State and explain Biot – Savart law.
Bio – Savart law
This law was deduced from a number of experiments on the effect of current carrying conductor on permanent magnets such as the compass needle experiment of oersted. This law provides the intimate relation between the magnetic field intensity \(\vec{H}\) at a point in space and the current I.
This low states that the differential magnetic field intensity \(d\vec{H}\) produced at a point P(x,y,z) due to a differential current element I\(d\vec{l}\) located at point \(Q(x^\prime,y^\prime,z^\prime)\) is proportional to the current element I\(d\vec{l}\), since of the angle Φ between the current element and the line joining P to the element, i.e., position vector \(\vec{R}=\vec{r}-\vec{r^\prime}\) and is inversely proportional to the square of the distance R from the differential element to the point P under consideration. The direction of \(d\vec{H}\) is normal to the plane containing the differential element and the line drawn from the element to the point P.
According to the Biot-Savart law
\(dH\alpha\frac{I.dl.sin\ \mathrm{\Phi}}{R^2}\)
Or,
\(dH=k.\frac{I.dl.sin\ \mathrm{\Phi}}{R^2}\) … (1)
In S.I units \(k=1/4\pi\), therefore equation (1) becomes
\(dH=\frac{I.dl.sin\ \mathrm{\Phi}}{4\pi R^2}\ A/m\) …(2)
The total magnetic field intensity at P(x,y,z) can be obtained by integrating the equation (2) along the current element, we have
\(H\left(x,y,z\right)=\int_{a}^{b}\frac{I(x^\prime,y^\prime,z^\prime)dl{(x}^\prime,y^\prime,z^\prime)\ \sin\funcapply\Phi}{4\pi R^2}\)
Fig. 10 Biot-Savart law for a current element Idl
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