Question

Find the mutual inductance between the straight wire and the square loop of figure.

Answer 1

The flux through the square frame is given by
$\varphi =Mi$
Let us first calculate the flux through the square frame.



Let us now consider an element of loop of length dx at a distance x from the wire.
Now,
Area of the element of loop, A = adx
Magnetic field at a distance x from the wire, $B=\frac{{\mu }_{0}i}{2\pi x}$
The magnetic flux of the element is given by
$d\varphi =\frac{{\mu }_{0}i\times a\mathrm{d}x}{2\pi x}$
The total flux through the frame is given by
$$\begin{gathered} \varphi =\int \mathrm{d}\varphi \\ ={\int }_b^{a+b}\frac{{\mu }_{0}iadx}{2\pi x} \\ =\frac{{\mathrm{\mu }}_{0}ia}{2\mathrm{\pi }}\ln \left[1+\frac{a}{b}\right] \end{gathered}$$
Also,
$\varphi =Mi$
Thus, the mutual inductance is calculated as
$$\begin{gathered} Mi=\frac{{\mathrm{\mu }}_{0}ia}{2\mathrm{\pi }}\ln \left[1+\frac{a}{b}\right] \\ \Rightarrow M=\frac{{\mathrm{\mu }}_{0}a}{2\mathrm{\pi }}\ln \left[1+\frac{a}{b}\right] \end{gathered}$$