Biot-Savart Law:

\\[ d\mathbf{B} = \frac{\mu_0 I}{4\pi} \frac{d\mathbf{l} \times (\mathbf{r} – \mathbf{r}’)}{|\mathbf{r} – \mathbf{r}’|^3} \\]

Magnetic Field Components:

\\[ B_x = -\frac{\mu_0 I y}{4\pi (x^2 + y^2)} \left[ \frac{z_2 – z}{\sqrt{x^2 + y^2 + (z_2 – z)^2}} – \frac{z_1 – z}{\sqrt{x^2 + y^2 + (z_1 – z)^2}} \right] \\] \\[ B_y = \frac{\mu_0 I x}{4\pi (x^2 + y^2)} \left[ \frac{z_2 – z}{\sqrt{x^2 + y^2 + (z_2 – z)^2}} – \frac{z_1 – z}{\sqrt{x^2 + y^2 + (z_1 – z)^2}} \right] \\] \\[ B_z = 0 \\]

Magnetic Field Magnitude:

\\[ B = \sqrt{B_x^2 + B_y^2} \\]

Where:

• \\( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \\): Permeability of free space
• \\( I \\): Current (A)
• \\( z_1, z_2 \\): Wire endpoints (m)
• \\( (x, y, z) \\): Observation point (m)
• \\( B_x, B_y, B_z \\): Magnetic field components (T)
• \\( B \\): Magnetic field magnitude (T)

\\[ B_x = -\frac{4\pi \times 10^{-7} \cdot 10 \cdot 0.05}{4\pi (0.05^2 + 0.05^2)} \left[ \frac{0.1 – 0}{\sqrt{0.05^2 + 0.05^2 + 0.1^2}} – \frac{-0.1 – 0}{\sqrt{0.05^2 + 0.05^2 + (-0.1)^2}} \right] \approx -2.828 \times 10^{-5} \, \text{T} \\] \\[ B_y = \frac{4\pi \times 10^{-7} \cdot 10 \cdot 0.05}{4\pi (0.05^2 + 0.05^2)} \left[ \frac{0.1 – 0}{\sqrt{0.05^2 + 0.05^2 + 0.1^2}} – \frac{-0.1 – 0}{\sqrt{0.05^2 + 0.05^2 + (-0.1)^2}} \right] \approx 2.828 \times 10^{-5} \, \text{T} \\] \\[ B = \sqrt{(-2.828 \times 10^{-5})^2 + (2.828 \times 10^{-5})^2} \approx 4 \times 10^{-5} \, \text{T} \\]

\\[ B_x \approx -1.414 \times 10^{-4} \, \text{T}, \quad B_y \approx 1.414 \times 10^{-4} \, \text{T} \\] \\[ B \approx 2 \times 10^{-4} \, \text{T} \\]

\\[ B_x \approx -1.96 \times 10^{-4} \, \text{T}, \quad B_y \approx 1.96 \times 10^{-4} \, \text{T} \\] \\[ B \approx 2.773 \times 10^{-4} \, \text{T} \\]