Magnetic field due to the current flowing in a bent wire

[Elder1994]

Homework Statement

A very thin linear conductor, which carries the current 𝐼, bends an angle 𝛼 with respect to its initial direction at some given point, as shown in the figure. Find the magnetic field 𝑩 along the line of the initial run.

Relevant Equations

$$\mathbf {B} = \int {\frac {dS \otimes r} {r^3}}$$

Hello, in this problem I'm supposed to calculate de magnetic field due to a bent wire at any point of the x-axis after the bending of the wires. It is obvious that the part of the wire that is parallel to the x-axis makes no contribution to the field so we can focus on the other part of the wire. Here I'm having some problems trying to define all the parts of the integral, I now that the dS = dx i + dy j, but I can not figure out how $$\mathbf {r}$$ looks like.

 

[Simon Bridge]

Helps to think about how the physics described by the equations works.

Can you sketch the field?

Taking the x-axis pointing in the direction of the current before the bend ... what happens is the current changes direction to an angle to the x axis.

So did you have a think about some easy angles ... ie. what if the bend was 90 degrees?

So what would the field along the x-axis look like if the wire was not bent at all, but, instead, was entirely at that angle?

Once you understand the situation, you'll find it easier to decide how to go forward.