No, change in energy per unit time is power, which is measured in watts. It's the P in P=IV. Voltage is change in energy per unit charge.

Yes, humans perceive changes in the kinetic energy of the eardrum as sound. However, if a tree falls in the wood, and no one is around to hear it, it still produces sound waves, so it's imprecise to describe sound as the way humans perceive kinetic energy.

The second sentence is the easier one. You're listing independent descriptions of the film. You could describe it as a "dark film" or as a "greasy film" without being wrong, therefore commas are appropriate.

The first sentence is more complex to analyze. The two descriptions aren't completely independent. You can drop "white" without damaging the sentence -- saying "middle-class Americans" would still be correct. However, if you were to say "white Americans" were affected, then you would be wrong, because poor white Americans (here the comma MUST be omitted to be correct!) were not affected due to not having TVs, and rich ones (notice that the pronoun "ones" replaces "white Americans" as a whole unit) were not affected due to already having access to everything they want.

I apologize for being imprecise. It doesn't require that there be more than one kind; that was meant as an example.

So I can sort of say that voltage is the difference between joules of PE used to move charge in Coulombs at two different points. But what does it mean that joules of PE get used? I mean, in a circuit, isn't PE used up as kinetic energy for electrons all the time? Or is it that because voltage is the difference in energy being used up, it means that only when there is a resistance, more PE is required to move electrons along because electron energy gets lost, then there's a voltage?


So..commas are used if two descriptions of something are independent?
Why would "middle-class Americans" be fine? Is it that if I use "middle-class" it implies white so I don't need to add it?

Voltage is the difference (in joules) in potential energy available to one coulomb of charge before and after moving through a part of the circuit.

Again, thinking of things in terms of cause-and-effect relationships isn't always accurate. A good scientist shouldn't think about the how or the why when making a measurement, because that can introduce bias in the results. The measurement simply is what it is. It's an empirical observation; it's data. That data can then be used in support of a hypothesis about the causes and the effects, but that data is not itself a description of either.

To be more concrete: don't necessarily think of it as "using" potential energy, because it could also be GAINING it -- if you force a charge backwards against the potential difference (for example, if you're recharging a battery) then that charge now has potential energy that can be released to flow back through the circuit.

To actually DISCUSS the hows and whys here: For a given, fixed voltage, it doesn't matter how much resistance there is; that voltage will always be capable of moving that coulomb of charge. You can see this in the examples you've been working: the voltage drop across a single resistor varies according to the total resistance of the circuit. Resistance affects how long it takes that unit of charge to pass through the circuit: V=IR shows us that.

This isn't WRONG, but it's insufficiently precise. To get into the nitpicky details, there are two ways to use adjectives: coordinate adjectives, and cumulative adjectives.

Coordinate adjectives each modify the noun separately. You can rearrange them without changing the meaning. These are written with commas. If you can replace the comma with the word "and" without making it sound weird, then this is the type of use you're looking at.

A cumulative adjective effectively creates a new compound noun, and as a result, you can't separate the adjective from the noun without changing the meaning. For example, the "hot" in "hot rod" is being used as a cumulative adjective. If you compare "the yellow hot rod" and "the yellow, hot rod" you will see they mean different things -- while both are brightly-colored, the first one is a fast vehicle; the second one is a heated bar. Apply the tests: "hot yellow rod" doesn't work to describe a car, nor does "yellow and hot rod".

There's a gray area in between, and I think the "white middle-class Americans" example is one of them even though it leans more towards cumulative. To make things more abstract... If I had a lot of marbles, and I asked you for the large, blue marble, then it could go either way, and the nuance is mostly one of grouping. I could be asking you "of the blue marbles, I want the large one," or I could be asking you "of all of the marbles, I want the one that is large and blue." The former (which wouldn't have a comma) would be more intuitive if the marbles were already sorted out by color, or if all of the marbles were blue in the first place, or for whatever other reason I considered "blue marble" to be a specific type of thing, and then I was being more specific within that grouping.

Not so much the implication as the generalization: black middle-class Americans were also affected the same way, I would imagine. But proportionally speaking the overwhelming majority of middle-class Americans were white, so if you took a snapshot of the affected people, you would see that most of them are white and middle-class.

Oh, so voltage is a measurement. Err, so at one point of a circuit voltage and potential energy is something. At another point in comparison to that point it might be something else. Voltage is a measurement of potential...but what is potential energy a measurement of?

Why is cross-sectional area used to compare with resistivity in R = pL / A? Is circumference ever used?

Ohh, I understand the different now. If there's a comma then the last bit without the comma is what is being described. If there's no comma then all the words after the first word is one thing. Thank-you!

Oh, but that means if someone says "bright yellow sun" it technically(?) means a bright yellow-sun :/ ? What about a bright-yellow sun? Do I just use a dash?

