Question

Two long straight wires of telephone circuit inside a house are coplanar with 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. What is the rms voltage per unit length of this induced emf ? Express your answer in $\mu V$. Take $d_2=2d_1,i_0=\frac{2\sqrt{2}}{\ln 2}\text{A},\omega =50\text{rad/s}$.

Answer 1

Magnetic field due to a current carrying conductor at a distance $x$ from it is given by
$B=\frac{{\mu }_{0}i}{2\pi x}$


$\therefore$ magnetic flux through a rectangular region of length $l$ and width $dx$ enclosed by the two wires of the telephone circuit is
$d\varphi =\frac{{\mu }_{0}i}{2\pi x}\times ldx$

Integrating over the complete region enclosed by the two wires
$\varphi =\frac{{\mu }_{0}i}{2\pi }l\ln \left(\frac{d_2}{d_1}\right)$

Induced emf,
$\in =\frac{d\varphi }{dt}=\frac{{\mu }_0}{2\pi }l\ln \left(\frac{d_2}{d_1}\right)\times i_0\omega \cos \omega t$

& rms value of induced emf
${\in }_{\text{rms}}=\frac{{\mu }_0}{2\pi }l\ln \left(\frac{d_2}{d_1}\right)\frac{i_0\omega }{\sqrt{2}}$
$=\frac{4\pi \times {10}^{-7}}{2\pi }\times 1\times \ln 2\times \frac{2\sqrt{2}}{\ln 2}\times \frac{50}{\sqrt{2}}$
[$l=1$ for unit length]

$=2\times {10}^{-5}\text{V}=20\mu \text{V}$