Thanks for the replies! For the first thing I was just curious about it. For the second one, someone mentioned not remembering what the teacher said about it so I think I missed that in class ^^;;

For math I had a homework question that asked for a sketch of: r(x) = (3x² + 6)/(x²-2x-3)
When there's an imaginary root, do I just ignore those? (I ended up simplifying the equation to (3(x² + 2)) / ((x-3)(x+1)) for the sketch)

Yeah, you can more or less ignore them when you're graphing.

As for WHAT an imaginary root is...

The real numbers aren't the only kinds of numbers you can work with using standard algebra. There's an extension of the real numbers called complex numbers: a complex number is the sum of a real number an an imaginary number.

Complex numbers allow you to have meaningful values for some things that the real numbers fail at. The most significant use is also the fundamental definition: The imaginary number i is equal to the square root of -1. And just as all real numbers are multiples of the basis 1, all imaginary numbers are multiples of the basis i, and so all complex numbers can be expressed in the form a + bi.

The square of any complex number is a real number. Unlike real numbers, though, where x² is always positive, the square of a complex number may well be negative.

The norm of a complex number (a + bi) is sqrt(a² + b²). This is an extension of the notion of absolute value: you'll notice it's equal to the absolute value for numbers without an imaginary part.

So what is an imaginary root?

Well, complex numbers work just fine and dandy in polynomials. You can multiply and add complex numbers just fine as long as you remember the identity i * i = -1. So if a real root is a solution to p(x) = 0 where x = a + 0i, then an imaginary root is a solution to p(x) = 0 where x = 0 + bi.

Let's consider p(x) = x² + 1. We know there are two roots, but none of them are real. But what about imaginary roots, now that we know about them?

Well, let's pull out our old friend the quadratic formula. a=1, b=0, c=1, so:

x = (-0 +/- sqrt(0² - 4*1*1))/(2*1)
x = +/- sqrt(-4) / 2
x = +/- 2sqrt(-1) / 2
x = +/- sqrt(-1)
x = +/- i

So x = i and x = -i are the imaginary roots of the function!



Now, if you want to think about what complex numbers LOOK like... It gets complicated!

If real numbers are one-dimensional, then complex numbers are two-dimensional. If functions of real numbers can be plotted in two dimensions... functions of complex numbers are plotted in FOUR. That's not easy to imagine on its own!

But you CAN think of it as a FIELD.

Imagine the real number line. If you were to map a function over it, then each point on that line would have a corresponding number for the value of the function at that point.

For complex numbers, imagine a plane. If you consider f(x + yi) -- that is, the real part of the number along the x axis, and the imaginary part along the y axis -- then you can evaluate the function at any point on the plane, and the result is a two-dimensional vector. For example, this is a graph of the function f(x) = (x + 2)(x - 2), if you allow x to take on complex values:

https://www.pacifict.com/images/cv16.gif

With a graph like this, the roots are the places where the length of the vector is zero. The real roots are places where this is true on the x axis. The imaginary roots are places where this is true on the y axis. There could also be complex roots: Consider the function f(x) = x⁴ + 1. We know it has to have four roots, but we can obviously see that none of them are real... Let's solve it anyway!

(Lots of algebra, put in a spoiler for brevity)


Therefore, the four complex roots of f(x) = x⁴ + 1 are +/-sqrt(1/2) +/- sqrt(1/2)i. No real roots! No imaginary roots! But if you take (x - sqrt(1/2) - sqrt(1/2)i)(x - sqrt(1/2) + sqrt(1/2)i)(x + sqrt(1/2) - sqrt(1/2)i)(x + sqrt(1/2) + sqrt(1/2)i) and multiply it out, sure enough you get back to x⁴ + 1 like we wanted!

Admittedly, a lot of that went over my head, but just to clarify...

>An imaginary number is anything with "i"
>A real number is anything without "i"
>A complex number is not an imaginary number nor a real number but a mix of the two?

Err..what do you mean by the "norm"? Is it the magnitude or something? Like the imaginary number is on the y-axis (I don't get why the y-axis but for some reason internet always uses it? Buuut they're likely just ignoring the y-axis and replacing it with a separate imaginary axis?) and then there's a point, say (3, 4i) and the magnitude is the same as sqrt(9 + 16) which is 5?
But I don't understand how the absolute value fits in..

Uhh, so what indeed? All I know about it is that i is the square root of -1, so I suppose it's all the multiples of the square root of -1?

When I have imaginary roots, and I make a graph, it it that I won't even see the imaginary root graphed?


As for the second post, I'm currently wondering what it means.
First, imaginary roots exist...
Real numbers are one dimensional because they're on a number line? And then having two number lines as the x and y axis they're...plotted as two dimensional??
Also, if real numbers are one dimensional and are plotted in two dimensions, then imaginary numbers should be one dimensional and plotted in two dimensions too! Maybe..?
Complex numbers are a combo of real numbers and imaginary numbers put together. How that works, I'm guessing, is by having a figure with four axis, two for imaginary and two for reals and somehow one point can be made from that? But how that works gets into the difficult visualization part.

