Question about magnetic field of a current

[homomorphism]

Homework Statement

a long hairpin is formed by bending a piece of wire. if the wire carries a current $$i = 1.15 A$$.

the figure looks like this:

$$\subset$$

point a is at the center of the semicircle part (so that there is a radius r from a to the outside of the semicircle. i flows counterclockwise. the point b is in the middle of the two parallel lines.

a) what are the magnitude and direction of $$\vec{B}$$ at point a?
b) at point b, very far from a?

Homework Equations

$$\vec{B}_{wire}=\frac{\mu_{0}i}{2d\pi}$$

where d = r in this case.

$$\vec{B}_{semicircle}=\frac{\mu_{0}i}{4d}$$

$$\oint{\vec{B}\cdot d\vec{s}=\mu_{0}i}$$

The Attempt at a Solution

I know that I have to add up the contributions of the semicircle, and the two wires to get the total magnetic field at a. However, when i looked at the solution to part a, they say that each wire contributes $$\frac{1}{2}\vec{B}_{wire}=\frac{\mu_{0}i}{4d\pi}$$. I understand the contribution of the semicircle. how come the total contribution the wires is not $$2\vec{B}_{wire}=\frac{\mu_{0}i}{d\pi}$$ ?? This seems to be the magnetic field contribution from both wires for part b) though. does this have to do with how they enclose the wires in an amperian loop?

 

[Redbelly98]

The "straight wire" formula is for a wire that extends for a long distance in both directions from the given point. Compare this description to the situation at point A.

 

[homomorphism]

ah i understand now. thank you