How to Maximize Inductance in a Coaxial Cable?

[JH2o]

hiya guys,

I'm having a large amount of trouble with a problem that I've been working through (currently revising for an uncoming exam in electromagnetism, along with a lot of other things). the problem is stated as follows (condensed):

coaxial cable - inner cylinder radius a, out cylinder radius 4a
region of radius 2a < r < 3a filled with relative permeable material mu(r) > 1

find the inducance L per unit length when cable is in a circuit with current in axial symmetry out along one cylinder and returning along the other.

suppose there is only enough magnetic material availble to this inductor, explain why L can be increased by distribution and find the largest inductance acheivable by doing so.

any light towards where to even start on this would be great!

thanks

 

[Astronuc]

This homework problem is dated - Mar 29, 2007.

PF requires students to write the appropriate/relevant equations and for one to demonstrate effort.

Here it would help for one to write the equation for inductance in a coaxial cable. If one can express the equation for L as a function of the radial geometry, i.e., L = L(r), subject to the geometric constraints/limits, e.g., 0 < r < a, 2a to 3a, up to 4a, then should be able to differentiate with respect to the independent variable and find the conditions to maximize L.

See an example for writing such an equation -
https://eng.libretexts.org/Bookshel...atics/7.14:_Inductance_of_a_Coaxial_Structure