Do you wonder why people get confused?

[maverick_starstrider]

Taking a break for a few days. Thanks for the help.

May I suggest you relax with an EM TEXTBOOK? Just a thought.

 

[maverick_starstrider]

Tac-Tics: But I thought the definition of the dot product IS |A||B| cos theta times some unit vector. How can we ignore the angle in the light of this and still consider the dot product?

So electric field is what it is without any charges dropped in and the flux is the interaction after a charge is dropped in? So without test charge there is a field but no flux. Once you throw in a charge, the you have flux?

Ok. Can I claim semantics on this one? A vector has direction but no angle. Ok. But a vector in 3D space makes angles to each of x hat, y hat and z hat. So can we say it has both? Or it has angle only in the context of a coordinate system? This one is losing me entirely. Ok so a vector has direction. Which is either away from or into a charge? There seem to be many variants of the away from when I look at the vector arrows. The all point in differing directions...but its still only away from?

My brain is fried again. Thanks for the help. I hope you are correct that it will be clear as glass soon. Do you see what I mean though. If all we've been discussing were made into a 3D movie, it would be easier to then look at the equations and comprehend them. All our banter is analogous to a movie of sorts where visualization plays a large part in understanding the math. I wonder do all you guys have visualizations of each concept or do you think only in term of the equations?

We're the sorry the universe does not obey laws that are sufficiently accessible to you. Take it up with your local deity of choice.

 

[diazona]

For when you get back:

Tac-Tics: But I thought the definition of the dot product IS |A||B| cos theta times some unit vector. How can we ignore the angle in the light of this and still consider the dot product?

The definition of the dot product is that you multiply corresponding components of the vectors and add them up.
\vec{A}\cdot\vec{B} = (A_x \hat{x} + A_y \hat{y} + A_z \hat{z})\cdot(B_x \hat{x} + B_y \hat{y} + B_z \hat{z}) = A_x B_x + A_y B_y + A_z B_z
That is equal to multiplying the magnitudes times the cosine of the angle between the vectors (not times a unit vector):
\vec{A}\cdot\vec{B} = |\vec{A}|\,|\vec{B}|\cos\theta

So electric field is what it is without any charges dropped in

Yes

and the flux is the interaction after a charge is dropped in?

NO. You're thinking of force.

So without test charge there is a field but no flux. Once you throw in a charge, the you have flux?

Without a test charge there is a field but no FORCE. Once you throw in a charge, then you have force.

Ok. Can I claim semantics on this one? A vector has direction but no angle. Ok. But a vector in 3D space makes angles to each of x hat, y hat and z hat. So can we say it has both? Or it has angle only in the context of a coordinate system?

Sure, there are angles only in the context of a coordinate system. Remember what I said before, that you only have an angle between two vectors. Any of \hat{x}, \hat{y}, or \hat{z} can be one of those two. So for a vector in 3D space, there is an angle between it and each coordinate axis, once you have defined the axes. But it doesn't have an angle by itself.

Ok so a vector has direction. Which is either away from or into a charge?

Generally a vector can have any direction. But the electric field of a single point charge at a particular point (which is an example of a vector) does have a direction either away from or into the charge.

There seem to be many variants of the away from when I look at the vector arrows. The all point in differing directions...but its still only away from?

Sure. At any given point, the field points away from the charge (if it's a positive charge). Look at some of the pictures http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html .

My brain is fried again. Thanks for the help. I hope you are correct that it will be clear as glass soon.

Gee, I hope so, but I'm not holding my breath...

Do you see what I mean though. If all we've been discussing were made into a 3D movie, it would be easier to then look at the equations and comprehend them. All our banter is analogous to a movie of sorts where visualization plays a large part in understanding the math. I wonder do all you guys have visualizations of each concept or do you think only in term of the equations?

Sure, we have visualizations. Just hunt around online and you can find a bunch http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/estatics/u8l4c.html http://qbx6.ltu.edu/s_schneider/physlets/main/efield.shtml . (The visuals come from the math and aid understanding of the math, though, not the other way around.)

 

[Tac-Tics]

We're the sorry the universe does not obey laws that are sufficiently accessible to you. Take it up with your local deity of choice.

If you've got an issue with the poster, use the report button.

But I thought the definition of the dot product IS |A||B| cos theta times some unit vector.

Math often presents us with several related definitions. This definition is given to most students first, because we all knows what an angle is, and it's easy to see that the dot product is related to the angle two vectors meet.

