# Biot and Savart Law

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;

1. the length of the current element
2. current flowing through the conductor
3. the sine of the small angle between current element and the line joining the element with the point
4. 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. $\therefore \text{Current element}=|I\overrightarrow{dl}|=Idl$