WHY does the sign for B1 and B2 keep CHANGING in this Bnet equation?

[riseofphoenix]

The two wires shown in the figure below are separated by d = 8.0 cm and carry currents of I = 4.85 A in opposite directions.

d = 8 cm = 0.08 m
d = 8 cm = 0.08 m
r = 0.08/2 = 0.04 m
2d = 21.6 cm
I (up) = 4.80 A
I (down) = 4.80 A

(a) Find the magnitude and direction of the net magnetic field at a point midway between the wires.

magnitude: ______ µT
direction: ______

(b) Find the magnitude and direction of the net magnetic field at point P₁, 8.0 cm to the right of the wire on the right.

magnitude: ______ µT
direction: ______

(c) Find the magnitude and direction of the net magnetic field at point P₂, 2d = 16.0 cm to the left of the wire on the left.

magnitude: ______ µT
direction: ______

_______________________________________________________________________________________________________________________

Solution:


Assume that the wire on the right is wire 1 and that on the left is wire 2. Also, choose the positive direction for the magnetic field to be out of the page and negative into the page.

(a) At the point half way between the two wires,

B_net = -B₁ - B₂
B_net = -[(µI₁)2πr₁ + (µI₂)/2πr₂]
B_net = -(µ/2πr)(I₁ + I₂)
B_net = [-(4π x 10^-7)/2π(0.04 m)](4.85 + 4.85)
B_net = (-0.000001257/0.251)(9.7)
B_net = -0.000012193/0.251
B_net = -4.85 x 10^-5 T
or
B_net = 48.5 µT into the page

QUESTION: the signs for B₁ and B₂ changed. WHY? I don't understand!


(b) At point P₁,

B_net = +B₁ - B₂
B_net = (µ/2π)[(I₁/r₁) - (I₂/r₂)]
B_net = [(4π x 10^-7)/6.283][(4.85 A)/(0.08 m) - (4.85 A)/(0.16)]
B_net = (0.0000002)(60.625 - 30.125)
B_net = (0.0000002)(30.5)
B_net = 0.0000061
B_net = 6 µT out of page

QUESTION: the signs for B₁ and B₂ changed. WHY? I don't understand!

(c) At point P₃

B_net = -B₁ + B₂
B_net = (µ/2π)[-(I₁/r₁) + (I₂/r₂)]
B_net = [(4π x 10^-7)/6.283][-(4.85 A)/(0.24 m) + (4.85 A)/(0.16)]
B_net = (0.0000002)(-20.208 + 30.125)
B_net = (0.0000002)(9.917)
B_net = 0.000001983
B_net = 1.983 µT out of page

QUESTION: the signs for B₁ and B₂ changed. WHY? I don't understand!

 

[TSny]

The key is to clearly visualize the magnetic field pattern of a long straight current .

 

[collinsmark]

In addition to TSny's link, make sure you understand the right hand rule, and how the magnetic field of a long wire fits into that.

And lastly, the sign convention used in the solution was stated at the beginning of the solution,

"Also, choose the positive direction for the magnetic field to be out of the page and negative into the page."​

That last part is an arbitrary convention. You could have chosen the opposite convention and let positive be into the page and negative out. You would end up with the same, final answers in the end. The important thing to do here is to pick a convention, and stick with it consistently throughout the problem.

 

[riseofphoenix]

Oh I understand the link TSny gave me, but I still don't understand why the three equations consist of the following..

a) B_net = -B₁ - B₂

For instance, for this one, how did they conclude that the magnetic field the wire on the right (B₁) is negative? and that the magnetic field for the wire on the left (B₂) is also negative? Did they just "say so"? the diagram itself doesn't indicate that nor does the "Right hand grip rule"

b) B_net = +B₁ - B₂
c) B_net = -B₁ + B₂

And so essentially, you're saying that I still could have gotten the right answer if I did the following??

a) B_net = +B₁ + B₂
b) B_net = -B₁ + B₂
c) B_net = +B₁ - B₂

 

[collinsmark]

Oh I understand the link TSny gave me, but I still don't understand why the three equations consist of the following..

a) B_net = -B₁ - B₂

For instance, for this one, how did they conclude that the magnetic field the wire on the right (B₁) is negative? [/color]

Is it going into the page or out of the page. Use the right hand rule, and the decided upon convention, where negative is into the page (and positive out of the page).

and that the magnetic field for the wire on the left (B₂) is also negative? [/color]

Once again, use the right hand rule. Is the magnetic field of B₂ into or out of the page?

