Discussion Question 9B
P212, Week 9
Magnetic Field due to an Asymmetric Hollow Cylinder
A conducting cylinder is oriented parallel to the z axis and carries a uniform current I in the
negative z direction (into the page). This cylinder is hollow, however, with a cylindrical bore
centered on the point Q shown in the figure. The radius of this bore is R1, while the outer radius
of the cylinder is R2.
(a) Calculate the magnetic field at the point P on the y axis.
(i) This appears to be a horrendous problem, with no symmetry! But really, the hollow
cylinder is a superposition of two solid cylinders ... and a solid cylinder of current is
something we can deal with. How could you treat this system so that it consists of two solid
cylinders?
Two separate cylinders, one w/ I into the page that has radius R2, and one centered at Q, w/
radius R1 and current I out of the page
(ii) Calculate the field at point P due to each of the two cylinders, then add the two
contributions together. Remember to work by components since you are adding field
vectors.
x
y
P
Q
R1
R2
I = 2.5 A into page
R1 = 5 cm
R2 = 12 cm
point P: y = 15 cm
point Q: y = -3 cm
2
22
21
2
outer 2
The current density is 66 87 A/m We think of this as outer
area
cylinder with current I 3 025 amps into the plane and an inner cylinder
with 2 5 3 025 0 5252
inner outer
II
J..
RR)
RJ .
II-I .. .
π
π
== =
(−
==
==−=−
()()( )()( )
or +0 5252 out of page .
We now use expression for infinite wire:
2 7 2e-7 3 025 2e-7 0 5252
B= 3 450
0.15 0.15+0.03
.
e- I . .
ˆˆ ˆˆ
xx x.xT
r
μ
=− =
∑
G
