# Biot-savart’s law

It states that the magnetic field intensity dH produced, at the point P by the differential current element I dl

1. is proportional to the product I dl and the $\sin&space;\alpha$  the angle between the element and the line joining the point P to the element.
2. and is inversely proportional to the square of the distance  R between P and the current element.

then the direction of $\overrightarrow{dH}$ can be determined by right hand rule with the right hand thumb pointing in the direction of the current and the fingers encircling the wire in the direction of  $\overrightarrow{dH}$ .

i.e,  $dH&space;\propto&space;\frac{I&space;\&space;dl&space;\&space;\sin&space;\alpha&space;}{R^{2}}$ .

$dH&space;=&space;\frac{&space;k\&space;I&space;\&space;dl&space;\&space;\sin&space;\alpha&space;}{R^{2}}$ .

where k is the constant of proportionality , $k=\frac{1}{4\pi&space;}$ .

$\overrightarrow{dH}&space;=&space;\frac{&space;\&space;I&space;\&space;dl&space;\&space;\sin&space;\alpha&space;}{4\pi&space;\&space;R^{2}}\&space;\overrightarrow{a_{R}}$  A/m.

$\overrightarrow{dH}&space;=&space;\frac{&space;\&space;I&space;\&space;\overrightarrow{dl}&space;X&space;\&space;\overrightarrow{a_{R}}&space;}{4\pi&space;\&space;R^{2}}$ A/m.

$\overrightarrow{dH}$   is perpendicular to the plane that contains $\overrightarrow{dl}$   and $\overrightarrow{a_{R}}$.

$\overrightarrow{dH}&space;=&space;\frac{&space;\&space;I&space;\&space;\overrightarrow{dl}&space;X&space;\&space;\overrightarrow{R}&space;}{4\pi&space;\&space;R^{3}}$  A/m.

then the total magnetic field strength  measured at a point P is given by

$\overrightarrow{H}&space;=&space;\oint&space;\frac{&space;\&space;I&space;\&space;\overrightarrow{dl}&space;X&space;\&space;\overrightarrow{a_{R}}&space;}{4\pi&space;\&space;R^{2}}$ A/m.

closed path is taken since the current can flow only in closed path and this is called as integral form of Biot-Savart’s law.

as similar to  different charge distributions in electro-statics , there exists different current elements like line, surface and volume in the study of  static magnetic fields.

$\overrightarrow{H}&space;=&space;\int_{l}&space;\frac{&space;\&space;I&space;\&space;\overrightarrow{dl}&space;X&space;\&space;\overrightarrow{a_{R}}&space;}{4\pi&space;\&space;R^{2}}$ A/m.  —-for a line current element.

$\overrightarrow{H}&space;=&space;\int_{s}&space;\frac{&space;\overrightarrow{k}&space;\&space;ds&space;X&space;\&space;\overrightarrow{a_{R}}&space;}{4\pi&space;\&space;R^{2}}$  A/m. —-for a surface current element.

$\overrightarrow{H}&space;=&space;\int_{v}&space;\frac{&space;\overrightarrow{J}&space;\&space;dv&space;\&space;X&space;\&space;\overrightarrow{a_{R}}&space;}{4\pi&space;\&space;R^{2}}$  A/m. —-for a volume current element.

the dot and cross products between dl and I represents either H is out of  (or) into the page(plane) .

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