Did they just "say so"? the diagram itself doesn't indicate that nor does the "Right hand grip rule"

The decided upon convention where positive is out of the page and negative into the page was one of "just said so." It's an arbitrary convention, and could have been the opposite.

But once it is chosen, consistency is vital. Once a convention is chosen, it must be followed consistently from start to finish.

b) B_net = +B₁ - B₂
c) B_net = -B₁ + B₂

And so essentially, you're saying that I still could have gotten the right answer if I did the following??

a) B_net = +B₁ + B₂
b) B_net = -B₁ + B₂
c) B_net = +B₁ - B₂

Yes, as long as you are consistent about how define positive/negative going into or out of the page. Consistency is the important part.

 

[riseofphoenix]

Is it going into the page or out of the page. Use the right hand rule, and the decided upon convention, where negative is into the page (and positive out of the page).

That's the thing though! I have NO idea if the magnetic field is going into the page or out of the page... :(:(

That diagram only indicates it going CCW.

Can't I just say this instead?

Magnetic field is going CCW: -B
Magnetic field going CW: +B

 

[TSny]

That diagram only indicates it going CCW.
Can't I just say this instead?
Magnetic field is going CCW: -B
Magnetic field going CW: +B

At point a in the diagram is the field coming out at you or away from you?

How about at point b?

 

[riseofphoenix]

At point a in the diagram is the field coming out at you or away from you?

How about at point b?

At point a, it's moving away from me. At point b, it's coming out at me.

 

[riseofphoenix]

is that right?

 

[TSny]

is that right?

No. You can see the little arrows on the circular magnetic field lines that tell you the direction of the field lines (CCW). Suppose a little bug walked CCW around the circular field line that contains points a and b. When the bug reaches point a would the bug be walking toward you or away from you?

 

[TSny]

Here's a pic that shows a top view. Note that the field line goes CCW in the top view, but the B field points in opposite directions at P₁ and P₂.

 

[riseofphoenix]

Toward me?

 

[TSny]

Toward me?

Yes, the bug would be walking toward you (out of the page) at point a.

 

[riseofphoenix]

Yes, the bug would be walking toward you (out of the page) at point a.

Ohh...whenever I come across a problem like this,

1) Pay attention to what direction the current is going (up or down)
2) Use the right hand grip rule.

3) Determine whether the magnetic field direction is clockwise or counterclockwise

4) Label 2 points on the magnetic field on each, opposite side of your right hand that you are (supposedly) using to hold the wire with

5) The magnetic field vectors (B) are tangent to these points on the circular plane/magnetic field lines and is either going OUT OF THE PAGE (towards you) or INTO THE PAGE (away from you)

 

[TSny]

Right. Magnetic field vectors are always tangent to magnetic field lines.

 

[riseofphoenix]

So back to the problem...
I know that B_net = B₁ (right wire) + B₂ (left wire)

PART A

Looking at B₁ (right wire)
1) Current is going down
2) Using right hand grip rule (with my thumb pointing down), the magnetic field is going clockwise
3) Looking at the wire from the top, I draw 2 points on the right side of my hand and on the left side of my hand on the outermost circle. On the right, B1 is going OUT of the page (to me) and on the left, B1 is going INTO of the page (away from me), But am I'm only interested in the right point?

Looking at B₂ (left wire)
1) Current is going up
2) Using right hand grip rule (with my thumb pointing up), the magnetic field is going counterclockwise
3) Looking at the wire from the top, I draw 2 points on the right side of my hand and on the left side of my hand on the outermost circle. On the right, B2 is coming OUT of the page (to me) and on the left, B2 is going INTO the page (away from me), But am I'm only interested in the LEFT point?

4) Choose direction of B
If B is going INTO the page: -B
If B is coming OUT of the page: +B

5) Bnet = +B₁ (right wire) + -B₂ (left wire)

 

[riseofphoenix]

OK wait...im confusing myself...

