I'm trying to get my head around the derivation for the magnetic field of an infinite wire, in my notes I have the statement: Setup: Wire centred on the z axis, current has direction +z.
"Biot-Savart: $\vec B$ perpendicular to $Id\vec l\implies B_z = 0$
How is this so?
In your question, $Id\vec{l}$ is along the z-axis. If a vector is perpendicular to this vector it means that it has to lie in the plane perpendicular to it and cannot have a component in the z direction. Put in equations, Let $$ \vec{B} = B_x \;\hat{x} + B_y\;\hat{y}+B_z\;\hat{z} $$ Since $I\vec{dl}$ is along the z-axis, let, $$ I\vec{dl} = I\;dl\; \hat{z} $$ The dot product $I\vec{dl}\cdot \vec{B}=0$, thus, $$ B_z\times I\;dl\; = 0 \implies B_z =0 $$ Which is the conclusion
If the field is perpendicular to the $z$-axis, it can have $x$ and $y$ components but it cannot have a $z$ component.