It's possible to think of it as a "field." What is a field..?

From this point forward I'm confused.
What is f(x + yi) and what is the vector referring to?
In the graph picture, I guess the x-axis turned into an axis that for complex numbers because of the problem with how difficult it is to visualize three dimensions? But that can't be it, surely? I'm definitely guessing here. And besides, why have just one axis take on complex values? Why not both? Is that even possible?
As for the last bit I still need to follow it on pen and paper, but essentially it's possible for something to just have complex roots!


Not math question: What is a "pacifying issue"? I read somewhere that someone thought that, for sure, the government was thinking (1970) that Earth Day or something environmental was a "pacifying issue." Does it mean an issue the government wants to just ignore..? Thinks will just die after a while?

A complex number is a real number (which might be 0, so all imaginary numbers are complex numbers) plus an imaginary number (which might be 0i, so all real numbers are complex numbers).

Yes, norm = magnitude = absolute value.

Imaginary numbers are put on the y axis by convention purely because the real number line is usually drawn horizontally. It doesn't actually matter, but the convention makes it possible for people to look at a graph together and agree on what's being plotted.


Yes, if you look at JUST imaginary numbers, then they are indeed one-dimensional. However, the main reason you can't plot a function using imaginary numbers in two dimensions is because if you multiply two imaginary numbers together, you get a real number, so you wouldn't have a way to plot it. You COULD meaningfully plot it in THREE dimensions, and you'd get a single curve, not a surface.

Yes, you're completely right. This is sometimes graphed using color to represent some of the axes (often hue for one of them and brightness for the other), or it can be roughly sketched using the picture like I showed above.

You're familiar with electric fields, magnetic fields, and gravitational fields already. You can correspond every point within those fields to some useful value, such as "the force an electron would feel at this point". The same is true here: You can correspond every point on the xy plane with the value of the function at that complex number, and draw some representative examples, like in the picture I showed.

In fact, the example I showed could be interpreted as a magnetic field, with a bar magnet laid upon the x axis. You can see the vectors tracing out curves that go from one pole of the magnet to the other. There's an actual physical realization of this -- if you put a sheet of paper over such a magnet, and sprinkle iron filings over the paper, they will actually align themselves with that picture!



Just like you're used to seeing f(x) referring to a function over the real numbers, f(x + yi) is referring to a function over the complex numbers. The vector refers to the value of the function at that point -- a vector pointing straight right might represent 1+0i, one pointing straight up might represent 0+1i, one pointing up-right (and a little longer than the other two) might represent 1+1i, et cetera. The length of the vector corresponds to the norm (= magnitude = absolute value) of that complex number.

No, one axis isn't representing complex numbers. It does indeed require BOTH axes. The x axis is the real part of the input to the function, and the y axis is the imaginary part. That's why it's written f(x + yi).

Yep, that's correct. They can still be called imaginary roots, from the perspective of them not being real roots, but complex roots is technically more precise.

It's an issue that is brought up to attract focus to something that they think will bring peace to the public. Earth Day wasn't there for the purposes of ACTUALLY protecting the environment. It was created to make people feel good about thinking about the environment.

By calling it a "pacifying issue" the implication is that it's distracting away from more controversial issues.

Ohh, so f(x) means, the function where real number x is put into it, whereas f(x + yi) means the function where a complex number is put into it, in which the real number is on the x axis and the imaginary numbers are represented by the y axis?

So..basically for things like y=x² you can't plot imaginary numbers because when every x is an imaginary, x squared would be a real and thus not on a graph where both axis are imaginary..? But then y = x would be graphable and maybe y = x³ would be too..? But then there is a way to plot it--I think I'm misunderstanding something :/

Err, what's the difference between a curve and a surface? In fact, what is a curve and what is a surface?


Thanks for the definition of the pacifying issue! I wasn't seeing it while googling it. The person speaking about it was saying that the environmental issue was not just a pacifying issue but in fact serious, and I'd been confused about what he meant.

Yup, bingo.


You can of course find specific examples that you would be able to graph. (And yes, I think x³ works, because i³ = -i. The graph ends up being upside down relative to the real-valued version.)

Geometrically, this means you're taking that four-dimensional graph and taking a cross-section of it.

Whee, this starts getting into some crazy abstract geometry, and I've gotta be careful in defining this or else we're going to dive into another rabbit hole. XD

Okay, so... A point is a zero-dimensional object. No matter how many dimensions of space you have, a point in it is infinitely small.

A curve is a stretched-out point. It's infinitely thin, but it has a defined length. Imagine an infinitely thin piece of wire. It can bend around in however many dimensions you have available in the model -- in a one-dimensional system (for example, the real number line) all curves are straight line segments, but in a two-dimensional system (a graph), curves can bend around as long as they stay flat (because if they didn't stay flat then they wouldn't fit in the two-dimensional space). In a three-dimensional system (space), a curve can bend around in all three dimensions, et cetera. We say that a curve is a one-dimensional object embedded in an n-dimensional space.

