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A current I flows in a circuit shaped li...

A current I flows in a circuit shaped like an isosceles trapizium. The ratio of the bases is 2. The length of the smaller base is l. Calculate the megnetic field induction at point P located in the plane of the trapizium, but at a distance a from the midpoint of the smaller base,

Text Solution

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From similar triangles APF and CPE, we find `d=2a` and
`tan alpha=1/(2a)`. Magnetic field at P due to wire 1 is
`B_1=(mu_0I)/(4pid)[sin alpha-sin(-alpha)]=(mu_0Isin alpha)/(2pid)=(mu_0Isin alpha)/(4pia)`
The direction of the magnetic field is into the plane of the paper.
`alpha` is the half-angle subtended by the wire 1 at point P and d is the
perpendicular distance between the wire and point P.

Magnetic field at P due to wires (2) and (4) is zero.
Megnetic field at P due to wire 3 is
`B=B_3-B_1=(mu_0Isin alpha)/(2pia)-(mu_0Isin alpha)/(4pia)=(mu_0Isin alpha)/(4pia)`
or `B=(mu_0I)/(4pia)[1/(sqrt(l^2+4a^2))]`
out of the plane of the paper.
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