What exactly do the terms near-field and far-field mean?

[s3a]

What exactly do the terms "near-field" and "far-field" mean?

What exactly do the terms "near-field" and "far-field" mean (especially in the context of the two problems attached)?

Other than the confusion about those terminologies, I fully get how to do these problems (1.1 and 1.2).

Any help in understanding what these terms mean would be greatly appreciated!

 

[Redbelly98]

Near field means at distances comparable to or closer than the relevant dimension characterizing the charge distribution -- d in this case.
Far field means at distances much larger than the relevant dimension d.

(Note, in optics diffraction problems, it's more complicated and there is a different meaning.)

 

[s3a]

What exactly does “[a] relevant dimension characterizing [a] charge distribution” mean? Like, in this case, how is d a relevant dimension which characterizes a charge distribution and, also, what does it mean to “characterize a charge distribution”?

Having said that, I like the conciseness of your answer :) but I'd just like the above clarified, please.

Also, thank you very much for clarifying about the case with diffraction in optics since I will eventually be covering that.

 

[Redbelly98]

By a "relevant dimension characterizing [a] charge distribution", I essentially mean the size of the distribution. In this case -- with two point charges -- it's pretty straightforward to use the distance between the two charges. In other cases, it could be something else: the radius or diameter of a charged sphere, for example.

 

[s3a]

Okay now that makes more sense but I think I'm still missing something since I'm having a bit of trouble applying that to the problems.

Am I correct in assuming that what I want is a distance much larger than d for the far-field as well as comparable to or closer than d for the non-test charged particles such that I minimize or maximize the electric field at the point the positive test charge is located on?

Problem 1.1:
Near-field: r = d/2 (Because that is the closest distance, smaller than d, that has no effect by an electric field at the point where the positive test charge is located?)

Far-field: r >> d (Because that is a significantly larger distance than d that yields the maximum strength equivalent to a single 2Q charge acting on the positive test charge? - I say maximum because it's the sum of the two individual charges making an electric field for the positive test charge.)

Problem 1.2:
Near-field: r = d/2 (Because that is the closest distance, smaller than d, that yields the maximum strength equivalent to a single 2Q charge acting on the positive test charge?)

Far-field: r >> d (Because that is a significantly larger distance than d that has no effect by an electric field at the point where the positive test charge is located?)

If something I said is unclear, tell me and I will improve the phrasing according to what you did not understand and then repost.

 

[Redbelly98]

The near field is not a single location, it is just a way of saying you are relatively close to the charge distribution. There is no requirement that the electric field be minimized, maximized, or anything else.

The far-field is r >> d because that is a significantly larger distance than d, period. It is often the case that some simplifying approximation can be made to express the field strength for r >> d; for example that it is nearly the same as that due to a single charge of 2Q.

Hope that helps clear things up

 

[s3a]

Sorry for asking so many questions but, each time you tell me something, I have more questions (in a good way though – as in I am narrowing the issue down further every time).

Okay so, to reiterate and expand a bit in my own words to see if I get it, for the far-field we take r >> d no matter what and that sometimes will lead to a simplified expression where in my case with problem 1.1 simplifies to the equation for a charge of 2Q but what about the near-field? Is it always the case that any r smaller than d which happens to simplify to something arbitrarily nice (such as having an expression depict a single non-test charge of 2Q) is chosen to be the near-field?

What if there is no value of r smaller than d for which there is some arbitrarily nice expression? Is it then the case that the near-field is the situation with any arbitrary r << d? Also, I am assuming that the far-field for problem 1.2 is r >> d even though that doesn't simplify to any convenient/fancy expression. Is this a correct assumption?

 

[s3a]

The rest of my previous post still stands but, I wanted to change the following:

"Is it then the case that the near-field is the situation with any arbitrary r << d?"

to

"Is it then the case that the near-field is the situation with any arbitrary r <= d?".

Sorry for being pedantic but, I need to know every little detail about things in order to not get confused by them.

 

[Redbelly98]

Sorry for asking so many questions but, each time you tell me something, I have more questions (in a good way though – as in I am narrowing the issue down further every time).

Okay so, to reiterate and expand a bit in my own words to see if I get it, for the far-field we take r >> d no matter what and that sometimes will lead to a simplified expression where in my case with problem 1.1 simplifies to the equation for a charge of 2Q but what about the near-field? Is it always the case that any r smaller than d which happens to simplify to something arbitrarily nice (such as having an expression depict a single non-test charge of 2Q) is chosen to be the near-field?

The guidelines are rather approximate and not always clear-cut. But to my understanding, the near-field just means comparable to, or less than, some characteristic size or distance parameter.

What if there is no value of r smaller than d for which there is some arbitrarily nice expression? Is it then the case that the near-field is the situation with any arbitrary [STRIKE]r << d[/STRIKE] r ≤ d?

