Question

State Biot Savart law.

Answer 1

Biot savart law:

1. Biot savart law states that “ magnetic field due to a current-carrying conductor at a distance point is inversely proportional to the square of the distance between the conductor and point, and the magnetic field is directly proportional to the length of the conductor, current flowing in the conductor”.
2. It can be mathematically expressed as: $\mathrm{dB}=\frac{{\mathrm{\mu }}_0}{4\mathrm{\pi }}\times \frac{\mathrm{Idlsin\theta }}{{\mathrm{r}}^2}$

Derivation:

• Consider a point charge P placed at a distance r from an infinitely small length of wire dl.
• The distance vector r makes an angle $\mathrm{\theta }$ from the direction of the current.
• As the current flow into the wire the current along the wire can be expressed as: $\mathrm{dB}\propto \mathrm{I}$
• It can also be observed that as we move away from wire toward P the magnetic field decreases. i.e $\mathrm{dB}\propto \frac{1}{{\mathrm{r}}^2}$
• Since $\mathrm{\theta }$ is the angle between $\overrightarrow{\mathrm{r}}$ and $\mathrm{I}$. so the expression of magnetic field can be given as: $\mathrm{dB}\propto \mathrm{I}\mathrm{dl}\mathrm{sin\theta }$
• Combining the equation we get, $\mathrm{dB}\propto \frac{\mathrm{I}\mathrm{dl}\mathrm{sin\theta }}{{\mathrm{r}}^2}$

$\Rightarrow \mathrm{dB}=\mathrm{k}\frac{\mathrm{I}\mathrm{dl}\mathrm{sin\theta }}{{\mathrm{r}}^2}$; where k is the constant of proportionality

$\Rightarrow \mathrm{dB}=\frac{{\mathrm{\mu }}_0}{4\mathrm{\pi }}\times \frac{\mathrm{I}\mathrm{dl}\mathrm{sin\theta }}{{\mathrm{r}}^2}$