1. Use Biot–Savart law for a current element

For a small current element, the magnetic field magnitude at a point is

$dB=\frac{\mu_0}{4\pi}\frac{I\,\Delta l\sin\theta}{r^2}$

where:

• $I=10\,\text{A}$
• $\Delta l=\Delta x=1\,\text{cm}=10^{-2}\,\text{m}$
• $r=0.5\,\text{m}$
• $\theta$ is the angle between $\Delta \vec l$ and $\vec r$

2. Find the angle $\theta$

The current element is along the $x$-axis:

$\Delta \vec l=\Delta x\,\hat i$

The field point is on the $y$-axis, so the position vector from the origin to the point is along $\hat j$. Thus, the angle between $\hat i$ and $\hat j$ is

$\theta=90^\circ$

Hence,

$\sin\theta=1$

3. Substitute values

Using

$\frac{\mu_0}{4\pi}=10^{-7}\,\text{T m/A}$

we get

$dB=10^{-7}\cdot \frac{10\times 10^{-2}}{(0.5)^2}$

4. Simplify

First,

$10\times 10^{-2}=10^{-1}=0.1$

and

$(0.5)^2=0.25$

So,

$dB=10^{-7}\cdot \frac{0.1}{0.25}$

$\frac{0.1}{0.25}=0.4$

Therefore,

$dB=0.4\times 10^{-7}\,\text{T}=4\times 10^{-8}\,\text{T}$

5. Match with options

$4\times 10^{-8}\,\text{T}$

This corresponds to Option C.