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i wanted to calculate $\mathbf{B}$ of these two parallel wires, and i've taken a glance at the correction and i've found out this equation, i tried to understand it but in vain, where do $(x + \frac{d}{2})$ and $(x - \frac{d}{2})$ come from? also why is the infinitesimal element is the vector product of $(\mathbf{e}_z \times \mathbf{e}_x)$ and not $(\mathbf{e}_x \times \mathbf{e}_z)$?

Any help even an advice is extremely appreciated. I can link the whole assignement for more clarification. Many thanks.

Schematics of the two parallel wires

enter image description here

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For an area vector between the wires in the direction of the B field, ez x ex is correct for a cross product in a right handed system. I think the given expression for Bx is incorrect (assuming e is a unit vector pointing from the origin to P). It appears they used the tangent of the angle (between r and x) instead of the sine.

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