91Ó°ÊÓ

The$\mathrm{Ó\neg }$ is the side of square loop.

The$z$ is the distance above the center of square loop.

The$I$ is current through loop.

The${\mathrm{θ}}_1$ is the angle made by initial point of the side with the normal.

The${\mathrm{θ}}_2$ is the angle made by end point of the side.

Write the formula ofmagnetic field reduces to the field due to a dipole.

$\mathit{B}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}\mathbf{I}}{\mathbf{2}\mathbf{Ï€}}\frac{{\mathbf{Ó\neg }}^{\mathbf{2}}}{{\mathbf{z}}^{\mathbf{3}}}\stackrel{\mathbf{Áåœ}}{\mathbf{z}}$ …… (1)

Here,${\mathit{μ}}_{\mathbf{0}}$ is permeability, $\mathit{Ó\neg }$is the side of square loop,$\mathit{z}$ is the distance above the center of square loop and role="math" localid="1657686397925" $\mathbf{I}$is current through loop.

Draw the circuit diagram for given provided condition.

Consider a point$\mathrm{P}$at a distance$z$above the center of the square loop.

${\mathrm{θ}}_1$and${\mathrm{θ}}_2$signs will be the opposite since they are rotating in the opposite directions with respect to the usual vertical line. Therefore,

$-\sin {\mathrm{θ}}_1=\sin {\mathrm{θ}}_2$

Magnetic field at$\mathrm{P}$due to one side of square loop is,

role="math" localid="1657686862591" $B_1=\frac{{\mathrm{μ}}_{0}I}{4\mathrm{Ï€³\S }}\left(\sin {\mathrm{θ}}_2-\sin {\mathrm{θ}}_1\right)$

Here, $I$is the current flowing through the loop and $S$is the distance of point from the side.

Substitute $\sin {\mathrm{θ}}_{2}for-\sin {\mathrm{θ}}_1$into above equation.

$B_1=\frac{2{\mathrm{μ}}_{0}I\sin {\mathrm{θ}}_2}{4\mathrm{Ï€³\S }}$

Refer to figure, $S$is the hypotenuse, $\frac{\mathrm{Ó\neg }}{2}$is the perpendicular and $z$is the base. So from Pythagoras theorem,

$S=\sqrt{z^2+{\left(\frac{\mathrm{Ó\neg }}{2}\right)}^2}$

It is understood that the ratio of the perpendicular side to the hypotenuse is the sine of an angle. So,

role="math" localid="1657687065877" ${\sin }_2=\frac{{\displaystyle \frac{\mathrm{Ó\neg }}{2}}}{\sqrt{z^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2}}$

Substitute $\sqrt{z^2+{\left(\frac{\mathrm{Ó\neg }}{2}\right)}^2}$for $S$and $\frac{{\displaystyle \frac{\mathrm{Ó\neg }}{2}}}{\sqrt{z^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2}}$for $\sin {\mathrm{θ}}_2$into equation (1).

$$\begin{gathered} B_1=\frac{2{\mathrm{μ}}_{0}I\left({\displaystyle \frac{{\displaystyle \frac{\mathrm{Ó\neg }}{2}}}{\sqrt{z^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2}}}\right)}{4\mathrm{Ï€}\left(\sqrt{{\mathrm{z}}^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2}\right)} \\ =\frac{{\mathrm{μ}}_{0}I}{4\mathrm{Ï€}}\frac{\mathrm{Ó\neg }}{z^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2} \end{gathered}$$

Similarly, magnetic field at$\mathrm{P}$due to all the sides of square loop is,

$B_1=4\left(\frac{{\mathrm{μ}}_{0}I}{4\mathrm{Ï€}}\frac{\mathrm{Ó\neg }}{z^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2}\right)$

From the figure, horizontal components of magnitude field will cancel each other.

Determine the vertical component of magnetic field at point P is,

$B_{1ver}=B_1\sin \mathrm{ϕ}$ …… (2)

Here,$\mathrm{Ï•}$is the angle formed by the line from a point P to the centre of a side and the normal on the plane of a square is such that$\sin \mathrm{Ï•}=\frac{{\displaystyle \frac{\mathrm{Ó\neg }}{2}}}{\sqrt{z^2+{\displaystyle \frac{{\mathrm{Ó\neg }}^2}{4}}}}$

Substitute $\frac{{\displaystyle \frac{\mathrm{Ó\neg }}{2}}}{\sqrt{z^2+{\displaystyle \frac{{\mathrm{Ó\neg }}^2}{4}}}}$for $\sin \mathrm{Ï•}$and $4\left(\frac{{\mathrm{μ}}_{0}I}{4\mathrm{Ï€}}\frac{\mathrm{Ó\neg }}{z^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2}\right)$for$B_1$into equation (2).

$B_{1ver}=4\left(\left(\frac{{\mathrm{μ}}_{0}I}{4\mathrm{Ï€}}\frac{\mathrm{Ó\neg }}{z^2+{\left({\displaystyle \frac{\mathrm{Ó\neg }}{2}}\right)}^2}\right)\right)\left(\frac{{\displaystyle \frac{\mathrm{Ó\neg }}{2}}}{\sqrt{z^2+{\displaystyle \frac{{\mathrm{Ó\neg }}^2}{4}}}}\right)$

Hence, magnetic field at point P is$B=\frac{{\mathrm{μ}}_{0}I}{2\mathrm{Ï€}}\frac{{\mathrm{Ó\neg }}^2}{{\left(z^2+{\displaystyle \frac{{\mathrm{Ó\neg }}^2}{4}}\right)}^{{\displaystyle \frac{3}{2}}}}\stackrel{Áåœ}{z}$

For $z≫\mathrm{Ó\neg }$, write the above equation as,

$B=\frac{{\mathrm{μ}}_{0}I}{2\mathrm{Ï€}}\frac{\mathrm{Ó\neg }}{z^3}\stackrel{Áåœ}{z}$

Determine the value of magnetic field reduces to the field due to a dipole, when $\mathrm{Ó\neg }$is

Magnetic dipole moment is,

$m=IA$

Here, $A$is the area of square loop.

Substitute ${\mathrm{Ó\neg }}^2$for $A$(as$\mathrm{Ó\neg }$is the side of the square loop) into above equation.

$m=I{\mathrm{Ó\neg }}^2$

Substitute $m$for $I{\mathrm{Ó\neg }}^2$into equation (1).

$B=\frac{{\mathrm{μ}}_{0}m}{2{\mathrm{Ï€³ú}}^3}\stackrel{Áåœ}{z}$

Hence, the value of magnetic field reduces to the field due to a dipole, when $z≫\mathrm{Ó\neg }$is $B=\frac{{\mathrm{μ}}_{0}m}{2{\mathrm{Ï€³ú}}^3}\stackrel{Áåœ}{z}$.