Biot Savart's Law and Equation

Biot-Savart Law: Biot-Savart law was discovered in 1820 by two physicists Jeans-Baptiste Biot and Felix Savart. According to this law:

1. The magnetic field is directly proportional to the length of the current element.
   
   
   $dB \propto dl \qquad (1)$


2. The magnetic field is directly proportional to the current flowing in the conductor.
   
   
   $dB \propto i \qquad (2)$


3. The magnetic field is inversely proportional to the square of the distance between length of the current element $dl$ and point $P$ (This is that point where the magnetic field has to calculate).
   
   
   $dB \propto \frac{1}{r^{2}} \qquad (3)$


4. The magnetic field is directly proportional to the angle of sine. This angle is the angle between the length of the current element $dl$ and the line joining to the length of the current element $dl$ and point $P$.
   
   
   $dB \propto sin\theta \qquad (4)$

From equation $(1)$,$(2)$,$(3)$,$(4)$ :

$\qquad dB \propto \frac{i dl sin\theta}{r^{2}}$

Now replace the proportional sign with the constant i.e. $\frac{\mu_{0}}{4 \pi}$. Therefore the above given equation can be written as

$ dB = \frac{\mu_{0}}{4 \pi} \frac{i dl sin\theta}{r^{2}}$

The magnetic field at point $P$ due to entire conductor:-

$ B =\frac{\mu_{0}}{4 \pi} \int \frac{i dl sin\theta}{r^{2}}$

Case$(1)$: If $\theta=0^{\circ}$ then the magnetic field will be zero from the above equation i.e.

$B=0$

Case$(2)$: If $\theta=90^{\circ}$ then the magnetic field will be maximum from the above equation i.e.

$B =\frac{\mu_{0}}{4 \pi} \int \frac{i dl}{r^{2}}$.

The vector form of Biot-Savart magnetic field equation is:-

$ \overrightarrow{B} =\frac{\mu_{0} i}{4 \pi} \int \frac{ \overrightarrow{dl} \times \overrightarrow{r}}{r^{3}}$