Question

(a) Using Biot-Savart's law, derive the expression for the magnetic field in the vector form at a point on the axis of a circular current loop
(b) What does a toroid consist of? Find out the expression for the magnetic field inside a toroid for N turns of the coil having the average radius r and carrying a current I. Show that the magnetic field in the open space inside and exterior to the toroid is zero.

Answer 1

I $\to$ current
R $\to$ Radii
X $\to$ Axis
x $\to$ Distance of OP
dl $\to$ Conducting element of the loop
According to Biot-Savart's law, the magnetic field at P is
$db=\frac{{\mu }_{0}I|dl\times r|}{4\pi r^3}$
$r^2=x^2+R^2$
$|dl\times r|=rdl$(they are perpendicular)
$\therefore dB=\frac{{\mu }_0}{4\pi }\frac{Idl}{(x^2+R^2)}$
dB has two components $-d{B}_x$ and $d{B}_{\perp }.d{B}_{\perp }$ is cancelled out and only the x-component remains $\therefore d{B}_x=dBcos\theta$ $cos\theta =\frac{R}{(x^2+R^2{)}^{1/2}}$

$d{B}_x=\frac{{\mu }_{0}Idl}{4\pi }fr\frac{R}{(x^2+R^2{)}^{1/2}}$
Summation of dl over the loop is given by $2\pi R$
$\therefore B=Bxi=\frac{{\mu }_{0}I{R}^2}{2(x^2+R^2{)}^{3/2}}i$
(b) Toriod is a hollow circular ring on which a large number of turns of wire are closely wound.Three circular Amperian loops 1,2 and 3 are shown by dashed lines.
By symmetry, the magnetic field should be tangential to each of them and constant in magnitude for a given loop.
Let the magnetic field inside the toroid be B. We shall now consider the magnetic field at S.By Ampere's Law,
$\int \overline{B}.\overline{dI}={\mu }_{0}IBL={\mu }_{0}NI$
Where L is the length of the loop for which B is tangential I be the current enclosed by the loop and N be the number of turns.
We find, $L=2\pi r$
The current enclosed I is NL
$B\left(2\pi r\right)={\mu }_{0}NI,therefore,B=\frac{{\mu }_{0}NI}{2\pi r}$
For a loop inside the toroid, no current exists thus, I=0 Hences, B=0
Exterior to the toroid :
Each turn of current coming out of the plane of the paper is cancelled exactly by the current going into it. Thus $I=0,and,B=0$