Do the current elements I1L1 and I2L2 produce a force on each other?

[martinezwsu]

Homework Statement

Two current elements are perpendicular to each other. Wire 1 is oriented vertically,has a current I₁ flowing in the +Y direction, and has a length L₁. Wire 2 is situated above wire 1, has a current I₂ flowing in the -X direction, and has a length L₂. Do the current elements I₁L₁ and I₂L₂ produce a force on each other? If so, what is the magnitude and direction of each? Do they violate Newton's 3rd law? Below is my best reconstruction of the diagram my professor drew on the board today. Its kind of hard to draw this way but the only thing wrong with my diagram is that wire 1 should be centered with wire 2(not intersecting!)



Wire 2
----------<<I₂L₂<<-------------

l​

l​

Wire 1

l​

^​

^​

I₁L₁

l​

l​

l​

Homework Equations

F=I*L*B

The Attempt at a Solution

Beginning the first part of the problem, I tried to find the forces that the wires produced on each other. I believe that the Force on wire 1 due to wire 2(F₁₂)=I₁L₁*B₂. Using the right hand rule the B field caused by wire 2 is coming out of the page, below itself, and into the page above.

Now when it comes to the force on wire 2 due to wire 1(F₂₁), I know the equation for F₂₁=I₂L₂B₁. The B field produced by wire 1 is out of the page to the left of wire 1, and into the page to the right.

Now that I have the magnitude of the Forces the wires produce on each other, and the direction of their respective magnetic fields, I believe my next step is to find the direction of the forces. Using the right hand rule for F₁₂, I point my fingers in the direction of I₁L₁ and sweep them towards B₂. This gives a direction of F₁₂ in the +X direction.
Now for the direction of F₂₁ I point my fingers in the direction of I₂L₂ and sweep them towards B₁, but I don't know which direction this is since I can't decide how B₁ interacts with wire 2. Does B₁ interact with wire 2 at all? I'm stuck here and don't know if what I've done so far is even right.

 

[Ajwrighter]

I know I'm not a mentor or anything but.. Write down Newtons third law.. what does it say aka what does it really mean? Now write down your wire equation again.. what does it say? Are they the same or are they different? What does your intuition say regarding the difference between the two?

 

[Ajwrighter]

Additionally you might want to get into the habit of writing down the x (cross)-format. All that X (cross) means is that one part is crossing over to another part, like a swinging door that crosses from one room to another each time it is pushed open or closed.

 

[martinezwsu]

I know I'm not a mentor or anything but.. Write down Newtons third law.. what does it say aka what does it really mean? Now write down your wire equation again.. what does it say? Are they the same or are they different? What does your intuition say regarding the difference between the two?

Should F₁₂=-F₂₁ according to Newtons 3rd law?


Also I'm still perplexed at how the B field caused by wire 1 effects wire 2. I know that the field lines make a circle around the wire but I don't know if the field lines caused by wire 1 interact with wire 2 at all. I don't know, and haven't been able to find any information on what the magnetic field lines look like above wire 1.

 

[Ajwrighter]

Should F₁₂=-F₂₁ according to Newtons 3rd law?


Also I'm still perplexed at how the B field caused by wire 1 effects wire 2. I know that the field lines make a circle around the wire but I don't know if the field lines caused by wire 1 interact with wire 2 at all. I don't know, and haven't been able to find any information on what the magnetic field lines look like above wire 1.

Step1: what does Newton's third law state?

Step2: Draw the field lines radiating outward as if you could visually see the field itself. Start at one end and to the other using your right hand rule. Do this for both wires as they are described in the problem. Do the field lines touch?

Step3: Draw a second picture with both wires only this time draw the direction of the force for the wire on the x-axis. What does it tell you in regards to the wire on the y-axis

Here is a hint: pretend 'you' are the wire on the x-axis, stretch your right hand to the right and let that be considered the direction of the magnetic field. Now stretch your left hand in front of you and let that be the force. Now spin your self around without bending your elbows. What would happen if your left hand came into contact with something? Would your hand go right through the object?

 

[martinezwsu]

Step1: what does Newton's third law state?

Step2: Draw the field lines radiating outward as if you could visually see the field itself. Start at one end and to the other using your right hand rule. Do this for both wires as they are described in the problem. Do the field lines touch?

