Magnitude of the Magnetic Field at point P

[Potatochip911]

Homework Statement

Two long wires (in cross section) that each carry a current $i$ directly out of the page. What is the magnitude of the field at point $P$, which lies on the perpendicular bisector to the wires?

Homework Equations

$B_{wire}=\frac{\mu_0 i}{2\pi r}$

The Attempt at a Solution

I'm confused as to how the magnetic fields are adding up. The magnetic fields produced by a wire are given by the right hand rule, thumb in direction of current, then the way your fingers curl is the direction of the magnetic field. However, I just can't see whether or not they will cancel or add at this point.

Here is a picture I drew of the currents however this doesn't give the correct answer :

 

[blue_leaf77]

If you were to decompose the magnetic fields to x and y components where x is horizontal and y is vertical, which component would cancel according to that picture?

 

[Potatochip911]

If you were to decompose the magnetic fields to x and y components where x is horizontal and y is vertical, which component would cancel according to that picture?

The x components would cancel and the y components would add, my textbook has the opposite of this happening oddly enough.

 

[blue_leaf77]

my textbook has the opposite of this happening oddly enough.

May be they use different definition for the axes. Anyway, the y component will double and therefore it's sufficient to calculate only this component.

 

[Potatochip911]

May be they use different definition for the axes. Anyway, the y component will double and therefore it's sufficient to calculate only this component.

Here is their solution:

For some reason they are using $\sin\theta$ instead, it really doesn't make any sense to me. This is the result you would get if you considered the magnetic fields to be acting like electric fields in that they are traveling straight towards the point $P$ since then the y components would cancel and the x components would be kept.

 

[blue_leaf77]

The solution is correct, it should be sine. According to the solution, $\theta$ is the angle subtended by the connecting line between one of the wires and point P and the $d_1$ line. Now, draw a small arrow representing the magnetic field from one of the wire (let's take the lower one) at point P. This vector should be perpendicular to the connecting line between the lower wire and point P. From there, decompose this vector into its x and y components and figure out where the angle $\theta$ should be placed among the components.

 

[Potatochip911]

The solution is correct, it should be sine. According to the solution, $\theta$ is the angle subtended by the connecting line between one of the wires and point P and the $d_1$ line. Now, draw a small arrow representing the magnetic field from one of the wire (let's take the lower one) at point P. This vector should be perpendicular to the connecting line between the lower wire and point P. From there, decompose this vector into its x and y components and figure out where the angle $\theta$ should be placed among the components.

Thanks it makes sense now, I completely forgot that $\vec{B}$ is perpendicular.

 

[Potatochip911]

The solution is correct, it should be sine. According to the solution, $\theta$ is the angle subtended by the connecting line between one of the wires and point P and the $d_1$ line. Now, draw a small arrow representing the magnetic field from one of the wire (let's take the lower one) at point P. This vector should be perpendicular to the connecting line between the lower wire and point P. From there, decompose this vector into its x and y components and figure out where the angle $\theta$ should be placed among the components.

Actually I'm not sure what I was thinking before because I still got it being $\cos\theta$, maybe there is something wrong with my geometry? Edit: Actually I think I just used a different angle

 

[blue_leaf77]

No, $\theta$ is the angle at the lower wire.