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 `phi=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

\[d\phi = \frac{\mu_0 i \times adx}{2\pi x}\]

The total flux through the frame is given by

\[\phi = \int d\phi\]

\[       =  \int_b^{a + b} \frac{\mu_0 iadx}{2\pi x}\]

\[       = \frac{\mu_0 ia}{2\pi}\ln\left[ 1 + \frac{a}{b} \right]\]

Also,

\[\phi = Mi\]

Thus, the mutual inductance is calculated as

\[Mi = \frac{\mu_0 ia}{2\pi}\ln\left[ 1 + \frac{a}{b} \right]\]

\[ \Rightarrow M = \frac{\mu_0 a}{2\pi}\ln\left[ 1 + \frac{a}{b} \right]\]