Magnetic field at P, between six parallel wires.

[arrowface]

Homework Statement

Six infinitely long parallel wires, each carrying current +I in z direction, perpendicular to the (x, y) plane. They are spaced at angles 2pi/7 around a circle of radius d. A seventh wire, which would make the arrangement symmetric, is missing. What is the magnitude and direction of the magnetic field at point P at the center of the circle?

Homework Equations

∫B*dl = μ*I

The Attempt at a Solution

The solution to the problem is:

B*2pi*d = μ*I

B = μ*I / 2pi*d, in the negative x direction.My question is, why is B in the negative x direction? My method of understanding this is using the right hand rule, am I right with this method of thinking?


.......y
.......^
.....x......|
...o...o......|
...o...o....z.o---->x
...o...o

Looking at the drawing as seeing as the currents are coming out of the page, this would indicate your thumb pointing out of the screen. Force being your middle finger points down, because the magnetic attraction is heavier from the down region do to the lack of symmetry. Therefore leaving B, magnetic field(pointer finger) to point in the negative x direction. Am I correct in this method in determining why it is pointing in the negative x direction?

 

[TSny]

Hello arrowface. Welcome to PF!

Sounds like you might be trying to use a right-hand-rule for magnetic force. But, here you are dealing with magnetic field produced by a straight current. What right-hand-rule should you use in this case?

 

[arrowface]

Thank you! I have been stalking for quite some time. I figured it was time to get more involved...



The two right hand rules that I am aware of is the three finger one, which I'm trying to apply. Where your thumb is I, pointer is B, and middle finger/palm is Force. The only other right hand rule I am aware of is the corkscrew method where your thumb is I, and your curled fingers point in the direction of the B field. The corkscrew method has only helped in the past when dealing with one wire, but with 6 wires I would be unsure on using it.

 

[TSny]

The superposition principle says that the net field will be the vector sum of the individual fields from each wire. You can use the right-hand-rule (corkscrew) for each wire. You then get 6 fields at P that need to be added.

However, there is a slick way to avoid actually having to add the 6 individual field vectors. Imagine you were to add a 7th wire at the missing position also carrying current in the +z direction. Can you figure out what the net field would be at P? (7 wires is easier than 6!) Then you can make up for adding the 7th wire by superposing an 8th wire on top of the 7th wire but carrying current in the negative z-direction.

 

[arrowface]

I see what you are saying, however I'm confused on how it leads to finding which direction the B field is going in. From your given explanation how would you determine that the B field is going in the negative x direction?

 

[TSny]

First see if you can see what the net field will be if all 7 currents are in place.

Suppose you assume that the 7 currents produce some non-zero B field at P that points in some direction. Note that the system of currents would not change if you rotated the whole system by 2Pi/7. So, the magnetic field vector at P must still point in the same direction as before the rotation. But if you rotate the system, that should cause the B vector at P to rotate along with the system. So, you have a contradiction. To avoid the contradiction, what must be the field at P due to the 7 currents?

 

[arrowface]

If there was a 7th current pointing in the positive z direction, the B-field at point P would be zero.

 

[TSny]

If there was a 7th current pointing in the positive z direction, the B-field at point P would be zero.

So, now add an 8th wire on top of the 7th wire with current in -z direction. This will cancel the effect of the 7th wire to get you back to 6 wires. So, the total of the 8 wires is the same as the 6 wires. But 7 of the 8 wires add to zero, so the total of the 8 wires is just the 8th wire alone. So, you just need to find the field of the 8th wire alone.

 

[arrowface]

I think I may be missing something trivial. The field is not hard to find since you can simply solve for it using an equation. The direction that it is pointing is the part that is confusing me. I understand how to attain the field, but I do not fully understand why it is pointing in the negative x direction.

I was under the assumption that one would be able to find out which way the field points by simply using the right-hand-rule?

 

[TSny]

If you had a single wire at the missing location (x) with current going in -z direction, what would be the direction of the B field at P produced by this current? You should be able to use the corkscrew right hand rule to answer that.

 

[arrowface]

I see(i hope)! Can you please let me know if my visualization is correct:

Since the 8th wire has a current pointing into the screen, the magnetic field around the current is clockwise. Therefore as the field hits point the P, the field is going in the negative x direction?

 

[hms.tech]

I see(i hope)! Can you please let me know if my visualization is correct:

Since the 8th wire has a current pointing into the screen, the magnetic field around the current is clockwise. Therefore as the field hits point the P, the field is going in the negative x direction?

Looks Ok,

In a more general case, the magnetic field(At point P) would be directed anti parallel to the tangent at the point x (the position of the wire)

This was an intuitive way of solving this problem, I am curious about the math.
The individual magnetic fields can be added as vectors but is there a way to find the direction and magnitude of the $B^{\rightarrow}$(ie a magnetic field vector) produced by a wire?

 

[TSny]

I see(i hope)! Can you please let me know if my visualization is correct:

Since the 8th wire has a current pointing into the screen, the magnetic field around the current is clockwise. Therefore as the field hits point the P, the field is going in the negative x direction?

Yes, that's exactly right.

 

[TSny]

Looks Ok,

In a more general case, the magnetic field(At point P) would be directed anti parallel to the tangent at the point x (the position of the wire)

This was an intuitive way of solving this problem, I am curious about the math.
The individual magnetic fields can be added as vectors but is there a way to find the direction and magnitude of the $B^{\rightarrow}$(ie a magnetic field vector) produced by a wire?

I'm not sure I understand your question. See here for how to get the B vector for a long straight wire.

 

[hms.tech]

I'm not sure I understand your question. See here for how to get the B vector for a long straight wire.

Very well, we have the "magnitude" of the B field vector but do we know its direction ?

(Please don't say its tangential to the circle because as I said earlier, that really doesn't give us the direction in abstract form)

 

[TSny]

Very well, we have the "magnitude" of the B field vector but do we know its direction ?

(Please don't say its tangential to the circle because as I said earlier, that really doesn't give us the direction in abstract form)

I don't understand what abstract form means (but that's my fault ) Of course, you can start with the Biot-Savart law and deduce the direction of the B-field of a long straight wire

 

[arrowface]

TSny, thank you for hanging in there with me. I appreciate the explanation as I have had trouble with a few related problems, but this helped clarify the logic behind them.


I was hoping you could help me figure out the direction of the force(in the same case) if there was a wire running through point P, parallel to the rest and in the +I z-hat direction. The problem asks for the force, however as last time I understand the math but not the direction.

The answer is that it points in the negative y direction. Is there a method of explaining why the force points down this way using the right-hand-rule?

 

[TSny]

Yes. Use your right hand rule that you mentioned previously where your thumb is with I, pointer is with B, and middle finger with F.

 

[arrowface]

If you have a moment could you please explain why your middle finger(force) points down?

Is this because we found which way B points from your previous help(-x), and since current points out of the screen in this problem that would mean force is pointing down?

 

[TSny]

That's right. Thumb in +z direction and pointer in -x direction. So middle finger ends up pointing down in -y direction. Good.