Explain Biot Savart's Law. - Vidyadip

Question

Explain Biot Savart's Law.

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Answer

Consider an arbitrarily shaped wire carrying a current I.
Let dl be a length element along the wire. The current in this
element is in the direction of the length vector $\overrightarrow{ dl }$ which produces differential magnetic field $\overrightarrow{ dB }$ directed into the plane of the paper as shown in the figure below:

Current carrying wire of arbitrary shape
3. Consider point P at distance r from element dl. The net magnetic field at point P can be obtained by integrating i.e.,
summing up of magnetic fields $\overrightarrow{ dB }$ from these length elements.
4. Experimentally, the magnetic fields $\overrightarrow{ dB }$ produced by current I in the length element $\overrightarrow{ dl }$ is dB = $\frac{\mu_0}{4 \pi} \frac{ Id d \sin \theta}{ r ^2}$ ....(1)
where $\theta$ is the angle between the directions of $\overrightarrow{ dl }$ and $\vec{r}, \mu_0$ (permeability of free space) =
$4 \pi \times 10^{-7} T m / A \approx 1.26 \times 10^{-6} T m / A$
5. The direction of $\overrightarrow{ dB }$ is dictated by the cross product $\overrightarrow{ dl } \times$ $\overrightarrow{ r }$.
Vectorially, $\overrightarrow{ dB }=\frac{\mu_0}{4 \pi} \frac{\overrightarrow{ Id } \times \vec{r}}{r^3}$ ....(2)
Equations (1) and (2) are known as the Biot-Savart law.

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