A steady current I flows through a wire loop | vedclass

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(D) The magnetic field at point $P$ due to the segments $PQ$ and $PR$ is zero because point $P$ lies on the line of these conductors.The magnetic fiel

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$7$

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(D) The magnetic field at point $P$ due to the segments $PQ$ and $PR$ is zero because point $P$ lies on the line of these conductors.The magnetic field at $P$ is only due to the segment $QR$.Let $PD$ be the perpendicular distance from $P$ to the side $QR$. Using the area of the triangle:Area $= \frac{1}{2} 	imes PQ 	imes PR = \frac{1}{2} 	imes QR 	imes PD$$\frac{1}{2} 	imes 3x 	imes 4x = \frac{1}{2} 	imes 5x 	imes PD$$12x^2 = 5x 	imes PD \implies PD = \frac{12x}{5}$.The magnetic field due to a finite wire segment at a perpendicular distance $d$ is given by $B = \frac{\mu_0 I}{4 \pi d} (\sin \phi_1 + \sin \phi_2)$.In $	riangle PQR$, $\sin \phi_1 = \frac{PQ}{QR} = \frac{3x}{5x} = \frac{3}{5}$ and $\sin \phi_2 = \frac{PR}{QR} = \frac{4x}{5x} = \frac{4}{5}$.Substituting these values:$B = \frac{\mu_0 I}{4 \pi (12x/5)} \left( \frac{3}{5} + \frac{4}{5} ight)$$B = \frac{5 \mu_0 I}{48 \pi x} \left( \frac{7}{5} ight)$$B = \frac{7 \mu_0 I}{48 \pi x}$.Comparing this with $k \left( \frac{\mu_0 I}{48 \pi x} ight)$, we get $k = 7$.

$A$ steady current $I$ flows through a wire loop $PQR$ having the shape of a right-angled triangle with $PQ = 3x$, $PR = 4x$, and $QR = 5x$. If the magnitude of the magnetic field at $P$ due to this loop is $k \left( \frac{\mu_0 I}{48 \pi x} \right)$, find the value of $k$.

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