Not quite. Voltage is a measurement of potential difference. You don't measure voltage at a single point. You measure voltage across two points.

Potential energy as a general concept is a measurement of how much energy could be released if the subject were free move on its own. It is, in a sense, stored energy. You're familiar with gravitational potential energy. The other kinds are no different.

The cross-sectional area is used because not all wires have a circular cross-section. Obviously for a cylindrical conductor you could use R = pL / (pi*r²), but cross-sectional area is a more general concept.

Usually the distinction DOESN'T matter, because "(bright yellow) sun" and "bright (yellow sun)" are both valid parses of the sentence. If the distinction truly matters, then yes, you can use a hyphen (not a dash, they're technically different). Though usually if the distinction matters you just reword the sentence ("The sun was bright yellow as it crossed the sky" instead of "the bright yellow sun crossed the sky").

That said, "bright yellow" is a somewhat unusual case -- technically it ought to be "brightly yellow" if you're treating it as a modified adjective (because adjectives don't modify other adjectives), but instead it's actually a modified noun ("yellow" as a thing instead of a description) being used as an adjective.

Ohh, it hadn't occurred to me that a wires could be not circular.

Hmm, so when there is PE, it basically means if there were charge at a certain position with PE, it would move some way. This means voltage measures the difference in how much a charge would get pulled at different points? But isn't current the same throughout the entire wire? Why would there be a difference in PE? Oh actually, I think I'm getting messed up between potential different and change in potential energy.
So..voltage measures difference in required potential energy per charge between two points? And as electrons move along, they require more potential energy per charge because of resistors using up energy? Or if electrons are forced the other way around..more PE per charge is required the other way around?
..Am I talking about cause-and-effect again? But without talking about cause-and-effect what does PE per charge mean?

If "brightly yellow sun" means the sun is brightly yellow, then does "starry blue sky" mean that technically, the sky is starry blue? Although, maybe people normally put a comma between starry and blue :/

No, voltage measures how much work is done BY a unit charge going through those points (or conversely, how much work must be done to force the charge to go through those points in the opposite direction).

No, that's not the confusion you're having, because there ISN'T a difference.

It's either "required potential energy" or "difference in available potential energy".

Take some time to reflect on the points I've made above, and if you still have questions, ask again, but I think the change in perspective might answer a lot for you.

"Starry" is an adjective, not an adverb. If it were "starrily blue" then you'd be right, but I just made that word up right now.

So..is it like this?
Voltage is either the work it takes or would take to move one coulomb from one place to another.
Work is a convenient thing that relates force and distance to each other.
Hence, work in voltage is how much force and distance it takes to reach a certain point in space.
PE is how much work it would take for a given amount of charge reach that certain point in space.
It takes force and distance, or work, to move negative charges across a resistor because..how else does something move? Or, in the case of charging a battery, it takes work to move negative charges the other way around for the same reason, maybe.

Ohh, starry isn't even an adverb :/. I'd forgotten that things like "blue green" were different from "bluish green."

No, it's the work it would take to move the coulomb upstream, or the work the coulomb can do while moving downstream.

Imagine a weight. Pick it up off the ground; you put work into the weight to lift it against the gravitational field. You've charged the battery.

But if you tie the weight to a rope on a pulley and let go of it, then that weight can itself DO work to lift up something on the other end of the rope. How MUCH work can it do? Disregarding friction in the rope and pulley, EXACTLY the same amount of work it took to lift it up that high. You've discharged the battery and used it to do something.

Work measures a change in energy. Applying a force over a distance is one way to get a change in energy, but it's not the only way -- you are now learning about another way using electricity. Heat is another way.

Voltage measures change in energy (that is, work) per unit charge. Equivalently, for each volt of potential, each coulomb of charge that passes through the circuit can do a joule of work.

Does that mean that voltage is both the change in potential energy per every positive coulomb of charge and the change in work per every coulomb of charge, because even though work is what has been done and potential energy is what could be done, the work that could be done is equal to work done if the work was done?

I guess in the context of electric fields work is applying a field force over a distance?

Yep, exactly, although that's a big statement to try to have in your head all at once. XD

Yes, but don't generalize that too far -- you can't expand W = Fd into W = (ma)d naively; you have to make sure that expansion makes sense in the context.

I finally understand that there's a difference between potential energy and potential energy per coulomb! Thank-you for explaining things all those times.

I have a circuit diagram with a ground that looks like this in one homework sheet:
http://i.imgur.com/wWOTcBX.png
What does the ground mean?

We actually discussed that one previously while we were talking about other stuff. The ground is an infinite current sink! It can accept as much charge as you want to push at it.

But if you're working with numbers in a circuit diagram... what does that mean?

Well, it's one of the few times when you can deal with an absolute potential instead of a potential difference! The ground defines a zero potential in the circuit.