A surface is a stretched-out curve. It has a defined area, but it's infinitely shallow, so it has no volume. Imagine a rubber balloon -- you can stretch it and bend it into any shape you want, but the rubber itself is still two-dimensional. It gets hard to visualize taking this into four dimensions, but you CAN bend it in four dimensions, and that's what counts here.

Anything higher gets called a hypersurface or an n-surface as long as it's embedded in a space with at least one dimension higher than itself. If it's not, then it's called a hypersolid / n-solid (or just a solid, if it's in three dimensions), because it's completely space-filling. (Yes, you could call a two-dimensional object in a two-dimensional space a 2-solid.)

I'm... pretty good at higher-dimensional geometry. >.> I have a reasonably easy time visualizing 2-surfaces embedded in 4-space. Klein bottles are fun. So are tesseracts. (Though tesseracts are 4-solids, not 2-surfaces.)

A cross section of a 3d object is 2d. A cross section of a 2d object might as well be 1d. Therefore a cross section of a 4d object a probably 3d?
And then therefore both quotes are saying imaginary numbers are plottable in 3 dimensions?
Why three dimensions though? What happened to the fourth one? What if something like x³ was plotted in four dimension? Would it not remain a curve? Is something like x² any different in the third dimension compared to with a fourth dimension? I guess f(x) = x where x and y are both imaginary works on in one dimension though?

--

So...basically every dimension is the other dimension stretched out? Like a point(0) stretches into a line(1) which stretches into a flat thing(2) which stretches into a 3d thing like a cube or a marble or whatever real-life thing(3) which stretches into err..some weird looking object..

Hmm, the problem for me with klein bottles (I googled it) and tesseracts, is that I'm not sure how to see it as not 3d.
With klein bottles, since it's 2-surfaces in 4-space, I guess maybe the tube thing on the inside is a 2-surface..? (edit: after writing the bottom, I guess a surface is literally anything that has an area but no volume, so a klein bottle is just showing a 2d surface existing in 4 dimensions?) As for what the 4-space is, I'm confused about what makes it a 4-space.

With tesseracts, since it's a 4-solid it means that it is a 4d object in 4 dimensions? What even is the 4 dimensions that I'm supposed to look at? Am I imagining that, say, the center of the tesseract would be like this 3d world (the world is 3d, right xD? Or is it just one way to look at it x'D?) then lots of other worlds were to be put all around it. And then, to see in 4d would to be able to see all those 3d worlds at the same time xD? Like, if a rubix cube were transparent? ...does that mean a klein bottle is 4-space for having 3-d things technically stretched out because the middle spout is a surface wrapped around 3d air and so is the outside surface of the bottle?

Occasionally in video games a 3d character would clip together. That's about as far as I'm understanding what's beyond 3d. If I held up a piece of paper and looked at it exactly from the side, that would be two lines clipping together and I guess if I were 2d I could "see" 3d things that way. But that's not really what 3d is. If two 3d objects had the chance to clip together, that would be looking at a 4d world from the side xD?

Oh, and with tesseracts, I forgot about them rotating. It looks like a klein bottle were constantly having its surface moved around? But not really...


...Whenever something comes up that I slightly don't understand there's a chance I'll ask about it depending how much I can think about it. I think I understand a surface is a 2-d thing and a curve is a 1-d thing and just because something is 2-d or 1-d doesn't mean they can't be bent in higher dimensions.
Though, can they be bent into a lower dimension?

You can take a cross-section of any dimension lower than the space. In geology, you can study a core sample -- drill out a long, narrow, practically one-dimensional cylinder so you can inspect the layers of the larger three-dimensional sphere that is the Earth.

That's a good question and I actually second-guessed myself while reviewing my post. The reason it's three dimensions is because you yourself constrained it thus by only considering pure imaginary numbers. So instead of having a plot that's got two-dimensional values over a two-dimensional field, you've got a plot that has two-dimensional values over a ONE-dimensional field. The three dimensions aren't two inputs and one output, it's one input and two outputs.

If it helps visualize, consider a cylinder; its central axis would be the imaginary number line, and at each point along that axis you have a two-dimensional flat cross-section that the value of the function at that point must be located in -- and only ONE such point in that cross-section can be the value of the function, or else it wouldn't be a proper function.

No, it would be a surface, because the input into the function has two dimensions at that point. A real number to the third power stays real, and an imaginary number to the third power stays imaginary, so that means the output of your function has to accommodate both of those.

Any three-dimensional view on this function is necessarily a cross-section or projection* of the complete four-dimensional graph. If you constrain the values of x to be purely imaginary, as we discussed earlier, then you're looking at a cross-section of the graph: if we consider (x + yi) to be the input and (z + ti) to be the output, then this constraint is looking at the slice of the output that intersects the hyperplane* defined by the function x = 0.

