Magnetic flux density between two wires

[moenste]

Homework Statement

Two identical wires R and S lie parallel in a horizontal plane, their axes being 0.10 m apart. A current of 10 A flows in R in the opposite direction to a current of 30 A in S. Neglecting the effect of the Earth's magnetic flux density calculate the magnitude and state the direction of the magnetic flux density at a point P in the plane of the wires if P is (a) midway between R and S, (b) 0.05 m from R and 0.15 m from S.

Answers: (a) 1.6 * 10^-4 T, (b) zero.

2. The attempt at a solution
(a) 0.1 / 2 = 0.05 m. B = μ₀ I / 2 π r = 4 π * 10^-7 * 10 / 2 π * 0.05 + 4 π * 10^-7 * 30 / 2 π * 0.05 = 1.6 * 10^-4 T.

(b) Because the distance between them is 0.1 m and the given distance (0.05 m and 0.15 m) is greater than the 0.1 m, therefore the magnetic flux density at a point P is zero? Not sure on this part.

And how do we determine the direction of the magnetic flux density at P?

30 - 10 = 20 A, so current at S is stronger so we need to take the P line with our current finger pointing downwards and so the field will be on the left side of P into the paper and on the right side it will be out of paper?

 

[Avalanche_]

Everything is correct in a) part. About b) part, that's true that each wire generates magnetic flux in opposite direction so it will cancel, but you got to calculate that just like in a) part. B coming from (lets say) S wire will have + sign and B from R wire will have - sign (depends on orientation of your coordinate system but it doesn't really matter here, what's important is that they have opposite signs). And you determine the direction of magnetic field with ''right hand rule''. Thumb goes in direction of current and 4 other fingers point at the direction of magnetic field.

 

[moenste]

About b) part, that's true that each wire generates magnetic flux in opposite direction so it will cancel, but you got to calculate that just like in a) part. B coming from (lets say) S wire will have + sign and B from R wire will have - sign (depends on orientation of your coordinate system but it doesn't really matter here, what's important is that they have opposite signs).

Ha, that's indeed so. I actually did calculate it before posting, but I calculated using the calculator right away and summed the numbers and so I missed the fact that they are the same.

I assumed that since the point P is 0.15 m from S so I decided to take it as negative (since it is larger than the 0.1 m distance).

B = 4 * π * 10^-7 * 10 / 2 * π * 0.05 - 4 * π * 10^-7 * 30 / 2 * π * 0.15 = 4 * 10^-5 - 4 * 10^-5 = 0 T.

And you determine the direction of magnetic field with ''right hand rule''. Thumb goes in direction of current and 4 other fingers point at the direction of magnetic field.

Yes, this part I know. You can see it on the image (the circles at the top). I don't know what to do with this part:

state the direction of the magnetic flux density at a point P in the plane of the wires if P is (a) midway between R and S, (b) 0.05 m from R and 0.15 m from S.

As I understand I need to show the direction of B at point P. Point P in (a) is the middle line between the R and S lines. How do I find the direction of B of the P "line"? I mean I don't know it's current, right?

I thought maybe I need to subtract the currents of the S and R lines, like 30 A - 10 A = 20 A so current is facing downwards like the "stronger" wire S... Not sure whether this is correct thinking.

 

[Avalanche_]

Oh, sorry, I didn't see that circles on top of your picture. And about that last part, you already did that. If P is in between wires [ a) part], you calculated the magnetic field, and just look at your circles in between the wires, both pointing in the same direction, so that got to be the direction of magnetic flux density at point P in a) part. And about b) part, you already said it cancels and B=0, so it's not pointing anywhere, it's simply zero, it has no direction at that point.

 

[moenste]

just look at your circles in between the wires, both pointing in the same direction, so that got to be the direction of magnetic flux density at point P in a) part

Hm, both magnetic fields at P are pointed into the paper (X). So the magnetic field will be into the paper at P?

But if we had a situation when the currents would go in the same direction (not like here in different ones), then the circles would've been into the paper on the left side of the P "line" and on out of the paper on the right side of it. What would we be the magnetic field at P?