Download Physics 435 Homework #10: Magnetic Fields and Currents and more Assignments Guiding Electromagnetic Systems in PDF only on Docsity!
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Physics 435
Spring 2009
Homework
- An infinite wire which passes through the origin and lies parallel to the z axis
carries a current I A magnetic material with μ = κμ 0 is present in the
region s > a. Answer all parts of this problem in terms of a I , , κ , andμ 0.
a) Find H s ( )
G
and B ( ) s
G
in all regions. b) Find the bound currents Kb
G
and Jb
G
everywhere c) Find the total enclosed current (free and bound) within a circle with s > a and check that your B ( s > a )
G
is consistent with Ampere’s law with this current.
- Consider an infinitely long solenoid of radius b. A magnetic material exists from a < s < b. The magnetization is of the form M = M z 0 ˆ
G
in the region a < s < b. Air is everywhere else. We define 3 regions: #1 r < a , #2 a < r < b
and #3 b < r. The only free current is a free surface current K = K φˆ
G
at radius
b. At s = b there is also a bound surface current K = K ' φˆ
G
. Another bound surface current exists at s = a. Answer all parts of this problem in terms of a,b ,
K , K ', and physical constants such as μ 0
a) Find M
G
in terms of a,b , K , K ', and physical constants such as μ 0.
b) Find K s (^^ = a )
G
c) Find B #3^ ,^ B #2^ , and B # 1
G G G
in terms of K and K’.
a) Find A #3 ( ) s , A #2 ( ) s , and A # 1 ( ) s in terms of K ,K’ using the ∫ A d =∫ B da
G G G G
v i^ A^ i method. d) Show that A is continuous at r = a and r = b e) Verify the discontinuity BC at the boundary s=a
interface interface b f o
A A K K
μ n n
G G G G
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- Consider a permanent magnet in the shape of a long cylinder of radius R with
a position dependent magnetization of M = β s z ˆ
G
that only exists in the magnet. There are no free currents anywhere.
a) Compute the bound current density ( J )
G
and all bound surface currents ( K )
G
b) Use Ampere’s law to compute B
G
everywhere and show your answer implies H = 0
G
everywhere. Hint—assume that B = 0
G
outside of the magnet and place one side of your Ampere loop outside the magnet.
c) Verify the boundary condition Bout − Bin = μ o ( K (^) free + Kbound (^) ) × n ˆ
G G G G
at s^ = R.
- The magnetic field of a spherical permanent magnet of radius R with
M = M z 0 ˆ
G
and no free currents is given by: 2 0 0 3
B r^ G(^ < R ) =^ μ M^ z^ ˆ and
( ) 0 0 3 ( ) 3 3 2 cos ˆ sin^ ˆ B r R M R r r
G > = μ θ + θ θ.
a) Show that these magnetic fields satisfy the boundary condition:
Br = R + δ − Br = R −δ = μ 0 K × r ˆ
G G G
where = = M ׈
G G G
K K (^) b r.
b) Compute H > ≡ H R ( + δ ,θ)
G G
and H < ≡ H R ( − δ ,θ)
G G
( δ → 0 ) and comment on
which components of H
G
are continuous across the boundary at r = R.
c) Find S ∫ B da
G G
i over a surface bounded by a “cap” consisting of the northern hemisphere of spherical shell of radius a > R and a “plate” consisting of the circular disk of radius a in the x-y plane. You should get 0 S ∫ B da^ =
G G
i since there are no magnetic charges but I want explicit integrals using the magnetic field forms given in the problem.
cap
z
M 0
R
a
plate
