A steady current goes through a wire loop having the shape of a right angle triangle with and If the magnitude of the magnetic field at due to this loop is find the value of . [2009]

Question

A steady current $I$ goes through a wire loop $P Q R$ having the shape of a right angle triangle with $P Q = 3 x, P R = 4 x$ and $Q R = 5 x .$ 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 $k$ . [2009]

Smart Achievers · Accepted Answer

(7)

The magnetic field B due to wires PR and PQ = 0. Only wire QR will produce magnetic field at P. From point P, $P Q R$

$Magnetic field at P due to wire R Q$

$B = \frac{μ_{0}}{4 π} \frac{I}{P M} (\cos θ_{1} + \cos θ_{2})$ ...(i)

$k$ $9 x^{2} = P M^{2} + a^{2}$ ...(ii)

$In ∆ P R M,$ $16 x^{2} = P M^{2} + {(5 x - a)}^{2}$ ...(iii)

$\Rightarrow 16 x^{2} - 9 x^{2} = {(5 x - a)}^{2} - a^{2}$

$\Rightarrow 7 x^{2} = 25 x^{2} - 10 a x$ $\Rightarrow 10 a x = 18 x^{2}$

$\Rightarrow a = 1.8 x$ ...(iv)

From eq. (ii) and (iv),

$9 x^{2} = P M^{2} + {(1.8 x)}^{2}$

$∴$ $P M = \sqrt{9 x^{2} - 3.24 x^{2}} = \sqrt{5.76 x^{2}} = 2.4 x$ ...(v)

Also $\cos θ_{1} = \frac{a}{3 x} = \frac{1.8 x}{3 x} = 0.6$ ...(vi)

$\cos θ_{2} = \frac{5 x - a}{4 x} = \frac{5 x - 1.8 x}{4 x} = \frac{3.2}{4} = 0.8$ ...(vii)

$Therefore, from eq. (i), (v), (vi) and (vii),$

$B = \frac{μ_{0}}{4 π} \times \frac{I}{2.4 x} (0.6 + 0.8)$ $= \frac{μ_{0}}{4 π} \times \frac{I}{2.4 x} \times 1.4$ $= \frac{7 μ_{0} I}{48 π x}$

$Comparing it with B = k (\frac{μ_{0} I}{48 π x}), we get k = 7$

Solution

Q.
A steady current $I$ goes through a wire loop $P Q R$ having the shape of a right angle triangle with $P Q = 3 x, P R = 4 x$ and $Q R = 5 x .$ 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 $k$ . [2009]

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Q. A steady current I goes through a wire loop PQR having the 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. [2009]

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Q.
A steady current $I$ goes through a wire loop $P Q R$ having the shape of a right angle triangle with $P Q = 3 x, P R = 4 x$ and $Q R = 5 x .$ 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 $k$ . [2009]