I allow for the possibility that there are exceptions to the general rule, though I do not personally know of any.

Also, I am assuming that the far-field for problem 1.2 is r >> d even though that doesn't simplify to any convenient/fancy expression. Is this a correct assumption?

Yes, that is the far field. But the expression does simplify, though not nearly as simply as the +2Q case. Scroll down to the expression for E at this link:

http://en.wikipedia.org/wiki/Electric_dipole_moment#Potential_and_field_of_an_electric_dipole

 

[s3a]

Just to confirm, r >> d is the far-field for problem 1.2 just because r is much larger than d and not because r >> d happens to simplify the equation, right? I know you mentioned this for the near-field but, I just want to make sure it applies here too.

Also, I've been trying to do algebraic manipulations to get from the wikipedia equation for the electric field that you gave me to the equation of problem 1.2 but, I've been having trouble doing so. Could you please show me show?

 

[Redbelly98]

Just to confirm, r >> d is the far-field for problem 1.2 just because r is much larger than d and not because r >> d happens to simplify the equation, right?

I don't know. The two always seem to go together in cases I am familiar with. At this point I would ask your professor what definition he or she prefers to use.

I know you mentioned this for the near-field but, I just want to make sure it applies here too.

Also, I've been trying to do algebraic manipulations to get from the wikipedia equation for the electric field that you gave me to the equation of problem 1.2 but, I've been having trouble doing so. Could you please show me show?

Since the equation of problem 1.2 is only for points along the y-axis, \vec{p} \cdot \hat{R} is zero, which helps simplify the expression from the wikipedia article,

Also, the electric dipole moment p is a vector with magnitude Qd, and points from the negative toward the positive charge. So this gives you the expression given in problem 1.2.

 

[s3a]

I don't know. The two always seem to go together in cases I am familiar with. At this point I would ask your professor what definition he or she prefers to use.

I'm actually studying this by myself and am not taking this course in a school so, I don't have any professors for this but, I'll now take this as “Near and far fields need to simplify to something nice.” unless and until I find a better definition.

Since the equation of problem 1.2 is only for points along the y-axis, \vec{p} \cdot \hat{R} is zero, which helps simplify the expression from the wikipedia article,

Also, the electric dipole moment p is a vector with magnitude Qd, and points from the negative toward the positive charge. So this gives you the expression given in problem 1.2.

When you say that the equation for problem 1.2 is only for points in the y-axis, I'm assuming you're referring to distances (since the equation states that the electric field is along the x-axis). Secondly, the distance r is not solely along the y-axis so, are you treating is as such by assuming that r >> d that way r approximates itself to seem perfectly vertically above each of the two charges kind of like how the line for the tagent function in the unit circle approaches infinity as the angle approaches 90 degrees?

 

[Redbelly98]

Yes, we are talking about the electric field at locations along the y-axis, and the electric field direction is parallel to the x-axis. And yes, since r>>d, it can be thought of as either the distance to the origin or the distance to either charge, since those two distances are approximately equal. I don't quite understand your last comment about the tangent function, but you seem to have the right idea.

 

[s3a]

I got what I needed to know to understand the problems fully (because of you) so, thank you very much!

If you care, just to explain what I was trying to say with the unit circle and the tangent function, I'm attaching an image I made for you. The radius of the circle is 1 unit. The blue lines are the axes. The black line needs to be touched by the green line eventually and it describes the tangent function. What I mean by the previous sentence is: notice that tan(pi/4) = 1. Now, notice that when theta = pi/4, the green line will touch the black line at a height of exactly 1. If theta = pi/2, then the green line will simply be parallel to the black line but, if the angle is less than pi/2 but approaches it, the height at which the green line will intersect the black line approaches infinity. I was associating the height of the black line as r in problem 1.2 and the radius of the circle from the centre to the black line as d. (Notice that d would equal 1 and r would be infinite or basically much larger than d.)

You are probably aware of this and I had just explained it badly earlier or something but, I explained it above just in case, you didn't learn that before.

 

[s3a]

Also, the electric dipole moment p is a vector with magnitude Qd, and points from the negative toward the positive charge. So this gives you the expression given in problem 1.2.

I realize this is an old thread, but I just wanted to ask/confirm if, for problem 1.2, you meant that the far-field equation for the electric field is the same as the electric field equation for the regular cases (i.e. for the cases that are neither near-field nor far-field cases) for all electric dipoles (unlike the situation in problem 1.1 – where the two charges are of the same sign – which allows one to approximate the two charges as one charge with double the magnitude of the charge).

 

[Redbelly98]

I meant that the far field equation for the charges in problem 1.2 is the same as that for an electric dipole. Period.

An electric dipole can be thought of as having only a far field, since it represents a limit as d→0. I.e., the "characteristic size" of a dipole is zero.

 

[s3a]

I see.

Thanks. :)