I guess defining a zero potential would be useful when there's more than one battery..?
---
What does it mean when when a current shows up negative using Kirchhoff's Laws? I heard something about how it means that the direction of current chosen was wrong in class, and after searching online it seems that it means the current is flowing the other way? But what does that mean?

More than one battery, or when you want to make sure current flows in a specific direction, or as a voltage divider, or as a safety mechanism... there are a bunch of reasons.


I THINK that what you're seeing there is that the battery on the right is so much weaker than the one on the left that it's getting charged! The current is not only passing through R2 backwards, it's passing through V2 backwards!

And conveniently, you know what the voltage across the battery is: 1.0V! It's written right there on the paper. So it's acting like a resistor!

That said, I've never worked this kind of problem before and right now I'm not in a great position to do the research, so I can't follow up on this at the moment.

So..a ground can end a circuit, I suppose?

Thanks for the response! I don't know where to envision the current so I'll ask my teacher tomorrow if I get the chance (if not then on Monday).

Today I learned that you were right about the battery being charged!

How do you solve for sequences?
I have this homework problem to solve for the nth of a partial sum:
http://i.imgur.com/2uUaJeG.jpg?1
And I'm not sure what I'm supposed to be thinking. So far, I've just been trying to come up with random ideas for everything (except when things alternate from negative to positive, since we pretty much learned how to do that).

Welcome to your first step into calculus! This stuff you're doing right now is the fundamentals that eventually lead into integrals (on the continuous side) and recurrence relations (on the discrete side, which you probably WON'T do in high school).

I can't quite tell what you're trying to do there. Are you trying to find the sum of a_n for n = 1 to ___? If so then your work looks right, although you could have made life a little easier for yourself by simplifying the expression before summing it:

1/(n+1) - 1/(n+2) = ((n+2) - (n+1))/((n+1)(n+2)) = 1/((n+1)(n+2))
n=1 -> 1/(2*3) = 1/6 = 10/60
n=2 -> 1/(3*4) = 1/12 = 5/60
n=3 -> 1/(4*5) = 1/20 = 3/60
n=4 -> 1/(5*6) = 1/30 = 2/60
Σa_n from 1 to 4 = 20/60 = 1/3

Same results!

I'm not sure what S_n there means, though, unless that's supposed to represent the closed form of the sum? I know how to do that but it's... a much more advanced technique than what I would have guessed to you to use at this level. (PROVING that a given formula -- if you already know it -- is the closed form of a sum is much easier.)

Doing it that way looks a lot cleaner!

Err, I don't know what the closed form of the sum means, but the exact question was: "Find the four partial sums and the nth partial sum of the sequence a_n." Then I was given a_n = 1/(n+1) - 1/(n+2).
I haven't learned techniques for anything complicated though.

I barely know what a "partial sum" is since class ended the moment it was introduced. The previous question had been "Find the four partial sums and the nth partial sum of the sequence a_n" and I was given a_n = 2/3_n. I sort of gave up on finding the partial sum, so I turned to the answer key (which only answers odd questions) and it said "2/3, 8/9, 26/27, 80/81; S_n = 1 - 1/3" so I assumed the "partial sum" was basically an equation for whatever the sequence of partial sums was like.

A closed form is a simple equation that directly describes any element of a sequence without having to reference any other element or evaluate a sum.

For example, the closed form of f(n) = 1 + 2 + ... + n is f(n) = n(n+1)/2.

That answer you got doesn't make sense... Why would it ever write "1 - 1/3" instead of just "2/3"? There has to be an n in there somewhere... Maybe it's "1 - (1/3)ⁿ"?

Oh woops, I forgot to write the n, it is 1 - (1/3)ⁿ

I think that the homework is asking for that? Except without a technique but just by somehow reasoning it..?

Well, the general technique is pretty difficult. XD I studied it my junior year of college, if I recall correctly.

BUT!

Let's take a look at something:

S_n = S_n-1 + 1/(n+1) - 1/(n+2)
S_n = (S_n-2 + 1/(n-1+1) - 1/(n-1+2)) + 1/(n+1) - 1/(n+2)
S_n = S_n-2 + 1/(n) - 1/(n+1) + 1/(n+1) - 1/(n+2)
S_n = S_n-2 + 1/(n) - 1/(n+2)
S_n = (S_n-3 + 1/(n-2+1) - 1/(n-2+2)) + 1/(n) - 1/(n+2)
S_n = S_n-3 + 1/(n-1) - 1/(n+2)

See a pattern there? That might be something you can work with.

I didn't manage to get it in the end, except for a S_n = S_n + 1/(n+2) - 1/(n+2), which makes 0 = 0.

