Electricity and magnetism
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Electrostatics (part 1): Introduction to Charge and Coulomb's Law
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Electrostatics (part 2)
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Proof (Advanced): Field from infinite plate (part 1)
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Proof (Advanced): Field from infinite plate (part 2)
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Electric Potential Energy
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Electric Potential Energy (part 2-- involves calculus)
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Voltage
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Capacitance
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Circuits (part 1)
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Circuits (part 2)
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Circuits (part 3)
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Circuits (part 4)
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Cross product 1
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Cross Product 2
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Cross Product and Torque
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Introduction to Magnetism
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Magnetism 2
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Magnetism 3
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Magnetism 4
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Magnetism 5
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Magnetism 6: Magnetic field due to current
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Magnetism 7
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Magnetism 8
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Magnetism 9: Electric Motors
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Magnetism 10: Electric Motors
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Magnetism 11: Electric Motors
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Magnetism 12: Induced Current in a Wire
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The dot product
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Dot vs. Cross Product
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Calculating dot and cross products with unit vector notation
Capacitance Introduction to the capacitance of a two place capacitor
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- We learned several videos ago that if I had an infinite
- uniformly charged plane-- let me draw one right here, and I
- won't draw it infinite and I'll tell you why in a
- second-- that if we had an infinite uniformly charged
- plane, and let's say this one's positive, that the
- electric field generated by it is constant.
- Those are the field lines.
- They should all be the same size.
- And the strength of the field, or the magnitude of the field,
- is equal to 2 times Coulomb's constant times pi times the
- charge density of the plate.
- So if this is infinite-- so what was charge density?
- We defined it when we proved that this truly is a uniform
- electric field, but what is charge density?
- Charge density is just the total amount of charge divided
- by the area, or charge per area.
- Well, if we have an infinite plane, the area's going to be
- infinite, and so if this is a constant number, this is also
- going to be infinite, so it's kind of hard to work with.
- But what we also know is that when we have a non-infinite
- plane that has some finite area, that near the center of
- it and fairly close to it, it approximates an infinite
- uniformly charged plane.
- So with that said, let's see if we can figure out some of
- the properties of the voltage and how the voltage relates to
- the charge.
- If we were to have two parallel-- let me draw it
- before I say it, because I think saying it'll just
- confuse it.
- So let's say I have two plates, that plate-- and then
- I'll do this one in a different color-- and I have
- that plate, and let's say they're the same size and they
- both have area A.
- Let's say that I place plus Q worth of charges here.
- So this is plus Q so this is positively charged, right?
- I could draw a bunch of charges here.
- Let's say this is minus Q, so this is negatively charged.
- So what's the electric field going to look like
- between these two?
- Well, it's essentially going to be the combination of the
- electric field generated by this plate on top of the
- electric field generated by this plate.
- And they're both going to be constant close to the center,
- assuming that they're reasonably-- and let's say
- that they're d apart.
- Assuming that d isn't too big, near the center we're going to
- have a constant electric field.
- For example, this green one is going to be generating-- its
- field lines are going to look something like this.
- Near the center, it's constant.
- These are meant to look constant.
- Near the center, it'll look like that, and it'll start to
- bulge out when you get to the edges.
- Once again, near the center, it's constant.
- That one should've been at an angle.
- They start to bulge out, and it'll look
- something like that.
- And similarly, this purple plate will generate a constant
- electric field, and since it's negatively charged, the field
- lines will be going towards it, not away from it, so its
- field lines are going to look something like this.
- Near the center, they'll be constant, and its field lines
- are going to look something like that.
- As you can see, they're going to be of the same magnitude
- and in the same direction, and they also
- will bulge out there.
- So the big picture is that you just kind of have twice the
- electric field as you would have if you just had one of
- these plates.
- So let's say we're operating near the center of these,
- where we have roughly a constant electric field and
- see if we can figure out the relationship between the
- voltage across these two plates and the area, and maybe
- the distance between the two plates.
- So we know that the electric field generated by any one of
- these charge plates-- I'll do it in the blue of this color.
- So for the bottom plate right here, what is the electric
- field generated?
- It's 2K pi times sigma.
- Sigma is just the total charge divided by the area.
- So Q/A, right?
- And we know that the total electric field generated by
- this one is going to be essentially the same thing.
- I mean, we could say it's a minus, a negative, because
- it's going towards it, but it's
- essentially the same thing.
- Because we see that they overlap just drawing the field
- line, so the electric field from that one, and we know
- that they go in the same direction.
- If this was somehow-- well, this is negative, so the field
- lines go towards it.
- So plus 2K pi and this is Q/A.
- We could have said minus and then had a minus Q/A, but we
- know that they go into same direction, so we know that
- they're going to be additive.
- And so we know that the total electric field is
- going to be 4K pi Q/A.
- So now we know the exact strength of
- the electric field.
- Let's see if we can figure out the voltage difference between
- this point and this point.
- What was voltage difference, just as a review?
- Well, voltage difference is the electrical potential
- energy per charge if the charge was here versus here.
- So how much more potential energy per coulomb is there
- for a charge to be here relative to here?
- So another way to view it is a charge here, a positive charge
- here, because by default we're always assuming a positive
- charge when we talk about positive numbers and the
- direction of the field lines or what the positive
- charge would do.
