis simplification, as you mentioned. This could involve assuming that the problem is only in the x-direction and ignoring the y and z components. This would essentially reduce the problem to a 1-dimensional case and make it easier to solve.

Another approach you could try is using Ampere's law, which relates the magnetic field to the current enclosed by a closed loop. This may be a more straightforward approach in this case and could potentially lead to a simpler solution.

Overall, my suggestion would be to simplify the problem as much as possible and then see if you can apply any other laws or principles to solve it. Don't be afraid to make assumptions or approximations as long as they are reasonable and don't significantly affect the results.

I hope this helps and good luck with solving the Hairy Electro-Magnetics problem!