* I discuss projections way down below.

* A hyperplane is a generic term for something that divides the entire space into two regions. In two dimensions, hyperplanes are lines -- if you plot x = 0, then you've divided the space into a region where y > 0 and a region where y < 0. In three dimensions, hyperplanes are planes -- if you stand up with your arms and legs outstretched and imagine the infinite two-dimensional sheet aligned with your body, you've divided the whole universe into "in front of you" and "behind you". In four dimensions, a hyperplane is three-dimensional. This gets quite a bit harder to visualize, so I'll take the usual metaphor of assigning the fourth dimension to time. If you imagine time as a filmstrip, then one frame of that filmstrip -- the state of the entire universe at that point in time -- then you've divided the four-dimensional space into "past" and "future".

The identity function (f(x) = x) is elegantly simple. Whatever dimension you consider for the input, the output will have the same dimensionality. The oddball quirks of complex numbers don't come into play.

Bingo! And every time you stretch it, you stretch it in a direction that's perpendicular to all of the other directions you've stretched it before -- a direction that, by definition, didn't exist before. (Now, this assumes you're working with nice ordinary rectilinear Euclidean space... I won't break your brain with spaces that violate that assumption, but know they exist, and know that you live in one.)

Looks like the light went on about what a 2-surface is!

What makes it a (minimum of) 4-space is that a Klein bottle can't exist in 3-space. If you tried to make one, the surface would intersect with itself -- which is exactly what the Klein bottle says it DOESN'T do.

You can make a lower-dimensional analogue of a Klein bottle quite easily, though: a mobius strip. Take a long, narrow piece of paper, twist it once, and tape the ends together. Now, if you tried to draw this object on a two-dimensional piece of paper, the lines would cross each other. But in three dimensions, you have an extra direction you can move things around in so you can avoid that problem.

You can't! XD We as humans are distinctly incapable of properly visualizing four-dimensional space. We can fake it moderately well by hijacking time as a dimension, but then you're only ever looking at a three-dimensional cross-section of it at any given time.

It's a non-Euclidean 4D manifold, if you want to get nitpicky about it. I refer you to the filmstrip metaphor above.

[EDIT: The human-visible parts of the world are a non-Euclidean 4D manifold, I should say. Kaluza-Klein theory (no, different Klein) suggests that it's at least 5D, and some theories predict that it's 11D, and others suggest other numbers.]

Hmm... Well, not quite. For starters... A square, which is a stretched-out line, has 4 sides, and each side is a line. A cube, which is a stretched-out square, has 6, and each side is a square. A tesseract, which is a stretched-out cube, has 8... and each side is a cube.

You could paint a 3D object on each of the eight faces of a tesseract. If you looked at the tesseract straight along one side, you'd only see that one object. If you looked at it from a corner, you'd see four 3D objects, each at an odd angle. If the tesseract isn't made of glass, you wouldn't be able to see through it; it IS a solid object.

What you described is actually a fairly accurate representation of looking at a three-dimensional object in a four-dimensional space. Let's bring it down a dimension for ease of visualization. If you make a figure with a piece of string on the 2D surface of a piece of paper, you -- a 3rd-dimensional viewer -- can see the whole thing at once, inside and outside. But a 2nd-dimensional viewer living on the paper could only see it along the edge; he could see the colors and texture of the side of the string, but he could never get the whole picture at once.

So yes, four-dimensional superbeings looking at our universe would be able to see your intestines without opening you up. They could also reach in and pull your intestines out of your body without hurting you -- but little ol' three-dimensional you wouldn't be able to see it unless they pushed a loop of it back down into your plane of awareness.

Similarly, you could pick up a loop of that string on the paper and it would disappear from our little Flatlander's line of sight, and you could put the middle of the string down on the paper and the Flatlander could see that part of it but not the part you still held in your hand. If you let go of it, the Flatlander would see that somehow you had managed to overlap two solid objects without breaking them -- something he would have imagined would be impossible!

Aha! Nope! Gonna break your brain here:

Klein bottles DON'T HAVE an "outside" surface! They don't have an "inside" surface, either.

Now, this isn't a general property of closed 2-surfaces. A (infinitely-thin) balloon is a closed 2-surface, and it clearly divides the universe into three-dimensional "inside" and "outside" regions.

But Klein bottles only have one side, not two.

Take that mobius strip we constructed earlier. Start at the tape mark and draw a line down the middle of it in one direction. Eventually you'll get to the OTHER SIDE of the tape mark and KEEP GOING because you won't have intersected with your starting point yet. Instead, you'll be drawing on what you would have assumed was the other side until you come back around to where you started. Any individual piece of a mobius strip looks like it has two sides, but taken as a whole, it only has one.

Sounds like you kinda independently derived what I was describing above!

This might not be the most productive avenue of discussion, because what you looked at was a TWO-dimensional projection of a FOUR-dimensional object. Animating it only gives you one more dimension, and it's not any of the right ones because the fact that it's rotating requires that time be elapsing for the four-dimensional object (that is, a fifth dimension)!

