91影视

There is a square loop which carries a steady current Iwith Rdistance from center to side.

There is a regular n-sided polygon carrying a steady current Iwith Rdistance from the center to any side.

The magnetic field at a distance R from a straight wire carrying current l is$\mathbf{B}\mathbf{=}\frac{{\mathbf{渭}}_{\mathbf{0}}\mathbf{l}}{\mathbf{4}\mathbf{蟺搁}}\left({\mathrm{蝉颈苍胃}}_2-{\mathrm{蝉颈苍胃}}_1\right)$

Here, ${渭}_0$is the permeability of free space and ${胃}_2$and ${胃}_1$are the angles made by the ends of the wire with the point at which the field is calculated.

For a square, the angles made by the vertices with the center is $45掳$.

Thus, from equation (1),

$$\begin{gathered} B=\frac{{渭}_{0}l}{4蟺R}\left(\sin 45掳-\sin \left(-45掳\right)\right) \\ =\frac{{渭}_{0}l}{4蟺R}\sqrt{2} \end{gathered}$$

Including four sides of the square, the net field is

$$\begin{gathered} \mathit{B}\mathbf{=}\mathbf{4}\mathbf{脳}\frac{{\mathbf{渭}}_{\mathbf{0}}\mathbf{l}}{\mathbf{4}\mathbf{蟺}\mathbf{R}}\sqrt{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\sqrt{\mathbf{2}}{\mathbf{渭}}_{\mathbf{0}}\mathbf{l}}{\mathbf{4}\mathbf{蟺}\mathbf{R}} \end{gathered}$$

Thus, the field is $\frac{\sqrt{\mathbf{2}}{\mathbf{渭}}_{\mathbf{0}}\mathbf{l}}{\mathbf{4}\mathbf{蟺搁}}$.

For a regular n sided polygon, the angles made by the vertices with the center is $\frac{蟺}{n}$. Thus, from equation (1),

$$\begin{gathered} B=\frac{{渭}_{0}l}{4蟺R}\left(-\sin \left(\frac{蟺}{n}\right)\right) \\ =\frac{{渭}_{0}l}{2蟺R}sin\frac{蟺}{n} \end{gathered}$$

Including n sides of the polygon, the net field is

$$\begin{gathered} B=n脳\frac{{渭}_{0}l}{2蟺R}\sin \frac{蟺}{n} \\ =\frac{n{渭}_{0}l}{2蟺R}\sin \frac{蟺}{n} \end{gathered}$$

Thus, the field is $\frac{n{渭}_{0}l}{2蟺R}\sin \frac{蟺}{n}$.

As n becomes infinitely large, $\frac{\mathrm{蟺}}{n}$becomes infinitely small and thus,

$\sin \frac{蟺}{n}鈮�\frac{蟺}{n}$

Equation (2) thus becomes,

$$\begin{gathered} B=\frac{n{渭}_{0}l}{2蟺R}\sin \frac{蟺}{n} \\ 鈮�\frac{n{渭}_{0}l}{2蟺R}\sin \frac{蟺}{n} \\ =\frac{{渭}_{0}l}{2R} \end{gathered}$$

This is the magnetic field at the centre of a circle.