Question

A thin insulated wire forms a plane spiral of $N=100$ tight turns carrying a current $i=8mA$. The radii of inside and outside turns are equal to $a=50mm$ and $b=100mm$. 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:
$B_o$ = ${\displaystyle \frac{\mu I}{2r}}$
The radius of the circular loop varies from $a$ to $b$. Therefore, total magnetic induction at the centre is:
$B_r$ = $\int {\displaystyle \frac{\mu I}{2r}}dN$....................(1)
(where ${\displaystyle \frac{\mu I}{2r}}$ 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.
$dN={\displaystyle \frac{N}{b-a}}dr$
Substituting value of dN in eq (1) and then integrating between a and b, we obtain
$B_o$ = ${\int }_a^b{\displaystyle \frac{\mu I}{2r}}{\displaystyle \frac{N}{b-a}}dr$
$\Rightarrow B_o$= ${\displaystyle \frac{\mu IN}{2(b-a)}}\ln {\displaystyle \frac{b}{a}}$
$\Rightarrow B_o$ = ${\displaystyle \frac{4\pi \times {10}^{-7}\times 100\times 8\times {10}^{-3}}{2(50\times {10}^{-3})}}\times 2.303$
$\therefore B_o$= $7\mu T$

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

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.