Electricity and magnetism
Magnetism 7 The magnetic force that two current-carrying wires exert on each other.
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- We've now learned that a current or a stream of moving
- charges can be affected by a magnetic field.
- And we've also learned that it can induce a magnetic field.
- So that begs the question, what is the effect of one
- current carrying wire on another current carrying wire?
- Let's do that.
- Let's draw my first current carrying wire in green.
- That's the first current carrying wire.
- And let's say that the current is-- it's in magenta-- and
- we'll call this current 1.
- And then I have another current carrying
- wire not too far away.
- And I will call that current I2.
- Now what else do we need to figure out?
- Oh, well, let me just tell you.
- Let's say that they are a radius of r apart.
- And I say radius because we learned in the last video that
- the magnetic field created by a current carrying wire is
- kind of a, you know, they're kind of these circular
- cylinders around the wire.
- So let's say the distance from this wire to that wire is r.
- That distance is r.
- And so my question to you is-- well, first, just before we
- break into the math-- what's going to happen?
- Well, we don't know the magnitudes of the currents or
- anything just yet.
- But what's going to happen?
- What will be the net effect on, let's say, this wire?
- Let's say for some reason this wire is fixed or we could say
- they're floating in space.
- Let's just focus on wire 2 for now.
- This is wire 2, this is wire 1.
- What's going to happen to wire 2?
- Well, let's think about it.
- Wire 1-- the current in it is going to
- generate a magnetic field.
- Now what's the shape of that magnetic field
- going to look like?
- Well, we could take our right hand, do that right hand wrap
- around rule.
- It's a little different than the cross product rule,
- although it's kind of a byproduct.
- So that's my right hand and I'm wrapping it around.
- So if I point my thumb in the direction of the current-- so
- that's the direction of the current, just like I did--
- then the magnetic field goes in the
- direction of my fingers.
- So they're going to go around this wire.
- And so if I were to just draw the magnetic field where it
- intersects with this screen, on the right hand side it will
- go into the screen.
- So we'll just see the rear ends of the
- magnetic field line.
- And I'll draw it in the same color as the current, so you
- know it's being created by I1.
- So I1-- its effect keeps going out to infinity, although it
- gets much weaker as we learned.
- It's inversely proportional with r.
- But this is the field of I1.
- I can draw these-- I don't want to crowd
- my page up too much.
- And then on this side of I1, what happens?
- Well, on this side, you can see the
- fingers come back around.
- So it pops out when it intersects
- with your video monitor.
- So on this side, the vectors-- this is the top of an arrow,
- coming out at you.
- Fair enough.
- So I1, by going in this direction, is generating a
- magnetic field that, at least where I2 is concerned, that
- magnetic field is going into the page.
- So what was our formula?
- And this all came from the first formula we learned
- about, the effect of a magnetic field soon. on a
- moving charge.
- But what was the formula of the net magnetic force on a
- current carrying wire?
- It was the force-- I'll do it in blue-- it's a vector, has a
- magnitude and direction-- is equal to the current.
- Well, in this case, we want to know the force on this
- current, on current 2, right?
- Caused by this magnetic field, by magnetic field 1.
- So it will be equal to I2, the magnitude of this current,
- times L-- where L is-- because you can't just say, oh well,
- what is the effect on this wire?
- You have to know how much wire is under consideration.
- So let's say we have a length of wire.
- And then of course, if you know the length of wire and we
- knew its mass and we knew the force on it, we could figure
- out its acceleration in some directions.
- So let's say that this distance is
- L, and it's a vector.
- L goes in the same direction as the current.
- That's just the convention we're using.
- It makes things simple.
- So that's L.
- So the force on this wire, or at least the length L of this
- wire, is going to be equal to current 2 times L.
- We could call that even L2, just so that you know that it
- deals with wire 2.
- That's a vector quantity.
- I could make it a full arrow.
- Doesn't matter.
- It's just a notation.
- I've seen professors do it either way, I've seen it
- written either way, as well.
- Cross the magnetic field that it's in.
- What's the magnetic field that it's in?
- The magnetic field-- I'll do it in magenta, because it's
- the magnetic field created by current 1.
- So it's magnetic field 1, which is this magnetic field.
- So before going into the math, let's just figure out what
- direction is this net force going to be in?
