Question

Two long straight wires of telephone circuit inside a house are coplanar with each other and run parallel to a third long straight conductor that carries a current i = $i_0$ sin $\omega$ t. The wires are at a distance of $d_1$ and $d_2$ from the conductor. The conductor induced a noise in the telephone circuit. The rms voltage per unit length of this induced emf is given as $\alpha \times {10}^{-5}$ V. Fill the value of $\alpha$.
(Take $d_2=2d_1,i_0=\frac{2\sqrt{2}}{ln2}A\omega$ = 50 rad/s)

Answer 1

$d\varphi =\frac{{\mu }_{0}i}{2\pi x}\times ldx;\varphi =\frac{{\mu }_{0}i}{2\pi }lln\left(\frac{d_2}{d_1}\right)$

$\epsilon =\frac{d\varphi }{dt}=\frac{{\mu }_0}{2\pi }l1n\left(\frac{d_2}{d_1}\right)\times i_0\omega cos\omega t$

${\epsilon }_{rms}=\frac{{\mu }_0}{2\pi }lln\left(\frac{d_2}{d_1}\right)\frac{i_0\omega }{\sqrt{2}}=\frac{4\pi }{{10}^{-7}}\times 1\times ln2\times \frac{2\sqrt{2}}{ln2}\times \frac{50}{\sqrt{2}}=2\times {10}^{-5}V$

Therefore, $\alpha =2$