MetaFrizzics

I believe that is some kind of engagement or wedding ring.

It is being worn on the left-hand second finger.

 

But seriously, it looks like some part of a bow & arrow or spear part.

Except recall that with one end closed, the reflected standing wave is 2x the pipe length...

It looks to me like you're choosing odd harmonics, but all harmonics should be considered. Perhaps you are trying to double the frequency by this method. But in this case the fundamental harmonic will be the wrong one.

It has been a long time since I read Helmholtz on sound waves and organs. I will go have a look for something and come back to you...

This is from another thread here:

In case someone missed it, they have apparently imaged an electron orbit too:

 

http://www.nrc-cnrc.gc.ca/highlights/0501electronimaging_e.html

what is the point of a buddy list on a forum such as this?

(1) A buddy system is a good idea when travelling from forum to forum.

 

(2) A handy reminder of who has taken your side in an argument in the past.

 

(3) Shows who knows certain code-words, like 'glort'.

 

(4) The exciting hobby of elitism.

I hope you lot are better scientists than comedians

mmmwah he ha ha haaa!

Hey! This guy's a comedian. I used to be a standup comedian myself, but they told me to sit down.

Think of it as a not so cheap still.

Aren't we getting our shielded wires crossed?

I thought we were doing analysis before design.

In my opinion, the reason for the philosophical muddle regarding several of your points is that modern physics is in denial about a very obvious fact of life:

 

(1) Physics is about dead things, and brainless machines. These are deterministic, although the physics is beyond us at the moment and we have to settle for QM probability interpretations.

 

(2) People and higher animals are motivated by restricted free will. I can't fly, but I can choose to go left or right and select restaurants on any basis, including a coin flip. Physics will never be able to explain 'causality' for these events.

 

(3) There is a God, but unfortunately, He is smarter than us by such a huge margin that it's child's play for Him to fool us into believing anything He fancies. So just the possibility of His existance, and the probability that He won't jump when we call means we cannot actually practise 'true' science with any confidence in our findings whatsoever.

[math] 4\pi r^2 = 2(\pi r(2r)) [/math]

 

Surface of a Sphere = Surface of Cylinder of equal radius and height.

AND, any slice of a sphere between two parallel planes has the same area as a cylinder the same height with a radius equal to the original sphere.

 

quotes are not evidence...besides, that quote was from 1952, so it is probably outdated

Did you actually read my post?

The 1952 quote by Schroedinger was AGAINST seeing an atom, and he was proved wrong. Read what follows: How can historical events be 'outdated'?

Pray' date=' tell me how you isolate and trap a single atom (and confine it to a spatial width that is about the size of the atom itself) ? And how would you know that you have only one atom ?

What modes are you exciting here, and how do you go about this excitation process ?

Electronic transitions are typically in the far UV. If you want to see photons that are spit out by electronic transitions, you better get yourself a new pair of special eyeballs.

The box called 'reality' ? ... I wonder...[/quote']

 

I hate to pull you on the carpet for this, but it is so cool, I think people should know about it:

 

"We never experiment with just one electron or atom...any more than we can raise Ichthyosauria in the zoo." -Erwin Schroedinger, 1952

"Here, right now, in a little cylindrical domain... in the center of our Penning Trap resides positron Priscilla, who has been giving spontaneous and command performances of her quantum jump ballets for the last three months." - Announcement from Hans Dehmelt's lab in Seattle, 1984

 

"In 1979 in Heidelberg' date=' Germany, Dr. Werner Neuhauser looked up from the eyepiece of a low-power microscope and fancied he heard the ghost of Mach whispering: "Now I believe in atoms!". Neuhauser had just glimpsed what appeared to be a bright blue star floating in the void: it was a single barium ion, caught in an electromagnetic trap and flourescing in a laser beam. Thus transpired the first observation of an isolated atom using a lens; soon one would be glimpsed with the naked eye as well. (The common claim that no one would ever see an atom with the naked eye was an error, assuming smallness was the important issue. In fact, brightness and isolation from other atoms are what matters.)"

................................................. The Infamous Boundary by D. Wick pg 137ff [/quote']

Yggdrasil: Thank you for confirming my claim. As a matter of fact this has been done over and over again. I first heard about it many years ago.

 

'Seeing' is receiving light from an object into the retina.

 

'Plotting' is taking measurements and making a graph.

 

You can indeed see a single atom or even electron.

It may not look like much more than a magic star, glowing and pulsing,

but nothing beats the thrill of knowing you are actually seeing it,

and not through a microscope, scanner, or 'interpreter'.

Okay here was my starting point:

You have a wire with a mu-metal jacket (he he!)

There are basically three areas of interest as far as I can see,

Region C is of course the key area to be shielded.

 

I will slip on some LateX shortly.

 

OMG he's wearing a dental barrier!

[math]\left(\sqrt[5]{x + y}\right)[/math]

 

just testing...

Hey! I'm famous!

 

Even with truly elastic collisions (e.g. classical collision theory, the billiard ball model) there is an extended time of contact between the balls colliding, in which they compress and distort. The 'elastic' part comes in when they bounce back to their original shape, launching them away from each other. It is during this period of contact that energy is transferred between the balls.

 

Whatever you do, don't get him started talking about his balls!

