The Biot-Savart Law is
an equation that describes the magnetic field created by a current-carrying
wire and used to determine the magnetic field at any point due to a current
carrying conductor.
The magnitude of the magnetic
field $d\vec{B}$ at a distance $\vec{r}$ from a wire element $d\vec{l}$ carrying a current I is found to be proportional
to I, to the length $d\vec{l}$ and inversely to the square of the
distance $\left |\vec{r} \right |$ . The direction of the magnetic field is perpendicular to the line
element $d\vec{l}$ as well as the radius $\vec{r}$. The Biot-Savart
Law is given by ,
\[d\vec{B}=\frac{\mu _{0}I}{4\pi }\frac{d\vec{l}\times \vec{r}}{\left | \vec{r} \right |^{3}}\]
Magnetic field due to current flowing in infinitely long straight wire:
The magnitude of magnetic field for the wire element,
Integrating for infinitely long wire,
Magnetic field due to current flowing in infinitely long straight wire:
The magnitude of magnetic field for the wire element,
\[dB=\frac{\mu_{0}I}{4\pi }\frac{dlSin\theta }{r^{2}}\]
\[B=\int_{-\infty}^{\infty}dB=2\int_{0}^{\infty }dB=\frac{\mu_{0}I}{2\pi}\int_{0}^{\infty}\frac{Sin\theta}{r^{2}}dl\]
Now, from the figure r2=l2+R2 and $Sin\theta =\frac{R}{\sqrt{R^{2}+l^{2}}}$
\[\therefore B=\frac{\mu_{0I}}{2\pi}\int_{0}^{\infty }\frac{dl}{(R^{2}+l^{2})^\frac{3}{2}}\]
Therefore the magnitude of the field
\[B=\frac{\mu_{0I}}{2\pi R}\]
