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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2007
Problem Set 7 Solutions
Problem 1:
Quickies…
a)
Two semicircular arcs have radii
R
2
and
R
1
, carry
current
i
, and share the same center of curvature C.
What is the magnitude of the net magnetic field at C?
The inner semicircle makes a field into the page, the outer one out of the page.
The
inner is closer and hence stronger, and hence the net field is into the page.
In class you
calculated the magnetic field from a semicircle of radius
R
to be
0
4
B
iR
μ
=
.
So:
0
11
into the page
4
i
RR
⎛⎞
=−
⎜⎟
B
G
12
⎝⎠
le, what is
θ
?
b)
A wire with current
i
is shown at left. Two semiinfinite
straight sections, both tangent to the same circle with
radius
R
, are connected by a circular arc that has a
central angle
and runs along the circumference of the
circle. The connecting arc and the two straight sections
all lie in the same plane. If
B
= 0 at the center of the
circ
The straight portions both make a field out of the page at the center of the circle while the
arc makes one into the page.
These must be equal so that the fields cancel.
Two semi
infinite lines together make an infinite line, and we calculated (using Ampere’s law) that
the field from an infinite wire is
0
2
B
=
π
.
The arc is just a fraction of a circle so it
creates a fraction of the field that a whole circle does at its center:
()
0
22
Bi
R
=
θπ
.
For these to be equal we must have
00
24
2
r
a
d
i
a
n
s
RiR
μμ
=π
=
θ
π
⇒
θ
=
c)
The figure at left shows two closed paths wrapped around
two conducting loops carrying currents
i
1
and
i
2
. What is
the value of the integral for
(a)
path 1 and
(b)
path 2?
To do this you have to use the right hand rule to check whether the currents are positive
or negative relative to the path.
On path 1
i
1
penetrates in the negative direction while
i
2
penetrates in the positive direction, so
21
o
di
i
⋅
=μ
−
∫
Bs
G
G
v
.
On path 2 i
1
penetrates twice in the negative direction and i
2
once in the negative
direction so
2
o
i
⋅=
−
μ
+
∫
G
G
v
PS07 Solutions1
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View Full Document Problem 2: Two Currents
In the figure at left a long circular pipe with outside radius
R
carries a
(uniformly distributed) current
i
into the page. A wire runs parallel to
the pipe at a distance of 3.00
R
from center to center. Find the current
in the wire such that the ratio of the magnitude of the net magnetic
field at point
P
to the magnitude of the net magnetic field at the center
of the pipe is
x
, but it has the opposite direction.
The field at point P is due both to the pipe and the wire.
The field at
the center of the pipe is ONLY due to the wire.
Since the direction of
these two is opposite the current in the wire must create an opposite direction field from
the pipe at point P and hence it must also be
into
the page.
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This homework help was uploaded on 04/07/2008 for the course 8 8.02 taught by Professor Hudson during the Fall '07 term at MIT.
 Fall '07
 Hudson

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