Step3: Draw a second picture with both wires only this time draw the direction of the force for the wire on the x-axis. What does it tell you in regards to the wire on the y-axis

Here is a hint: pretend 'you' are the wire on the x-axis, stretch your right hand to the right and let that be considered the direction of the magnetic field. Now stretch your left hand in front of you and let that be the force. Now spin your self around without bending your elbows. What would happen if your left hand came into contact with something? Would your hand go right through the object?

Step one: Newton's third law states that for every Force there is a reaction force equal in magnitude in the opposite direction.

Step two: I've drawn the picture and the field lines only touch below wire 2. They flow in the same direction on the left of wire 1 and they oppose each other on the right.

Step 3: Since I know that F₁₂ is in the +X direction, given the Newton's third law, F₂₁ must be in the -X direction. I believe you are trying to tell to to "test" this. I am having difficulty testing this with your hint and I am having some difficulty understanding that. However I have tested it using my understand of the right hand rule. According to the rule my thumb is the product(F₂₁) and my fingers point towards the current/length vector. My fingers sweep towards the magnetic field. Without sweeping my fingers i have trouble understanding how the derived direction of the force can be in the same direction of the current. Does this mean that these wires do in fact violate the 3rd law? And also how can I determine the real direction of F₂₁?

I believe that(this is just a theory) I can add the field lines that intersect below wire 2 and since 3/4ths of them are directed out of the page, then that is the direction of the B field that I can use to find to "true" direction of F₂₁. Pointing my fingers with the -X direction of the IL vector of wire 2 and sweeping them in out of page(in the direction of the B field), my thumb points up. Does this mean that the "true" direction of F₂₁ is +Y?

I am sorry if I'm being difficult, but I really appreciate your help.

 

[martinezwsu]

I'm also adding this diagram that I've drawn in Paint to go along with your suggestions.

 

[martinezwsu]

Doing a search for cases in which Newton's 3rd law could be violated, I believe I found what the answer could be.

I found this explanation in this thread:

https://www.physicsforums.com/showthread.php?t=94627"

Scroll down to find this post by vanesch:

In a way, we can equate Newton's 3rd law with "conservation of momentum". Indeed, if the "force on A" is dp1/dt and the "force on B" is dp2/dt and one says that these forces must be equal and opposite, this means that:
d(p1+p2)/dt = 0, so p1+p2 is constant.
The examples of relativistic charges not exerting "equal and opposite" forces on each other is of course correct, but it is not because "Newton's third law" is failing, it is because we've missed a player in the field (): the EM field itself.
So we naively think that particle A is exerting a force on particle B (Newtonian action at a distance), but we're missing that particle A is in fact NOT interacting with particle B, but only with the local EM field, and that this local EM field has its own dynamics, which results in the end in it interacting with particle B.
So this is in fact equivalent with a totally "Newtonian" picture where particle A hits particle C, which flies with finite velocity towards particle B.
If you forget to include particle C, you will find a "violation of Newton's third law" when particle C is travelling.
cheers,
Patrick.

So it appears that the third law has been violated when only F₂₁ and F₂₁ are taken into account. But once the effect that the magnetic fields have on each other is taken into account, then it can be found that energy HAS been conserved and that a violation of the 3rd law has NOT taken place. What do you think?

 

[martinezwsu]

Bump.

I'm going to work out the math right now, but does anyone else have any more ideas? Is it in fact the case that the forces exerted by the wires on each other are not the only forces in the system, and that when the force that the magnetic fields have on each other is taken into account, that only then is energy conserved?

 

[ideasrule]

You can't use F12=I1L1*B2 because the magnetic field generated by either wire is not constant across the length of the other wire. Instead, you'll have to integrate to find the force on wire 2, and use the Biot-Savart law (as well as integrate) to find the force on wire 1.

 

[martinezwsu]

Whoosh. Way over my head with that one. I get what you mean about the magnetic field being different at different radii, but I am not in calculus based physics, and we haven't had any mention of Biot-Savart. Do you have any suggestions that would not involve integration?

 

[martinezwsu]

I figured out how to get the magnitudes of the fields working with integration but I cannot figure out what the direction of the resultant B field. Any help?

 

[martinezwsu]

Bump.