Electricity and magnetism
Magnetism 5 Magnetic force on a wire carrying current.
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- Let's explore the repercussions of this
- equation some more.
- So what was the equation?
- It was that the force of a magnetic field on a moving
- charged particle is equal to the charge-- that's not what I
- wanted to do-- is equal to the charge of the particle-- and
- that's just a scalar quantity-- times the
- velocity-- the cross product of the velocity of the
- particle-- with the magnetic field.
- Now, isn't the velocity vector just the same thing as the
- distance vector divided by time?
- So the velocity vector is equal to-- let's call the
- distance that the electron travels, l.
- Distance divided by time.
- So we could rewrite that, that the force vector is equal to
- the charge times-- and I'm doing this on purpose-- 1 over
- time, right?
- Times the distance vector taken-- you take the cross
- product with the magnetic field.
- All I did is I rewrote velocity as per time times
- distance, or distance per time.
- And this is a scalar quantity, at least for our purposes,
- time only has a magnitude.
- Maybe we could call it change in time.
- But it doesn't have a direction.
- We're not going at an angle in time.
- So we could take the scalar quantity out.
- It doesn't affect this vector cross product.
- So what we get left with is, force is equal to charge per
- time times-- and this is just a regular times, because this
- is just a number, it's not a vector-- times the cross
- product of the distance vector and the magnetic field.
- And what is charge per time?
- Coulombs per second?
- Well, that's just current.
- So we get that force is equal to current times the distance
- that the current is flowing along, taken-- and you take
- the cross product of that with the magnetic field.
- And sometimes this is written as a capital L because it's a
- vector and all that, but we started with a lower case l,
- so we'll stay with the lower case l.
- So let's see if we can apply this formula, which is really
- the same thing as this.
- We just took the division by time and took it out of
- velocity so we get distance.
- And we took it and we divided the coulombs, or we took the
- charge divided by that.
- So we took charge divided by time, or charge per unit of
- time, you get current.
- So this is really just another way of writing this.
- It's not even a new formula.
- You could almost prove it to yourself if
- you ever forget it.
- But let's see if we can use this to figure out the effect
- that a magnetic field has on a current carrying wire.
- So let me-- actually, I probably want to put this up
- at the top, just so that I have space to draw a current
- carrying wire.
- So let me rewrite it in green.
- So you're familiar with the formula in all colors.
- So now our new derivation is that the force of a magnetic
- field on a current carrying wire is equal to the current
- in the wire-- and that's just a scalar quantity, although it
- could be positive or negative depending on the direction.
- Well, current is always a positive number, but if this
- current is going in the opposite direction as our
- distance vector, then it might be negative.
- But I wouldn't worry about that for now.
- Let's just assume this is a current in the direction of
- the distance vector.
- So it's a scalar quantity current times our distance
- vector l, or maybe the length of the conductor.
- You take the cross product of l with the
- magnetic field vector.
- So let's see if we can apply that.
- Let's say that we have a wire.
- Actually, let's do the magnetic field first. I've
- been doing a lot of magnetic fields that
- pop out of the screen.
- Let's do a magnetic field that goes into the screen.
- And those are even easier to draw.
- They're just x's.
- Now, why is it an x?
- Because you're looking at the rear end of an arrow.
- That's why it's an x.
- And that's why a circle with a dot means a field or a vector
- coming out of the window.
- Because if an arrow was shot at you, all you would see is
- the tip of the arrow with maybe a little
- circle around it.
- But anyway, this shows us a vector going into the screen.
- So this is our magnetic field.
- That is B.
- I don't know, let's assign some value.
- Let's say that the magnitude of B is equal to 1 tesla.
- And let's say I have a wire going through
- that magnetic field.
- Let's say the wire is going along or it's in the plane of
- your computer monitor.
- Let me just draw a wire going through the magnetic field.
- And my question to you-- let me tell you a little bit of
- information about this wire.
- Let's say the wire is carrying a current.
- So I is going in that direction.
- And it is carrying a current of-- I'm just making up
- numbers-- 5 amperes, or 5 coulombs per second.
- My question to you is, what is the net force of this magnetic
- field on a section of this wire?
