JEE Advance - Physics (2009 - Paper 2 Offline - No. 14)
A steady current I goes through a wire loop PQR having shape of a right angle triangle wit6h PQ = 3, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is $$k\left( {{{{\mu _0}I} \over {48\pi x}}} \right)$$, find the value of $$k$$.
Answer
7
Explanation
The segments PQ and PR, in the triangle shown here, cannot produce $$\overrightarrow B $$ at point P since point P lies on them. Here, only QR creates $$\overrightarrow B $$ at point P.
Applying $${B_P} = {{{\mu _0}i} \over {4\pi R}}(\cos {\phi _1} + \cos {\phi _2})$$, where $${\phi _1} = 53^\circ $$ and $${\phi _2} = 37^\circ $$, we get the magnitude of the magnetic field as follows:
$${B_P} = {{{\mu _0}I} \over {4\pi (4\pi \sin 37^\circ )}}(\cos 53^\circ + \cos 37^\circ )$$
$$ = {{{\mu _0}I} \over {16\pi x(3/5)}}\left( {{{3x} \over {5x}} + {{4x} \over {5x}}} \right)$$
$$ = {{5{\mu _0}I} \over {48\pi x}}\left( {{7 \over 5}} \right) = {{7{\mu _0}I} \over {48\pi x}} = k\left( {{{{\mu _0}I} \over {48\pi x}}} \right)$$
where $$k = 7$$.
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