However, the inner product can also be defined in terms of orthogonal projection. It's something like one vector casting a "shadow" over another (to use fluffy, visual terms). Inner products can also be defined on other kinds of "vectors". Later on in physics, you often deal with function spaces -- sets where each "vector" is actually a complex-valued function. Unlike vectors in R^3, functions don't have a direction and don't meet at angles. But we can still define an inner product on them which preserves many useful properties. (The inner product between two functions is usually defined as the integral of the product of the two functions).

So electric field is what it is without any charges dropped in and the flux is the interaction after a charge is dropped in? So without test charge there is a field but no flux. Once you throw in a charge, the you have flux?

The electric field is what it is. The test charges are only there to show you the "shape" of the field. After you understand how the field looks, forget everything you know about test charges. Just understand that the field can point one way, and the particle might even move backwards! (If the charge on the particle is negative)

One way to think of test charges is like this. Imagine the field is invisible (because... it actually is...) Now, your test charge is like a microscope. Wherever you hold the test charge, you are allowed to see the field at that point in space. Take the test charge away, and the field is invisible, but it's still there.


There is no "flux" until you pick a surface. You have to have a field AND a surface, and then you talk about flux of the field through the surface.

Ok. Can I claim semantics on this one?

Take a silly example. I have the vector <3, 4, 0>. It's length is 5. What is it's angle? There's no good answer. You have to take the angle with respect to another vector... and with respect to a given plane (... and orientation on top of that!) Granted the notion of "direction" isn't great either. But intuitively, "angle" is more misleading.

My brain is fried again.

Your brain becoming fried is probably an indication you have learned something. I remember struggling with these ideas when I started too. Give your mind some rest, and when you're showering or ready to go to bed, it will just sort of click. Replay the definitions of everything and the visual examples in your head, and you'll get it eventually.

I wonder do all you guys have visualizations of each concept or do you think only in term of the equations?

Visualizations are a huge part of a lot of math. The mind works a great deal better in two dimensional images than it does in symbols on a page. The hard work you have to do is be able to translate back and forth between them. It's almost like reading and writing sheet music or tabs. A tab doesn't sound at all like the music. And the music doesn't look like a tab. But the two are in some way the same. Part of the job of a musician is learning to translate between the two.

Whenever you see an integral or a direction derivative or a vector, think back to the simplest examples. There's a joke that goes like this:

A physicist and a mathematician go to a string theory conference. After the conference, the physicist exclaims "This nonsense about 9-dimensional spacetime is impossible to visualize!" The mathematician turns to him and says "Oh, it's not so hard." The physicist asks him, "Well, how do you do it then?" The mathematicians explains, "It's simple, I just imagine n-dimensional space, then set n equal to 9".

Of course, no one can really "see" 9-dimensional space, much less "n-"dimensional space. But the basic ideas in n-dimensional space are the same as they are in 3. There are vectors. They have lengths. You can take the dot product between any two. You can rotate any vector in any plane by any angle. You can add vectors in the same way. You can scale them in the same way. They are very similar.

 

[maverick_starstrider]

Of course, no one can really "see" 9-dimensional space, much less "n-"dimensional space. But the basic ideas in n-dimensional space are the same as they are in 3. There are vectors. They have lengths. You can take the dot product between any two. You can rotate any vector in any plane by any angle. You can add vectors in the same way. You can scale them in the same way. They are very similar.

Cross product is actually only defined for 3 and 7 dimensions ;)

 

[flatmaster]

Cross product is actually only defined for 3 and 7 dimensions ;)

You're sure about that? I'm under the impression that in n space, you need n-1 non-orthogonal vectors to define a cross product. My tensor notation is shaky, but I believe wikipedia agrees. It's near the bottom of the article (where stuff gets hard)

http://en.wikipedia.org/wiki/Cross_product

 

[rockyshephear]

Right, I was thinking of the cross product.|A| |B| sin theta times u hat. My bad. ;)

 

[rockyshephear]

Summary: Without a test charge there is just a field and no FORCE. Does that include a totallyl empty universe? Once you throw in a charge, then you have force. And when you have a surface you get flux thru the surface, even when the surface is integrated down to a finite point. Sound right?
---
Right, I meant an angle between two vectors, not as a stand alone vector having an angle.
---
Another thought. The same way the sun's rays are depicted as parallel but are not, many diagrams show vectors that seem parallel impinging on a surface. This is not accurate I assume since they come from a point charge and eminate out radially?
---
Thanks for all the time answering my insipid drivel. lol