I wonder if anyone's made a VR app that lets you inspect a tesseract interactively -- that would give you three spatial dimensions and one time dimension.

I'd be curious to paint a solid tesseract instead of rendering a wireframe one -- color each visible square (there are 24 of them) a different color.

...... huh. I just grew a little.

I was going to say that I'd color each cubic face of the tesseract a different color (this part works) and then have each square face of those cubes be a different shade of that color... but you can't do that! Just like every 1-dimensional edge of a cube is shared by two 2-dimensional faces, every 2-dimensional edge of a tesseract is shared by three 3-dimensional faces!

Just goes to show you, this stuff is hard. XD

No. You can slice them into a lower dimension (cross-section) or you can project them into a lower dimension (a photograph is a 2D projection of a 4D world; a video recording is a 3D projection) but you can't squash them into a lower dimension without losing information about it.

Hmm, this makes me wonder, do things in lower dimensions even exist in higher dimensions? I mean, a square is technically a bunch of lines put on top of each other, but since lines are infinitely thin, it isn’t really. A square is made of lines, but to draw a line in 2d would be to make a really long thing with a really small width.
…Although, maybe just by definition that’s not possible because it’s trying to represent 1d in 2d T_T. I guess every object in a certain dimension has the dimensions for said dimension, but it just so happens that every object in one dimension contains parts from a lower dimension..?


I don’t understand what a “non-Euclidean 4D manifold means” and hence I’m not sure how to apply the filmstrip metaphor, except that since a line that moves across time because 2D on paper, but not to the line because it thinks it’s been in the same place the entire time, something 2D and 3D and 4D and all experience the same thing too, and that relates to 4D because time literally adds an axis to everything? As for the other theories, uh, I'm surprised that there are theories that the world has more than three dimensions!


Err, so basically a mobius strip has only one side because both sides got connected..? Although, in 2d a mobius strip couldn’t have twisted in the first place, and now it looks like lines of a rectangle that shouldn’t be intersecting with each other are intersecting in weird places and the point where the strip is taped together is either or a line, or lost?
And then klein bottles have only one side because, whatever “side” means in 4d was connected (what is a side even ‘~’)? So this bottle, in 3d, has some odd dimension where another bottle exists and those two bottles got connected in such a way that makes it look like, in 3d, the spout has mysteriously intersected with the bottle where it shouldn’t be able to, and is attached to the bottom when it shouldn’t be attached there. Probably, at the bottom spout area, breaking into 4d would make the thing make more sense, but humans can’t do that?


Could you have stripes or checkers of color where the faces intersected? Would that even help?
I think given how much I don’t understand, I’ll give up on tesseracts at least until/if I learn more. Buuut what’s a face? Anything on the outside of a thing?


So the world is 4D o_o? Why is a photograph 2D but a video 3D? Because videos have that extra axis of time?


School question: What is a "net field"? In physics class we had a homework saying:
Draw the electric field (E) vectors at points A, B, C, D, and E. Draw field coming from each charge and then the net field. Draw lengths proportional to the magnitude. A, B, and E are horizontally equidistant between the two charged particles and each charge is equidistant between A and C or D.
http://i.imgur.com/FgSQ4Ay.png

Apparently all I needed to do was draw somewhat ambiguous vectors. And somehow apply the inverse square law on each vector--I still need to practice that. I don't understand what "each charge" and "net field" means though. All I think I know is that "electric field vectors" means where and by how much force a proton would go at a certain point.

They don't exist as solid objects in higher-dimensional spaces. They exist as boundaries. Your table might be three-dimensional, but its surface is topologically two-dimensional. The world in front of you might be three-dimensional, but the image on your retina is a two-dimensional projection.

There are such things, mathematically speaking, as space-filling curves, where you have a one-dimensional object that touches every possible point in a two-dimensional square. They don't serve a whole lot of practical purpose because you can't actually MEASURE it -- the length is infinite -- but their existence serves an important theoretical role in a number of proofs. (One such theorem: There are the same number of rational numbers (that is, fractions) as there are whole numbers, but there are more real numbers than there are rational numbers... but there are as many complex numbers as there are real numbers!)

That's intentional. XD I picked that phrasing specifically to make the point that the real world is WEIRD and it can't be so neatly described with simple mathematical models.

I'm not quite sure how to interpret what you're saying here. I... THINK you're right? But you're conflating two separate points I was making.

The filmstrip metaphor is just a way to imagine a four-dimensional space: an infinite series of three-dimensional spaces, all linked together.

Without getting too deep into it, these additional dimensions are usually interpreted as being "compactified" -- sort of rolled up into tiny little circles, so that if you move in that direction you end up where you started.

Bingo.

Yep.

