MCQ
Two long current carrying thin wires, both with current $I$, are held by insulating threads oflength $L$ and are in equilibrium as shown in the figure, with threads making an angle '$\theta$' with the vertical. If wires have mass $\lambda$ per unit length then the value of $l$ is 
($g =$ gravitational acceleration)
  • $2$$sin$$\theta \sqrt {\frac{{\pi \lambda gL}}{{{\mu _0}cos\theta }}} \;\;\;\;\;\;\;\;$
  • B
    $2$$\sqrt {\frac{{\pi gL}}{{{\mu _0}}}tan\theta } $
  • C
    $\;\sqrt {\frac{{\pi \lambda gL}}{{{\mu _0}}}tan\theta } $
  • D
    $sin$$\theta \sqrt {\frac{{\pi \lambda gL}}{{{\mu _0}cos\theta }}} $

Answer

Correct option: A.
$2$$sin$$\theta \sqrt {\frac{{\pi \lambda gL}}{{{\mu _0}cos\theta }}} \;\;\;\;\;\;\;\;$
a
Let us consider $'l'$ length of current carrying wire. 

At equilibrium

$\mathrm{T} \cos \theta=\lambda \mathrm{g} \ell$

and $T \sin \theta=\frac{\mu_{0}}{2 \pi} \frac{I \times I l}{2 L \sin \theta}\left[\because \frac{F_{B}}{\ell}=\frac{\mu_{0}}{4 \pi} \frac{2 I \times I}{2 \ell \sin \theta}\right]$

Therefore, $I=2 \sin \theta \sqrt{\frac{\pi \lambda g L}{u_{0} \cos \theta}}$

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