Write the expression for the magnetic field due to a current element in vector form. Consider a $\displaystyle 1$ cm segment of a wire, centered at the origin, carrying a current of $\displaystyle 10$ A in positive x-direction. Calculate the magnetic field $\displaystyle \overrightarrow{\mathrm{B}}$ at a point ( $\displaystyle 1 \mathrm{~m}, 1 \mathrm{~m}, 0$ ).

Magnetic field due to a current element ( $\displaystyle \mathbf{d} \boldsymbol{l}$ ) at a distance $\displaystyle \mathbf{r}$, \[\begin{aligned} & d \vec{\mathrm{B}}=\frac{\mu_{0}}{4 \pi} \frac{I(\overrightarrow{d l} \times \vec{\mathrm{r}})}{\mathrm{r}^{3}} \\ & \mathrm{~dB}=\frac{\mu_{0} i d l \sin \theta}{4 \pi \mathrm{r}^{2}} \\ & \quad \mathrm{r}=\sqrt{2} \mathrm{~m} \\ & \mathrm{~dB}=10^{-7} \times \frac{10 \times 10^{-2} \sin 45^{\circ}}{2} \\ & \quad \mathrm{~dB}=3.53 \times 10^{-9} \mathrm{~T} \end{aligned} \] Direction : along +ve Z-axis