A very good question, and a difficult one to answer. There are multiple meaningful answers depending on what specific thing you're thinking about. The most useful definition would be the topological one, which is the one that you demonstrated using a pencil on the mobius strip -- two points are on the same side of the object if you can connect them with a smooth curve with no breaks or discontinuities. So a sphere has two sides, an inside and an outside, because you can't trace a curve from a point on the outside to the inside.

Pretty much, yes.

The part that doesn't make sense in three dimensions is the place where the handle appears to pass through the wall of the bottle. It SHOULD be a smooth, continuous, uninterrupted object -- there IS no hole in the wall of the bottle! -- but without being able to see it in four dimensions you just see the two parts clipping together.

If you want to get technical:
* Vertex: 0D points formed by the intersection of edges
* Edge: 1D lines formed by the intersection of faces
* Face: 2D surfaces that, taken as a whole in a 3D object, comprise the boundary of the object; in 4D.
AFAIK there aren't formal names for the higher-dimensional analogues. Some people call the cubes that comprise the exterior boundary of a tesseract "cells."

Yes.

It's the resulting field when you add up all of the fields in the area. Just like "net force" is the resulting force when you add up all of the force vectors acting on an object.

Remember that picture of the complex vector field I showed you a few posts back? Your result should look like that.

It's actually not ambiguous at all. You don't have to be precise about the magnitudes of the vectors as long as they're approximately right; this is a qualitative exercise. The important part is that they're pointing in the right direction and that you only draw dots for null (that is, zero-length) vectors.

I can try to answer specific questions you may have, but I'm afraid there's not a lot I can give you in general right now without giving you the answers to your homework outright (which I'm not going to do).

I'm not sure if the net field is meaning the net field of each specific point or all the points...although now that I think about it, specific points makes a whole lot more sense.
http://i.imgur.com/tKt0m95.png

Is a "charge" the imaginary protons which each point is?

EDIT: I should probably mention that we went over the above in class--so I'm worried about understanding the wording of the question should another one pop up somewhere on a quiz or test or something. I'm not certain about the 1/9th thing, but based on memory and based on the whole if something is a certain distance away from something else, the force is 1/r² it seems right.

You got B, A, and E right.

You're right that the repulsive force is 1/9 at D compared to A, but you've overlooked the fact that the left side one has 3x the charge, so your answers for points C and D are wrong.

Actual homework answers, don't open unless you think you've got it right:

Hmm..The electron exert a force of 1 to the right at A and the proton a force of 3 to the right at A. Then the proton's forced exerted at D would be 3*(1/9) or 1/3 to the right, while the electron would be 1 to the left so I made a mistake there and D should have a vector 2/3 to the left?
Err, why am I right about the repulsive force being 1/9 at D compared to A..? Did I miss something else?


Then for C, the proton should exert 3 to the left while the electron should exert 1*1/9 or just 1/9 to the right. So, for C it should actually be 3 - 1/9 to the left?
But then, that's what I did before too :/
But I also did think about the fact that the proton has 3x the charge of the electron.

You're right because that's the PROPORTION of the strength between the two points. At D it is 1/9 the strength it is at A. It's just the case that the strength at A is 3.

I suppose I might have misunderstood your notes.

I did get D wrong from forgetting about the 3*1/9 part though, so thanks for that!

What exactly is the "magnitude of an electric field"?

The magnitude probably means the magnitude of the vector at that point in the field.

I think the only information you wouldn't be using is the position. You need the charge and the mass and the orientation of the field. And I think that the position is meant to be a red herring -- part of the problem is to identify that you DON'T need it because it's a UNIFORM field (you WOULD need it if it were a point source directly above the object, due to the inverse square law).

That said, I actually don't understand why they gave two different positions, unless it's trying to suggest that it fell some distance before finding equilibrium, but that... doesn't make sense if the field is truly uniform.

EDIT: Oh, it should be noted. The magnitude of the electric field is NOT the force it applies. It's measured in newtons per coulomb.

Oh, I should have asked also: what is a uniform field? As in a big infinite line of field..? Why doesn't that follow the inverse square law? A point source means field coming out of one point..right?

What exactly is "magnitude" of an electric field?

Uniform means it doesn't vary. The strength is the same everywhere. It doesn't follow the inverse square law because it says it doesn't. :P (In the real world, a uniform field is an approximation of the field between two oppositely-charged flat plates held at a fixed distance from each other.)

The magnitude (sometimes called "strength" or "intensity") of an electric field is, as I said, force / charge. It measures how much force the field can exert per unit charge.

Two oppositely charged plates held at a fixed distance from each other make a field that doesn't really vary :o?

Also, I feel kind of bad for asking this question, but since I really don't understand it: What is a charge? I keep hearing/doing things with this thing "charge" and I actually don't get it.
Also, when I write out stuff like mass is 4.0 x 10^-6 kg, I write m = 4.0 x 10^-6 kg, but what's the symbol for charge?