- And let's make this section of the wire, I don't know, let's
- say it's a 2 meter section of wire.
- So obviously the more wire you have, the more charged moving
- particles you'll have. So the larger a section you have, the
- more of a force you'll have on that longer piece of wire.
- So we have to pick our length.
- So we want to know, what is the force of the magnetic
- field on this section of wire?
- From here to here.
- So let's just go to this formula.
- The force is equal to the current.
- So that's 5 amperes.
- And remember, just from what we learned about electricity,
- the current is the direction that notional positive charges
- would travel in, and suits us fine.
- Because when we did the first equation, we cared about the
- direction a positive charge would go in.
- And if it was an electron or a negative charge, we would put
- a negative sign there.
- So that works fine.
- But if you ever have to visualize things as they maybe
- are in reality, but when you talk about electrons it's hard
- to say that they really are reality, because they're
- almost more an idea than an object.
- But it's always good to remember that when the current
- is flowing in this direction, that would be true.
- Because if they were positive charges moving, but we know
- it's a negative charge moving in the opposite direction.
- Or you can think of it as, maybe, holes.
- Well, I don't want to get into that.
- But anyway, the current-- you could visualize it if you like
- as positive charges going in this direction.
- So the current is going this direction.
- So you could view this distance vector
- that we care about.
- Its magnitude is 2 meters.
- Because that's the length of wire in question.
- And its direction is the direction of the current.
- So let me-- this is l.
- Sometimes I get a little carried away on tangents.
- So that is l.
- It's 2 meters in that direction.
- I is 5 amperes.
- And we already figured out that the
- magnetic field is 1 tesla.
- So what's this going to be equal to?
- So the force is going to be equal to-- we're using all SI
- units, so we don't have to convert anything-- 5 amperes
- times 2 meters in that direction.
- I won't specify right now, let's just say that's a
- magnitude of l.
- Actually, let me write it.
- Well, 2 meters times the magnetic field, 1 tesla.
- And so when you take a cross product of something, this is
- just a reminder.
- l cross B.
- That's equal to the magnitude of l times the magnitude of B
- times the sine of the angle between them times some unit
- directional vector that we figure out with
- the right hand rule.
- So we already did the magnitude of
- the distance vector.
- That was 2 meters.
- We did the magnitude of the magnetic field.
- And what's the sine of the angle between them?
- Well, if the magnetic field is going into the screen, if it's
- going straight into the screen, you could imagine a
- bunch of arrows shooting into the screen.
- Those are the vectors.
- While our distance vector, or this l is in the screen, they
- actually are perpendicular, in 3 dimensions.
- So this angle is 90 degrees.
- So this actually just becomes 1.
- So in terms of the magnitude, we're done.
- The l cross B magnitude is 2 times 1 tesla.
- And then we multiply that times the current.
- And then we actually have the magnitude of the force.
- The magnitude of this force is going to be equal to 5 amperes
- times 2 meters times 1 tesla.
- Which is equal to 10 newtons.
- And then the only question left is, what is the direction
- of the force that the magnetic field is exerting?
- And this is where we break out the right hand rule.
- And it's no different.
- You could just imagine one of the positive particles moving
- in that direction, and just use the right hand rule.
- So let's take our hand out.
- And if we-- let me draw a hand.
- A right hand.
- So this is my right hand.
- If I have my thumb sticking out like that.
- So the l is going to be my index finger.
- The first thing in the cross product.
- And then the B is the magnetic field.
- That's going into the screen.
- So you can't see it.
- All you can take my word for it is that my middle finger is
- pointed downwards into the screen and then my other
- fingers are just doing something else.
- And there you have it.
- Your thumb is actually the direction of the force.
- Your index finger is the direction of-- we'll say l for
- these purposes.
- And then the magnetic field is going into it, so you can't
- see my middle finger but it's pointing downwards.
- I could draw a little x there, to show it's going downwards.
- And then the force is what my thumb is doing.
- So the force on this wire, or at least on that section of
- wire, is going to be perpendicular to the direction
- of the current.
- And that direction is going to be a 10 newton force.
- Anyway, I've run out of time.
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