Huh. Okay, maybe I'm going to have to work that one myself then. XD

Lol xD

I tried it again to study for my math test tomorrow...I only got as far as [blank]/((n+1)(n+2)). Is there a way to write a nth term for something that pluses by another integer each time, as in 1, 3, 6, 10, 15, 21, etc?

Yes, there is. It usually involves an n². The particular sequence you described there is (n² + n)/2, or equivalently, n(n+1)/2, which I mentioned a few posts back and which I would be very surprised if you hadn't covered in class.

I guess then #38 would be (n(n+1))/(2(n+1)(n+2))

I as far as I recall, class hasn't covered anything divided by 2.
..I'm not entirely certain what has been covered in class except what sequences/sigmas/partial sums mean and that and looking for ratios and addition of similar numbers is important.

EDIT: How does (n² + n)/2 work?
Add a number the the same number multiplied by itself makes it always even..
n(n+1) means that with every increase in n, the "1" part increases by n too.
And then dividing by 2 makes it increase by 1 instead of 2.
So I guess I can instead of dividing it by 2, replace that with anything 2x = number to increase each time by that number? Only the first number will always get higher or lower. If I wanted it to stay at 1, then I'd have to subtract or add something extra?

That's not really the right way to go about it. It's good that you're developing those insights, but it doesn't quite generalize -- your conclusions are too specific to be especially useful.

A better strategy is to look at the changes in the sequence.

If your sequence is 1 1 1 1 1 (or any constant number), then there are no n's.

Otherwise, find the differences between the steps. 1 2 3 4 5 -> 1 1 1 1, for example, or 1 3 6 10 15 -> 2 3 4 5. Repeat this process until you get to a constant number. Each time you go through this process, the order of the polynomial increases by one -- that is, 1 2 3 4 5 has an n¹ term, 1 3 6 10 15 has an n² term, and 1 16 19 39 69 91 has an n³ term (if I didn't screw up my math). And it sometimes helps to start from 0 instead of 1.

Obviously it's possible for you to get stuck in a place where this technique doesn't work -- it only works for polynomials! If you've got a 2ⁿ in there, you'll instead need to consider the ratios between successive terms instead of the difference.

Thank-you very much for the n(n+1) thing! I had a math test today where I had to use (n-1)² + (n-1) (or I guess that's just n(n-1)?) to find some sequence. That is how to start from 0 right?

The period before the test I learned from someone in physics class that a_i = a₁r^{something to do with n} and I guess that works because when things are in ratios, it means that every next number is just that ratio multiplied again (hence the power)? And I'm not sure exactly why it's multiplied by the a₁ except that if the power were n-1 then it could be the first number..?

I'm not sure how I could've passed the math test without the knowledge of these things which I haven't learned in class though o_o'

It's multiplied by a₁ because k⁰ = 1.

That's one way to start from 0, yes. It's far from the ONLY way. In fact it's usually easier to start from 0 once you wrap your head around the idea. The n(n+1) works fine from 0 as it is because 0(0+1) = 0; we can clearly see that the sum of 0..n for n=0 is (drumroll please) 0.

Oh, so it's possible to just start n at 0 :o. Is there a way to indicate what to start n at? n₁=0?

Formally, you do it this way:
http://www.sprinklernewz.us/Math/summation-rule-1.jpg

This means "the sum from i=1 to n of i is n(n+1)/2".

It would also be correct to write i=0 for this one.

Ohh I see, so S_n can be written using that funny looking sign to be more specific.

Thank-you very much!

The symbol is called "sigma" (it's an uppercase sigma, to be specific) and it represents "summation". An uppercase pi (Π) represents a product (that is, multiplication) used the same way.

The S_n notation is more general. You can use it to describe any sequence. It's also fairly common to use f(n) notation, especially for sequences that you want to find a continuous (in the sense of a continuous function) extension for.

The disadvantage of the more general notations is that they are also less useful. Summations and products have a number of very useful properties -- for example, Σkx = kΣx -- that do not hold for generalized sequences. (Note that this is true in general: the more generalized a form is, the fewer consistent properties that are available using it.)

Oh wow, I didn't know that uppercase pi (for multiplying do something to n numbers together(?)) existed.

I'm on spring break so I've not done any math xD (no homework)

Πx is just 1 x 2 x ... x n, just like the sum, except with multiplication.

So... This is relevant for what you were studying recently, Potironette:


A visual proof that 1/3+1/3^2+1/3^3+...=1/2
https://pbs.twimg.com/media/C9smcDxUQAE0tvk.jpg

Ohh I remember my teacher splitting up pieces of paper trying to show that, though I only partly understood it. But why equals one half? Doesn't it never reach it?

There are techniques to actually find what the value would be if you went on to infinity. You'll study these in calculus. One that you could do right now would be to find the closed form representation of the sum (that was the S_n stuff) and plug infinity in for n.

The infinite sum converges to 1/2. The more terms you add, the closer you get.