Question on magnetic fields from wire

[RoboNerd]

Homework Statement

Homework Equations

Biot-Savart's law and ampere's law expressing B = mu0 * I / (2 * pi * r)

The Attempt at a Solution

[/B]
I know that for the first drawing, we have magnetic fields going into the page from both of the wires, and we have them being (infinitely) long, so we just sum the two expression from ampere's law to get

B0 at the origin = 2 * mu0 * I / (2 * pi * r )

In the right-hand drawing, we have half of each wire placed on the x-axis, and the rest goes parallel to the y-axis vertically. I know that by biot-savart's law, which has a cross product between dl and the r-roof vector, the magnetic field contributed by wires on the x-axis is zero as the equation becomes zero for each part.

OK. I know that the wires on the x-axis do not contribute to a magnetic field.

I examine the vertical wires going up and down in the second drawing. They both contribute a magnetic field as can be seen by examining biot-savart.

However, I am confused... why is the right answer C? Is it because that only half of the length wires contributes a magnetic field compared due to half of them running vertically due to them being bent? Or is there another reason?

Many thanks in advance for your help, and have a good day.

 

[TSny]

I am confused... why is the right answer C? Is it because that only half of the length wires contributes a magnetic field compared due to half of them running vertically due to them being bent?

That's essentially the reason. But it's important to clear up any confusion.

Consider two infinitesimal current elements "a" and "b" as shown in red below.

The two elements in the left figure have the same length and are the same distance from the x axis. What is the direction of the B field produced by "a" and "b" at the origin O. How do the magnitudes of B compare?

Now repeat for the right figure.

 

[RoboNerd]

OK.

Thanks for the help.

My result for the left figure was that a and b are the same in terms of the magnitudes of dB.

My result for the right figure was that the magnitude of dB is equal to zero and that the magnitude of dB at point b is the same as the magnitude of dB at point b in the left figure.

Thus, I am still of the impression that the B0/2 is the answer.

I hope my reasoning is right, because I got the right answer thanks to you. Thanks a lot for the help.

 

[TSny]

Your reasoning is right. Good work.

 

[RoboNerd]

Thanks again.