91Ó°ÊÓ

Biot-Savart Law in physics is a law in the form of a mathematical expression that describes the capability of a constant electric field to produce a magnetic field.

Mathematically,

$\mathbf{d}\stackrel{\mathbf{→}}{\mathbf{B}}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}}{\mathbf{4}\mathbf{π}}\frac{\mathbf{Id}\stackrel{\mathbf{→}}{\mathbf{I}}\mathbf{×}\widehat{\mathbf{r}}}{{\mathbf{r}}^{\mathbf{2}}}$

Here,${\mathrm{μ}}_0$ is the permeability of free space,$\mathrm{d}\stackrel{→}{\mathrm{l}}$ is the length element, and r is the radius.

The magnetic field induction at the centre of a semi-circular loop is,

$B=\frac{{\mathrm{μ}}_{0}I}{4R}$

Here, ${\mathrm{μ}}_0$is the permeability of free space, I is the current, and R is the radius.

The two straight segments can produce a point of zero fields at point P. There are two semi-circular segments with radius a and b that produces a magnetic field at point P out of the page and inside the page respectively.

localid="1668340707409" $$\begin{gathered} B=B_a−B_b \\ =\frac{{\mathrm{μ}}_{0}l}{4a}−\frac{{\mathrm{μ}}_{0}l}{4b} \\ =\frac{{\mathrm{μ}}_{0}l}{4}\left(\frac{1}{a}−\frac{1}{b}\right) \\ =\frac{{\mathrm{μ}}_{0}l}{4a}\left(1−\frac{a}{b}\right) \end{gathered}$$

Hence, this magnetic field is directed out of the page.