A coil of n number of turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively. When a current of strength I is passed through the coil, the magnetic field at its centre is

• $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{nI}}{\left(\mathrm{b} - \mathrm{a}\right)} {\log }_{\mathrm{e}} \frac{\mathrm{a}}{\mathrm{b}}$

• $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{n} \mathrm{I}}{2 \left(\mathrm{b} - \mathrm{a}\right)}$

• $\frac{2 {\mathrm{\mu }}_{\mathrm{o}} \mathrm{nI}}{\mathrm{b}}$

• $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{nI}}{2 \left(\mathrm{b} - \mathrm{a}\right)} {\log }_{\mathrm{e}} \frac{\mathrm{b}}{\mathrm{a}}$

D.

$\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{nI}}{2 \left(\mathrm{b} - \mathrm{a}\right)} {\log }_{\mathrm{e}} \frac{\mathrm{b}}{\mathrm{a}}$

Consider an element of thickness (dr) at a distance 'r' from the centre of spiral-coil.

Number of turns in coil =  n

Number of turns per unit length = $\frac{n}{b - a}$

a is the inner radii and b is the outer radii of spiral winding.

Number of turns in element dr = dn

Number of turns per unit length in element

                dr = $\frac{\mathrm{n} \mathrm{dr}}{\mathrm{b} - \mathrm{a}}$

⇒              dn = $\frac{\mathrm{n} \mathrm{dr}}{\mathrm{b} - \mathrm{a}}$  

Magnetic field at its centre due to element dr is

                   dB  =  $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{I} \mathrm{dn}}{2 \mathrm{r}}$

                         = $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{I} }{2} \frac{\mathrm{n}}{\left(\mathrm{b} - \mathrm{a}\right)} \frac{\mathrm{dr}}{\mathrm{r}}$

By integrating on both side

∴                  B = ${\int }_{\mathrm{a}}^{\mathrm{b}} \frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{I} \mathrm{n} \mathrm{dr}}{2 \left(\mathrm{b} - \mathrm{a}\right) \mathrm{r}}$

                      = $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{I} \mathrm{n}}{2 \left(\mathrm{b} - \mathrm{a}\right)} {\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\mathrm{dr}}{\mathrm{r}}$

                     = $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{I} \mathrm{n}}{2 \left(\mathrm{b} - \mathrm{a}\right)} {\left[{\log }_{\mathrm{e}} \mathrm{r}\right]}_{\mathrm{a}}^{\mathrm{b}}$                              $\left(\because {\int }_{\mathrm{a}}^{\mathrm{b}}\frac{1}{\mathrm{r}} = {\left[{\log }_{\mathrm{e}} \mathrm{r}\right]}_{\mathrm{a}}^{\mathrm{b}}\right)$                                         

                    = $\frac{{\mathrm{\mu }}_{\mathrm{o}} \mathrm{I} \mathrm{n}}{2 \left(\mathrm{b} - \mathrm{a}\right)} {\log }_{\mathrm{e}} \left(\frac{\mathrm{b}}{\mathrm{a}}\right)$