Question

Figure (38-E27) shows a square frame of wire having a total resistance r placed coplanarly with a long, straight wire. The wire carries a current i given by $i=i_{0}sin\omega t$. Find (a) the flux of the magnetic field through the square frame, (b) the emf induced in the frame and (c) the heat 20x developed in the frame in the time interval 0 to $\frac{20\pi }{\omega }$

Answer 1

Considering an element dx at a distance x from the wire.

We have

$\left(a\right)\varphi =B.A$

$d\varphi =\frac{{\mu }_{0}i\times adx}{2\pi x}$

$\varphi =\int d\varphi$

$=\frac{{\mu }_{0}ia}{2\pi }ln[1+\frac{a}{b}]$

$\left(b\right)e=\frac{d\varphi }{dt}$

$=\frac{d}{dt}\frac{{\mu }_{0}ia}{2\pi }ln[1+\frac{a}{b}]$

$=\frac{{\mu }_{0}a}{2\pi }ln[1+\frac{a}{b}]\frac{d}{t}\left(i_{0}sin\omega t\right)$

$=\frac{{\mu }_{0}a{i}_0\omega cos\omega t}{2\pi r}ln[1+\frac{a}{b}]$

$\left(c\right)i=\frac{e}{r}$

$=\frac{{\mu }_{0}a{i}_0\omega cos\omega t}{2\pi r}ln[1+\frac{a}{b}]$

$H=i^{2}rt$

$={\left[\frac{{\mu }_{0}a{i}_0\omega cos\omega t}{2\pi r}ln(1+\frac{a}{b})\right]}^2\times r\times t$

$=\frac{{\mu }_0^2\times i^2\times {\omega }^2}{4\pi \times r^2}l{n}^2[1+\frac{a}{b}]\times r\times \frac{20\pi }{\omega }$

$=\frac{5{\mu }_0^2{a}^2{i}_0^2\omega }{2\pi r}l{n}^2[1+\frac{a}{b}][\therefore t=\frac{20\pi }{\omega }]$