French physicists Jean-Baptiste Biot and Felix Savart gave **Biot** **and** **Savart** Law in 1820. This law is used to find out the magnitude of magnetic field at a point due to **current element** of a current carrying conductor. The current carrying conductor is imagined to be divided into different short segments, each short segment is called current element.

Biot-Savart Law states that the magnetic field at any point in a medium due to current element is directly proportional to;

- the length of the current element
- current flowing through the conductor
- the sine of the small angle between current element and the line joining the element with the point
- inversely proportional to the square of the distance of the point from the element

## Mathematical Expression of Biot and Savart Law for a Conducting Wire

Let $I$ be the current carried by a conducting wire $AD$. Due to this current, magnetic field is produced around the wire. Let $dl$ be the length of the current element $BC$ and $dB$ be the magnetic field due to current element, at a point $P$ at distance $r$ from the midpoint $O$ of $BC$. Let $\theta$ be the small angle between the current element $BC$ and the line joining $OP$.

According to Biot-Savart Law, \[dB∝dl\] \[dB∝I\] \[dB∝\sin\theta\] \[dB∝\frac{1}{r^2}\]

Combining all these, we get, \[dB∝\frac{Idl\sin\theta}{r^2}\] \[dB=K\frac{I dl \sin\theta}{r^2}\]

Where, $K$ is a proportionality constant whose value depends upon the system of units selected and medium around the conductor. This constant is replaced by a new constant $\frac{\mu}{4π}$ where $\mu$ is called the permeability of the medium and \[\mu=\mu_o\mu_r\]

$\mu_r$ is the relative permeability of the medium and $\mu_o=4π×10^{-7} \text{ Tm A}^{-1}$ is the permeability of air/vacuum. \[\therefore dB=\frac{\mu}{4π} \frac{Idl\sin\theta}{r^2}\]\[=\frac{\mu_o\mu_r}{4π}\frac{Idl\sin\theta}{r^2}\]

In air/vacuum, $\mu_r=1$. \[\therefore dB=\frac{\mu_o}{4π} \frac{Idl\sin\theta}{r^2}\]

If the point $P$ lies on the conductor itself i.e $θ=0°$ or $180°$, then $dB=0$ ($dB$ is minimum) and there is no magnetic field induction at any point on the conductor. If $θ=90°$, then \[dB=\frac{\mu_o}{4π} \frac{Idl}{r^2} \text{ (dB is maximum)}\]

### Various Units of $K$

\[K=\frac{\mu_o}{4π} \text{ (in air)}\] \[=10^{-7} \text{ Wb A}^{-1}\text{m}^{-1} = 10^{-7} \text{ TmA}^{-1}\]\[= 10^{-7} \text{ N s}^2\text{C}^{-2}=10^{-7}\text{ N A}^{-2}\] \[=10^-7 \text{ H m}^{-1}\]

In vector form, \[\overrightarrow{dB}=\frac{\mu_o}{4π} \frac{I(\overrightarrow{dl}×\overrightarrow{r})}{r^3}\]

Therefore, $\overrightarrow{dB}$ is perpendicular to both $\overrightarrow{dl}$ and $\overrightarrow{r}$.

## Current Element

The current element is a small part of the current carrying conductor. It is the combined effect of the current and the small part of the conductor.

Current element is defined as a vector having magnitude equal to the product of current with a small part of the length of conductor and the direction in which the current is flowing in that part of the conductor. \[\therefore \text{Current element}=|I\overrightarrow{dl}|=Idl\]