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Physics 230 March 22 2002

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Magnetic field due to a triangular loop

A current I = 5 A flows around a loop in the shape of an equilateral triangle
of side 5 cm. What is the magnetic field at the center of the loop?
The magnetic field direction produced by each of the three straight
segments is into the page, and each segment produces field of the same
magnitude. Thus the total magnetic field is just three times the
result due to one segment.

Using the result obtained in class, or equation 28.5 in the text, we
need the two angles labelled by the arcs in the diagram. From the
properties of equilateral triangles, each angle is 60 degrees, and sin
(60) = sqrt(3)/2. However, one angle is positive and one negative,
so

sin(theta1) - sin(theta 2) = sqrt(3)/2 - (-sqrt(3)/2) = sqrt(3).
Then for one segment:
*B* = (mu 0)/(4 pi *d*) x *I*
where *d* is the length of the green line in the figure.
*d* = (5/2 cm) tan (30) = (5/2sqrt(3)) cm
and so
B = (10^{-7} N/A^{2} )(5 A)(6)/(.05 m)

=6x10^{-5} N/A.m
The magnetic field due to the whole loop is then:
B = 3B(one side) = 1.8x10^{-4} T
The direction is perpendicular to the plane of the triangle, and into the
page.