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 \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 \propto \frac{I \ dl \ \sin \alpha }{R^{2}} .

dH = \frac{ k\ I \ dl \ \sin \alpha }{R^{2}} .

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

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

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

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

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

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

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

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

\overrightarrow{H} = \int_{v} \frac{ \overrightarrow{J} \ dv \ X \ \overrightarrow{a_{R}} }{4\pi \ 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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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.