# A steady current I goes through a wire loop PQR having shape of a right angle triangle with ( mathrm{PQ}=3 mathrm{x} )

( P R=4 x ) and ( Q R=5 x . ) If the magnitude of the magnetic field at ( P ) due to this loop is ( kleft(frac{mu_{0} I}{48 pi x}right), ) find the value of ( k )

[

B=frac{u_{0}}{4 pi} frac{I}{P M}left(cos theta_{1}+omegaleft(theta_{2}right)-dot{i}right)

]

( operatorname{in} Delta operatorname{Pes} M )

[

9 x^{2}=p m^{2}+a^{2}-(i i)

]

[

begin{array}{l}

text { in } Delta P R m

qquad begin{array}{l}

16 x^{2}=P m^{2}+(5 x-a)^{2}-(iii)

=7 x^{2}=25 x^{2}-10 x a

quad=>10 x a=18 x^{2}-(iv)

end{array}

end{array}

]

from ( (ddot{i} i) ) and ( (i v) )

[

begin{aligned}

9 x^{2} &=p m^{2}+(1 cdot 8 x)^{2}

P m &=sqrt{9 x^{2}-3 cdot 24 x^{2}}

&=sqrt{5 cdot 76 x^{2}}=2 cdot 4 x

end{aligned}

]

( Also quad cos theta_{1}=frac{5 x-a}{4 x} )

[

begin{array}{l}

begin{aligned}

&=frac{5 x-1.8 x}{4 x}

&=frac{3.2}{4}=0.8

end{aligned}

begin{aligned}

B=& frac{u_{0}}{u pi} times frac{I}{2.4 x}(0.6+0.8)

&=left[frac{U_{0} I}{48 pi x}right] 7

end{aligned}

end{array}

]

So k =7