# Finding the magnetic field inside a material shell under external field

#### DaniV

Problem Statement
A uniform external field B_{ext} = B_{ext}xˆ is applied to an infinitely long cylindrical shell with inside radius a, outside radius b, and relative permeability κ = µ/µ_{0}. The rest of space is vacuum. Show that the field inside the shell is screened to the value :
Bin = 4κb^{2}Bext /((κ + 1)^{2}b^{2} − (κ − 1)^{2}a^{2})
Relevant Equations
B=µ*H,
B=µ_{0}(H+M) when M is the magnetisation of the material
I know that we need to use some boundry condition both on the a radius surface and the b radius surface and somehowuse the superposition on them both, the boundry condition most be for the tangential and the radial part,
the only things I got is that i dont know how to produce a magnetisation M from the magnetic material, and how tofind some H_{in} that is suitable
to the problem?

the only relations that I suceed to produce its:
µ_{0}*κ*H_{in}*r(hat)=B_{ext}sin(θ)
µ_{0}*κ*H_{in}*θ(hat)=B_{ext}cos(θ)

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#### timetraveller123

i am not sure if the provided answer is correct . i worked out a different answer and it matches with answer i see over the internet
for example in wikipedia,
i am getting this also
(from wikipedia)

#### TSny

Homework Helper
Gold Member
i worked out a different answer and it matches with answer i see over the internet
for example in wikipedia,
The Wikipedia result is for a spherical shell, not a cylindrical shell.

#### timetraveller123

edit:
sorry for the inconvenience caused yes the answer is correct
i worked it out again the method is almost exactly same as the spherical one

Last edited:

#### TSny

Homework Helper
Gold Member
That's OK. I do that sort of thing all the time.

"Finding the magnetic field inside a material shell under external field"

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