Question

owing in three infinitely long wires along positive x, y and z directions. The magnetic field would be

Answer 1

The question is incomplete as it does not specify the point where the magnetic field is to be calculated or the magnitude of the current. Assuming the current in each wire is I along the positive x, y, and z directions respectively, the magnetic field B at a general point (x, y, z) is:

$\vec{B} = \frac{\mu_0 I}{2 \pi} \left[ \frac{-z\hat{j} + y\hat{k}}{\sqrt{y^2 + z^2}} + \frac{z\hat{i} - x\hat{k}}{\sqrt{x^2 + z^2}} + \frac{-y\hat{i} + x\hat{j}}{\sqrt{x^2 + y^2}} \right]$

If the question implicitly refers to the magnetic field at a point symmetric with respect to the axes, such as (a, a, a) where a ≠ 0, the total magnetic field is 0.

Answer 2

The magnetic field due to each infinitely long straight wire is calculated using the formula $B = \frac{\mu_0 I}{2 \pi r}$ and the right-hand rule for direction. The perpendicular distance $r$ from each wire to the point $(x, y, z)$ is determined. The vector components of the magnetic field from each wire ($\vec{B_x}$, $\vec{B_y}$, $\vec{B_z}$) are found. The total magnetic field is the vector sum of these individual fields. For a point $(a, a, a)$ (where $a \neq 0$), the fields from each wire cancel out vectorially, resulting in a net magnetic field of zero.