A steady current I goes through a wire loop PQR having shape of a right angle triangle with PQ = 3x, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is k $(\frac{μ_{0} I}{48 π x})$ find the value of k.

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Solution

The correct option is B 7
Using the concept of area of triangle
$\frac{1}{2}$ X PD X 5x = $\frac{1}{2}$ X 3x X 4x
$∴$ PD= $\frac{12 x}{5}$
QD = $\sqrt{(P Q)^{2} - (P Q)^{2}} = \sqrt{9 x^{2} - \frac{144 x^{2}}{25}} = \frac{9 x}{5}$
and DR = $5 x$ - $\frac{9 x}{5} = \frac{16 x}{5}$
Magnetic field at P due to current elements PQ and PR is zero as the point P is on the conductor. Therefore, magnetic field at P due to current element QR is
$B = \frac{μ_{0} I}{4 π P D} (sin ϕ_{1} + sin ϕ_{2})$

$B = \frac{μ_{0} I \times 5}{4 π \times 12 x} (\frac{(9 x / 5)}{3 x} + \frac{(16 x / 5)}{4 x})$

$B = \frac{μ_{0} I 5}{48 π x} (\frac{3}{5} + \frac{4}{5})$

B= $\frac{7 μ_{0} I}{48 π x}$ $∴ k = 7$

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A steady current I goes through a wire loop PQR having shape of a right angle triangle with PQ = 3x, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is k(μ0I48πx) find the value of k.

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