Physics For Engineers - 2/Biot- Savarts' Law

Biot- Savarts' Law: Biot-Savarts' law provides an expression for the magnetic field due to a current segment. The field $\vec{dB}$at a position $\vec r$due to a current segment $I\vec{dl}$is experimentally found to be perpendicular to $\vec{dl}$and $\vec r$. The magnitude of the field is

  • proportional to the length $\mid dl\mid$and to the current $I$and to the sine of the angle between $\vec r$and $\vec{dl}$.
  • inversely proportional to the square of the distance $r$of the point P from the current element.
\begin{displaymath}\vec{dB} \propto I\frac{\vec{dl}\times\hat r}{r^2}\end{displaymath}
In SI units the constant of proportionality is $\mu_0/4\pi$, where $\mu_0$is the permeability of the free space. The value of $\mu_0$is
\begin{displaymath}\mu_0 = 4\pi\times 10^{-7} \ \ {\rm N/amp}^2\end{displaymath}
  • The expression for field at a point P having a position vector $\vec r$with respect to the current element is
For a conducting wire of arbitrary shape, the field is obtained by vectorially adding the contributions due to such current elements as per superposition principle,  \begin{displaymath}\vec B(P) = \frac{\mu_0}{4\pi}I\int\frac{\vec{dl}\times\hat r}{r^2}\end{displaymath}  where the integration is along the path of the current flow.