Question

A thin insulated wire forms a plane spiral of $N = 100$ tight turns carrying a current $i = 8\,mA$. The radii of inside and outside turns are equal to $a = 50\,mm$ and $b = 100\,mm$. Find
(a) The magnetic induction at the centre of the spiral
(b) The magnetic moment of the spiral with given current

Answer 1

Hint:In order to solve this question, you must be aware of the concept of Biot-Savart’s law which describes the magnetic field generated by a constant electric current.The Biot Savart Law is an equation describing the magnetic field generated by a constant electric current.

Complete step by step answer:
(a) From Biot-Savart’s law, the magnetic induction due to a circular current carrying wire loop at its centre is given by:
${B_o}$ = $\dfrac{{\mu I}}{{2r}}$
The radius of the circular loop varies from $a$ to $b$. Therefore, total magnetic induction at the centre is:
${B_r}$ = $\smallint \dfrac{{\mu I}}{{2r}}dN$....................(1)
(where $\dfrac{{\mu I}}{{2r}}$ is magnetic induction due to one turn of radius $r$ and $dN$ is the number of turns in the interval ($r$, $r+dr$)i.e.
$dN =\dfrac{N}{{b - a}}dr $
Substituting value of dN in eq (1) and then integrating between a and b, we obtain
${B_o}$ = $\int_a^b {\dfrac{{\mu I}}{{2r}}} \dfrac{N}{{b - a}}dr$
$\Rightarrow {B_o}$= $\dfrac{{\mu IN}}{{2(b - a)}}\ln \dfrac{b}{a}$
$\Rightarrow {B_o}$ = $\dfrac{{4\pi \times {{10}^{ - 7}} \times 100 \times 8 \times {{10}^{ - 3}}}}{{2(50 \times {{10}^{ - 3}})}} \times 2.303$
$\therefore {B_o}$= $7\mu T$

(b) Magnetic moment of a turn of radius $r$ is
$dM =\dfrac{{Ndr}}{{b - a}} \times i\pi {r^2}$
Total magnetic moment of all turns is
$M = \int {dM} $ (1)
Substituting value of dM in eq(1), we get
$M = \dfrac{N}{{b - a}}i\pi \dfrac{{{b^3} - {a^3}}}{3}$
$\Rightarrow M = \dfrac{{100}}{{(100 - 50) \times {{10}^{ - 3}}}} \times 8 \times {10^{ - 3}}4\pi \times {10^{ - 7}}(\dfrac{{{{0.1}^3} - {{0.05}^3}}}{3})$
$\therefore M =15\,mA$

Note:Biot-Savart’s law is applicable for very small conductors which carry current. It is an equation that gives the magnetic field produced due to a current carrying segment.It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. Biot–Savart law is consistent with both Ampere’s circuital law and Gauss’s theorem.