EDIT: Oh, I guess the symbol for "charge" is Q or q. Except Q is "the charge that's affecting" and q is "the charge that is being affected"

EDIT2: I found an equation on my equations sheet that says
For a uniform E-field*: ΔV/d
I think the thing did fall after all...maybe, but the wording's weird? I'm going to try plugging things into this.
EDIT3: Except there isn't time nor acceleration..so did the thing even move ><
EDIT4: I don't think that works..assuming my free-body-diagram is fine I'm going to guess it's fine to find the force, then plug everything into F_e = |q|E
*What is an "E-field"?

Well, think about it. If you're an electron, then the closer you get to the positive plate, the harder it'll push you away, but as you get farther away from it you're also getting closer to the negative plate, so it starts pulling harder to pull you towards it. The forces balance out in a way that can be modeled as uniform field strength. (In reality, it's never perfect, especially near the edges.)

The customary variable for charge is q.

Charge is basically just the electromagnetic analogue of mass. It's just a property that matter has that influences how strongly fields act on it, just like more mass means more gravitational force. It measures the imbalance between protons and electrons, but since we're not dealing with individual ions here we've got a unit for macroscopic charges called the coulomb instead of counting electrons. (After all, you don't count the protons and neutrons in a macroscopic object to determine its mass.)

Ohh, that makes sense. The fact that both charges on both sides were moving things in one direction had blanked to me.

Thanks for that explanation of charge! I think I need to acknowledge its existence better.

Well, good luck with your homework. I'm off to bed.

Good night!
I actually finished the worksheet already and had that as one of my questions I didn't understand ^^

Well, the answer you had written in the picture was wrong, so I hope you fixed it. :P

Err, a is fine right?
B I changed to:
E = 43,000 N/C
..which is a really big number o_o

Work:
m = 4.0 * 10^-6
q = 0.90 * 10^-9
F_N = F_g = F_e
F_g = ma
F_g = (4.0 * 10^-6)(9.81 m/s²) = 3.9 * 10^-5 N = F_e

F_e / |q| = E
(3.9 * 10^-5 N) / (0.90 * 10^-9 C) = E
E = 43,000 N/C

At a quick glance I don't see any particular issues with your work but I haven't worked it myself.

I don't know how realistic that number is, but a search finds me another class's homework where the answers are on the order of 10⁵ N/C so it looks reasonable.

I do recommend asking about those distances. It doesn't make sense to me the way they presented it, even IF it's a red herring.

Thanks for looking it up! I'd been worried about doing something wrong because of the sudden jump in numbers.

I'll ask my teacher about the distances when I have school tomorrow.

Checked it just now--it was a typo!

Oh, good, now my scientific intuition feels better. XD

That's good xD

--
In physics class we had to field or vector diagrams for different combinations of charges--I don't think I'm comfortable with those so can I ask a couple question about this? :
http://i.imgur.com/l7eavCI.png
The black color is a field diagram
The colored is one vector

>For field diagrams, we learned that "the number of lines leaving a positive charge or approaching a negative charge is proportional (but not equal to!) the magnitude of the charge". So that just means that every ball of charge is proportional? And the diagram I drew is fine in that respect because every circle of charge is just one charge and has five lines coming out/going to it?
>The teacher also said something about how in field diagrams, the distance between lines mattered. Does that mean something like this is wrong for being too sparse or incomplete?


>When I draw vectors, should I be thinking: Here's what charge circle no. 1 is doing as a vector...here's what charge circle no. 2 is doing...here's what charge circle no. 3 is doing..now I add them all up for a vector?
>When is a vector diagram complete?

The spacing of the field lines represents the field strength in that area. When the lines are spread farther apart, the field is weaker than in the areas where the lines are close together.

Imagine drawing field lines for an isolated point charge. It would just be straight lines extending out from the charge in all directions. The farther from the charge you get, the farther apart the lines become, which matches what you know must be true.

Now, unfortunately, the field line density is only a qualitative description of the field. It doesn't accurately give you anything you can turn into numbers, because lines spreading out on paper don't follow the inverse square law -- they spread out proportional to 1/x, not 1/x². But you can still get some rough comparisons out of it.

Yes, in the first diagram you drew, you correctly suggest that each charge has the same magnitude. And the purple vector is approximately correct (which is all that matters in such a rough diagram).

Your spoilered diagram is bad for a couple reasons and, yes, that's because it's too sparse. The first problem is that having two field lines per charge simply doesn't give enough information about the shape of the field. There are huge regions of the page where you simply don't have information. The second is that the diagram as drawn suggests some real oddball field strengths. As drawn, the field on the bottom side of the positive charge is 3x stronger than the field on the top side of the charge (90 degrees vs. 270 degrees between field lines) when in reality the field strength is hardly any different. And as drawn, the field below the two negative charges is REALLY REALLY strong (stronger than right beside one of the charges!) while it essentially vanishes farther out. (I should really write a program to generate these field diagrams. It wouldn't be that HARD, and I don't know why I can't get WolframAlpha, Google, or the macOS Grapher to do what I want.)

Your strategy for drawing vectors is sound.