I get the whole figuring about whether or not the magnetic field is going into the page or out of the page...
But to save time (since this is going to be on the test), all I need to know is that I can either do one of the following methods

First method
a) B_net = -B₁ - B₂
b) B_net = +B₁ - B₂
c) B_net = -B₁ + B₂

Alternative method
a) B_net = +B₁ + B₂
b) B_net = -B₁ + B₂
c) B_net = +B₁ - B₂

 

[collinsmark]

So back to the problem...
I know that B_net = B₁ (right wire) + B₂ (left wire)

PART A

Looking at B₁ (right wire)
1) Current is going down
2) Using right hand grip rule (with my thumb pointing down), the magnetic field is going clockwise
3) Looking at the wire from the top, I draw 2 points on the right side of my hand and on the left side of my hand on the outermost circle. On the right, B1 is going OUT of the page (to me) and on the left, B1 is going INTO of the page (away from me), But am I'm only interested in the right point?

I don't think that's the point you're concerned with. (Although I'm not quite sure I'm following you here.)

Since this is part a), you're concerned about the point that is "midway between the wires." (Because that is what was specified in part a).

So it's a point to the left of the rightmost wire (which is at the midpoint between the wires).

Looking at B₂ (left wire)
1) Current is going up
2) Using right hand grip rule (with my thumb pointing up), the magnetic field is going counterclockwise
3) Looking at the wire from the top, I draw 2 points on the right side of my hand and on the left side of my hand on the outermost circle. On the right, B2 is coming OUT of the page (to me) and on the left, B2 is going INTO the page (away from me), But am I'm only interested in the LEFT point?

Since we're still on part a, you are interested in a point to the right of the leftmost wire.

 

[collinsmark]

OK wait...im confusing myself...

I get the whole figuring about whether or not the magnetic field is going into the page or out of the page...
But to save time (since this is going to be on the test), all I need to know is that I can either do one of the following methods

First method
a) B_net = -B₁ - B₂
b) B_net = +B₁ - B₂
c) B_net = -B₁ + B₂

This is correct for this particular problem if you are following the convention that into the page is negative and out of the page is positive.

Alternative method
a) B_net = +B₁ + B₂
b) B_net = -B₁ + B₂
c) B_net = +B₁ - B₂

This is correct for this particular problem if (unlike the convention used in the posted solution), into the page is positive and out of the page is negative.

Both are correct, as long as consistency is maintained. Both methods yield identical answers about whether the net magnetic fields are pointing into our out of the page (again though, as long as consistency is maintained).

But I wouldn't bother memorizing those formula though. If the instructor changes the problem around, memorizing the relationship won't help.

Perhaps I can explain this better if I use an analogy. Do you remember in more basic physics, where you calculated the velocity of a rock falling off a cliff? If so, you may recall that it was up to you if you wanted to label the down direction as positive or negative. If you labeled down as negative and the final answer was a large negative number it meant the rock was falling down. If you chose the down direction to be positive, and the final [STRIKE]speed[/STRIKE] velocity of the rock was a large positive value, it still meant the rock was falling down. Either way, the final answer was the same, as long as you were consistent. It's the same idea here. You get to choose whether into the page is positive or negative (unless your instructor/textbook explicitly tells you to use a particular convention). Once you choose a convention just stick with it from beginning to end and it will all work out.

 

[riseofphoenix]

This is correct for this particular problem if you are following the convention that into the page is negative and out of the page is positive.

This is correct for this particular problem if (unlike the convention used in the posted solution), into the page is positive and out of the page is negative.

Both are correct, as long as consistency is maintained. Both methods yield identical answers about whether the net magnetic fields are pointing into our out of the page (again though, as long as consistency is maintained).

But I wouldn't bother memorizing those formula though. If the instructor changes the problem around, memorizing the relationship won't help.

Perhaps I can explain this better if I use an analogy. Do you remember in more basic physics, where you calculated the velocity of a rock falling off a cliff? If so, you may recall that it was up to you if you wanted to label the down direction as positive or negative. If you labeled down as negative and the final answer was a large negative number it meant the rock was falling down. If you chose the down direction to be positive, and the final speed of the rock was a large positive value, it still meant the rock was falling down. Either way, the final answer was the same, as long as you were consistent. It's the same idea here. You get to choose whether into the page is positive or negative (unless your instructor/textbook explicitly tells you to use a particular convention). Once you choose a convention just stick with it from beginning to end and it will all work out.

Ohhh! Ok. that makes sense!
Thanks a BUNCH! And sorry for all of the confusion xD