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PS07 Solutions1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2008
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
12
11
into the page
4
i
RR
⎛⎞
=−
⎜⎟
⎝⎠
B
G
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
θ
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
circle, what is
?
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
d
∫
Bs
G
G
i
v
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
−
∫
G
G
v
.
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On path 2 i
1
penetrates twice in the negative direction and i
2
once in the negative
direction so
()
12
2
o
di
i
⋅=
−
µ
+
∫
Bs
G
G
v
d) Four infinitely long parallel wires carrying equal current
I
are
arranged in such a way that when looking at the cross section, they
are at the corners of a square, as shown in the figure below.
Currents in
A
and
D
point out of the page, and into the page at
B
and
C
. What is the magnetic field at the center of the square?
The magnitude of the magnetic field a distance
r
from an infinite wire is
0
2
I
B
r
µ
π
=
The direction of the field is azimuthal in a sense given by using the right hand rule.
Thus, the magnetic field due to each wire at point
P
is
0
0
0
0
11
ˆˆ
ˆ
2(/ 2
)
2
2
ˆ
) 2
2
ˆ
)
2
2
ˆ
2
AA
BB
CC
DD
I
B
a
I
B
a
I
B
a
I
B
a
⎛⎞
==
−
−
⎜⎟
⎝⎠
−
−
−
−
Br
i
j
i
j
i
j
i
j
G
G
G
G
Adding up the individual contributions, we have
0
2
ˆ
ABCD
I
a
=+++=
−
BB B B B
j
G
GGGG
PS07 Solutions3
Problem 2: Compass needles
You may have noticed that the compass needles underwent simple harmonic motion
when you moved them from one place to another.
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This note was uploaded on 02/22/2009 for the course 8 02 taught by Professor Sciolla during the Fall '08 term at MIT.
 Fall '08
 SCIOLLA

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