- So by default, a positive charge here really wants to go
- up to this negative plate, although we later learned that
- most of the movement in electronics and electricity,
- it's actually the negative charge that's moving.
- It's the electrons moving.
- But let's say we did have a positive ion
- or a positive charge.
- The voltage is a measure of if any charge is here, how badly
- does it want to move to this point if it has a way to move?
- If we have air here or if we have a vacuum here, it might
- be difficult or impossible for it to move up here.
- But maybe if we were to connect a wire where the
- charges could freely conduct, then it will move.
- And the voltage is just kind of how badly
- does it want to move?
- You could almost view it as electrical pressure.
- And maybe I'll do a whole video on trying to get an
- intuitive understanding of voltage, because that really
- is probably the most important thing to get an intuitive
- understanding of, if you ever want to study electrical
- engineering or whatever.
- But anyway, back to the problem.
- We know that the combined electric field is this, right?
- It goes upwards in that direction.
- So what is the electric potential at this point
- relative to this point or the potential difference
- from here to here?
- Well, that's the amount of energy per charge it would
- take to move a positive charge from here to there, right?
- Remember, electric potential energy is the amount of work
- necessary to move a charge from there to there, and then
- the voltage is how much to do it per charge.
- Let me write that down.
- So the work necessary to move a charge from there to there--
- let's say a 1-coulomb charge, it will be 1 coulomb times the
- electric field, because we're always going to have to be
- going against the electric field.
- So we have to apply an equal and opposite force.
- So the force that is going to be the electric field-- so
- far, this just generates this force. coulomb times electric
- field, charge times electric field, tells us the force on
- the charge, right?
- That's force, and then we have to
- multiply that times distance.
- Force times distance.
- So we see the work necessary is going to be the electric
- field times d joules-- the J is joules-- and so what is the
- voltage difference or the electric potential difference
- between this point and this point?
- Let's call that point a.
- Let's call that point b.
- So Va minus Vb, which is the voltage difference, that's
- essentially the electric potential energy difference
- divided by the charge.
- Or, per charge.
- Well, here, the charge was just 1, so we can just divide
- by 1, and we see that it is equal to the electric field
- times the distance.
- And the units are going to be joules because we divided both
- sides by charge joules per coulomb, or volts, right?
- That's just the units.
- So what does that equal?
- So the voltage difference-- so we can say change in voltage.
- The voltage difference is equal to the electric field,
- which we know is constant 4K pi Q over A times distance.
- Or we could rewrite this.
- Let's see if we could write Q as a function of V.
- So if we just do a little bit of algebraic manipulation, we
- can get Q is equal to what?
- We would essentially divide both sides by 4 pi Kd and
- multiply both sides by A, so we would get A
- over 4K pi d voltage.
- And why is this interesting?
- Why did I go through all of this work to get this
- relationship?
- Well, what it shows you, if you look at this, if we assume
- that the area of the plates aren't changing-- that's a
- constant; this is definitely a constant-- and if we assume
- that the distance between the plates don't change, what we
- see is that there's a proportional difference
- between the voltage and the amount of the combined charge
- in the plates.
- And that's interesting because, before doing this,
- maybe voltage is somehow proportional to the square of
- the charges or to the square root, but now we know that
- it's directly proportionally.
- And actually, this term right here has a name, and it is
- called capacitance.
- And so another way of rewriting this, if we divide
- both sides by voltage, we get Q/V is equal to 1 over 4K pi
- area over distance.
- And so what it essentially says is that the amount of
- energy that-- well, actually, I don't want to
- go into that yet.
- But for a given configuration, and the configuration is
- defined by the area of the plates and the distance-- for
- a given configuration, if I know the amount of charge that
- I put onto the plates, if I did a minus Q here and a plus
- Q here, I know the voltage across the
- plates or vice versa.
- If I know the voltage across the plates and I know its
- configuration, I know how much charge there is, and this is
- called capacitance, and the unit for capacitance
- is called the Farad.
- And if you become an electrical engineer or even
- take a couple of electrical engineering courses, you'll
- become very familiar with this.
- And one other thing to point out; this term right here,
- just so you know a little bit of terminology.
- This term right here.
- This 1 over 4K pi, this is often called epsilon nought,
- or just epsilon, and that's called the permittivity of
- free space or permittivity of the vacuum.
- And maybe in a future lecture or a future video, I'll talk
- more about why it's called that.
- But anyway, I'm already well over the time limit.
- So I just wanted to give you a sense of, one, that you can
- calculate the voltage across what we call, in
- this case, a capacitor.
- It has capacitance.
- That voltage, you can kind of view it as
- the electric pressure.
- How bad does the charge here want to move here?
- And if you put a wire here, you'll learn in a second-- not
- in a second, in several videos-- that
- that charge will flow.
- Or actually the negative charges will flow this way and
- generate current.
- And we'll do that when we start learning a little bit
- more about electricity.
- For any given configuration, it has a corresponding
- capacitance, and then given that capacitance, if I put
- some amount of charge, I can figure out the voltage, or if
- I know there's some voltage, I can figure out the charge.
- Anyway, I will see you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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