A field drawing is complete when there's enough information in the picture to convey the relevant information accurately without anything obviously missing.

Ohh, so the distance between field lines coming out of a charge does indicate the strength of the field around the charge also!

Anything obviously missing would mean that for a vector diagram, I can tell what all the charges roughly are..?

In drawing a line diagram for more than one charge, can I say I first look at charges, then draw how they interact with each other--and if there's another charge, how they interact with each in an area furthest away from that charge being currently ignored--then move onto the interactions between another pair if there's more charges, and finally draw field lines for each charge depending on the strength of each charge?
...drawing field diagrams seems almost odd to me :/. The teacher said something about getting familiar and developing an intuition for them, so maybe there isn't an easy to follow process for it?

If it helps you feel any better, this kind of field diagram is are ALL ABOUT intuition. They're useless for specific analysis of field phenomena; they exist purely to give you a qualitative idea of what you're looking at. The process of drawing one is a way to translate your mental processes into something easier to visualize.

You use a vector field plot -- that is, the grid of arrows we were looking at a couple pages ago -- or a contour plot (curves of equal field strength instead of curves describing the path of a particle) if you need something more rigorous.

That strategy sounds pretty reasonable. Personally I would start off marking evenly-spaced points around each charge so that I have an idea of what the line density is going to look like before I start drawing and what direction they're going to start off in, and then I'd connect the dots where there's an obvious interaction between the charges, and finally I'd fill in the paths that diverge away from the charges since then I'd have an idea of where the lines need to be spaced for those.

(In reality, me being Coda, I'd hack up a quick rendering program that would throw out some test points and trace their paths on the screen.)

EDIT: Actually that sounds fun. Maybe I WILL do that.

That can be done quickly :o?

Oh! drawing out the points first instead of hoping to fill them out later is a lot more convenient.

If something's about intuition...Am I expected to be able to visualize the diagram before I draw it :/ ?

It can be done relatively quickly if you're me. ;) More seriously, it's one of those things where a competent programmer using familiar tools can get something close to correct pretty quickly and then spend who-knows-how-many hours fiddling with it to get the results you want.

No, you're not, that's the point of drawing it. If you COULD visualize the whole thing easily there wouldn't be a need for it. Making the drawing gives you a way to translate things you know must be true about the system into something you can actually look at.

Are the energy of fields completely different from that of moving objects? And that's why the force of a charge is not mass times acceleration? Also, since F_e = |q|E, what kind of force is F = ma? Oh. That's kinetic energy. Do all energies have forces specific to them then? Or maybe they don't, but all I need to know is that the energies are separate in that they each have their own kinds of PE and KE and force equations because they are different kinds of energies?

Whenever a field force is mentioned, am I supposed to assume it is based on the proton? That is, if in a uniform field, E = 20 N/C west, I can assume there are positive charges at the east and/or there are negative charges at the west? And that is why to increase the PE for positive charge, the positive charge must be moved against the field, whereas to increase the PE for a negative charge, the charge must be moved with the field?
Is there no such thing as a negative field force (E)? Also, for point charges to calculate what the field force (E) would be at some point, I suppose it is also based on moving in the direction of the repulsion of a proton because all field forces are defined that way, for whatever reason?

All fields of force have corresponding potential energy. This means it takes energy to move an object against the field lines, and that energy is released by the object moving along with the field lines. This is true regardless of what kind of field it is. The different potential energy formulas are descriptions of how that particular field works. (As far as I know, kinetic energy is just kinetic energy, and there aren't different forms of it.)

F = ma is not a field. It doesn't cause anything to happen. It's a description of a relationship, and it's true of ALL forces, masses, and accelerations. So it doesn't matter "what kind" of energy it is.

By convention, electric fields are described as moving from positive to negative. This is often considered to be a historical mistake -- Benjamin Franklin arbitrarily chose one of the charges he was able to create in his static electricity experiments to be positive, and the other to be negative, and all of the math from then on holds to that convention. It IS just a convention, and as such it doesn't actually MATTER because the whole point is that everyone knows the convention and therefore agrees on what things mean, but it means that the fields describe the motion of a positively-charged particle when in practice the particles in motion are negatively-charged. But to answer your question: Yes.

As for negative fields... No, those aren't possible. Vectors are a magnitude and a direction. While the math works out to consider a vector with negative magnitude, it's 100% congruent to a vector with a positive magnitude in the opposite direction. Negative lengths are considered undefined, because measures are positive by definition. And so, in the interests of keeping everything consistent and unambiguous and mutually understandable, it isn't done.

Then, can F=ma be F_e=ma?
Does it matter if I use F_e=ma then use v_f²=v_i²+2ad to find a velocity or I use Δ(PE)=qEd then KE=.5mv² to find the velocity?

(I'm given the field E, q, initial velocity, the distance moved, and the mass, and asked to find the final velocity.)
(For some reason solving using the two different methods gave me two different answers. I don't know if I made a calculation mistake or a problem solving mistake)