- So here we say, well, the current is a scalar, so that's
- not going to affect the direction.
- What's the direction of L2?
- This is L2.
- I didn't label it L2 on the diagram.
- What's the direction of L2?
- Well, it's up.
- And then the direction of B1, the magnetic field created by
- current 1, is going into the page here.
- So here we just do the standard cross product.
- Let me see if I can pull this off.
- This is actually an easy one to draw.
- So I put my index finger in the direction of L2.
- And then I put my middle finger in the
- direction of the field.
- So my middle finger's going to point straight
- down into this page.
- My other fingers just do what they would naturally do.
- And then my thumb would go in the
- direction of the net force.
- This is just the cross product.
- You'll see teachers teach the cross product other ways,
- where they tell you to put your thumb in the direction of
- the field, and this and that, your palm--
- those are all valid.
- They're just different variations of the same thing.
- I find this one easier to remember.
- Because when I take the cross product, index finger is the
- first term of the cross product.
- Middle finger is the second term of the cross product.
- Thumb is the direction of the cross product.
- So anyway, this is the direction of L2.
- The magnetic field, we already know, goes into the page.
- So my middle finger is going into the page.
- And my thumb is in the direction of the force on the
- magnetic field.
- So that's the direction of the force.
- So there you have it.
- If this current is moving in this direction and its field
- is-- we know from this wrap around rule that pops out here
- and it goes in here-- the effect that it has on this
- other wire is that where the current is going in the same
- direction, is that it will be attracted.
- So the net force you is going in that direction.
- We could say the force from 1 on 2.
- That's just my convention.
- Maybe other people would have written it the force
- given to 2 by 1.
- That's the force given by 1 to 2.
- That's how I'm writing it.
- Now what's going to be the force on current 1 from I2?
- Well, it's going to be the current-- well, it's going to
- be the force there.
- Well it's going to be the same thing.
- Let me draw I2's magnetic field.
- You do the wrap around rule, it's going to look the same.
- So I2, sure, on this side its field is going to be going
- into the page.
- But what's I2's field going to be doing here?
- It's going to be popping out.
- I just did the wrap around-- take this wrap around, wrap it
- around that wire.
- So that's the field of I2.
- So then we can write down that the force-- and let's take, I
- don't know, this is some distance.
- Let's call that L1.
- So the force from current 2 on wire 1 of length L1, from here
- to here, is equal to current 1 times L1-- which is a vector--
- cross the magnetic field created by current 2.
- And so we can do the same cross product here.
- Put our index finger in the direction of L1.
- That's what you do with the first element
- of the cross product.
- And then you put your middle finger in the direction of B2.
- And then your thumb is going to tell you what the net force
- is going to be.
- So let me draw that.
- So let me draw my hand.
- And just so you know, before I do any of these, I actually
- look at my hand, just to make sure I'm
- drawing the right thing.
- So my index finger in the direction of I1, my middle
- finger-- sorry, my index finger in the direction of L1,
- which is the same as I1, and then my middle finger is going
- to do what the magnetic field is doing.
- So my middle finger is actually going to point
- straight up.
- And then my other fingers are just going to do what they do.
- And so now you're looking at the palm of my hand.
- And my thumb-- let me make sure I'm doing this correctly.
- Oh, no.
- I was drawing my left hand.
- See, that's an error.
- You don't want to draw your left hand when you're doing
- the right hand rule with cross products.
- So let me draw it down here.
- My index finger going in the direction of L1.
- My middle finger's popping straight up, because the
- magnetic field created by I2 is popping straight
- out of the page here.
- So my middle finger goes straight up and my other
- fingers do what they need to do.
- Looking at the palm.
- And then my thumb will go in that direction.
- So the cross product of L with B2 popping out of this page,
- the net force is going to be in this direction.
- So there's a little bit of symmetry here.
- This wire's going to be attracted towards that wire,
- and this wire's going to be attracted to that wire.
- They're both going to-- eventually if they were
- floating in space, they would slowly get closer and closer
- to each other and their radiuses would get closer and
- closer and they would accelerate to each other, at
- ever increasing rates actually.
- Anyway, I'm out of time.
- In the next video I'll do this same principle, but we'll do
- it with some numbers.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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