It uses second-rate scientists and is shot on a low budget.

But seriously' date=' ....[/quote']

LMAO

 

So I say to the taxi driver, 'Show me where I can have a good time!'. And he takes me to *my* house!

Why don't we just meet half-way, and say the following:

 

(1) Strong magnets (> 1 T) are difficult to shield at close range.

 

(2) High permeability means quick saturation (@ < 1 T), which is a trade-off.

 

(3) Low strength fields are easy to shield.

 

(4) High permeability Mu-metal is ideal for low strength fields (< 1 T).

 

---------------------------------------------

At this point, it seems to me we are all being equally disengenious at the least:

 

In the example that's in my lab, you shield atoms in a vacuum system from having changing Zeemann shifts from the slow fluctuations in the external magnetic field of the earth.

Again, ridiculously low strength fields.

The complete set of shields used in the frst apparatus in the lab had a shielding factor of about 30,000. The new set I designed should be greater than 100,000.

Yeah, until you lean a 2T magnet against it.

 

As an atomic clock designer, I have to say that my experience contradicts yours.

I don't see how that can be, since we use the same equations and electromagnetic theory.

 

1 tesla isn't an ordinary magnet, so one shouldn't pretend that this is typical.

Every magnet I have on my shelf here, taken from ordinary modern industrial motors is at least 1 T, and many are almost 2 T. They don't make good motors with bad magnets anymore. Sure these magnets are 'strong'. But they are also typical surplus nowadays. Who's going to buy a crappy magnet when there's a bin full of killer magnets over here for 50 cents each?

 

I don't see a magnet strength nor a distance scale on the drawings.

Its hard to get more disengenious than that. Is the OP going to build a magnetic flytrap? Surely he is hoping to build something significant and useful.

 

Take a look at this device, used in optics labs.

I have to admit those are cool. They didn't have them back when I was auditing the first holography courses at M.I.T. in /70-72, and they were virtually inventing everything on the fly. I used sand and heavy blocks in one nice experiment with an early laser.

Face it - magnetic shielding works fine for a wide range of applications.

...with a *narrow* range of T.

 

Not al(ways). But hardly the "never works" scenario you were describing.

True enough. I admit being facetious at least once. Will you?

NdFeB and SmCo5 type supermagnets are the only magnets you will find that have field strengths of order 1T

...and find them you will by the thousands. In calling these now common motor magnets 'supermagnets' you can only be a referencing to the 60's, when there were no drugs, only airplane glue and Beatle-boots.

Mumetal saturates at about 0.8T.

Thank you. Useful and accurate.

 

mumetal will achieve better than 90% (and in most cases, I guess better than 99%) shielding for the geometry described in the OP - at a spot that's half a magnet length away, along the equatorial direction. This is my opinion.

And your opinion is pretty safe, with that qualifier. Why even bother to shield a magnet normal to it's poles half a length away? Because it's easy?

 

making it look like it was meant all along, that you were talking about DC fields much in excess of 1T.

As far as I know we were. Do you think the OP was hoping to shield a toy bar-magnet in his grade 7 class?

Perhaps this is how we ended up on opposite sides in the first place. If it is a misunderstanding based upon coming at the problem from opposite ends, I can totally understand that. You guys are coming from the standard applications such as shielding sensitive equipment from very low field fluctuations, which is actually a practical use of mu-metal. I was coming from the other end of the diagram wondering how anyone is going to shield a hefty magnet. Let's meet in the middle, I say again.

 

But this???:

shows that you do not understand the equation that you began an earlier post with.

Ouch, bum sore now. Please. I was doing us all a favour by not posting a 10 page derivation. I think I know what I meant by my own diagrams and equations. I'm happy to post the derivation if you give me two days to set up all the LateX by hand. I have no auto-program, and I've been reverse-engineering everybody else's LateX posts up until now.

 

Thanks for the scolding. But since I can pull the equation out of the sky, I can also derive it, although perhaps not as rigorously as a mathematics specialist and author with 5 T.A.s doing most of the slog-work.

Ummm' date='

 

Transformers are AC, not DC, so the points raised by DQW and myself are not addressed by this, and the source says that shielding works but is expensive, which would seem to contradict your point.[/quote']

Actually, it seems the other way around to me:

d.c. (stationary magnetic) fields don't do any real work, since there is no motion of the 'blocker' relative to the magnet. The magnetic flux leaks and extends into nearby space waiting for some unsuspecting (moving) object made of iron like another metal bar that it can pull on or magnetize.

If you could have the mu-metal rotate around the magnet, there might be hope of demagnetizing it and weakening the field leakage.

 

I find the whole model of 'faraday' lines repulsive (no pun intended) since everyone takes them too literally, and assumes they actually exist.

 

 

I see your Link (3) also agrees with what I said, when you read the fine print at the bottom (even though their units are different than mine):

 

hus, if [math]\mu\simeq 10^5 \mu_0[/math] for Mumetal, then we can reduce the magnetic field strength inside the shell by almost a factor of 1000 using a shell whose thickness is only 1/100th of its radius. Clearly, a little Mumetal goes a long way! Note, however, that as the external field strength, [math]B_0[/math], is increased, the Mumetal shell eventually saturates, and [math]\mu/\mu_0[/math] gradually falls to unity. Thus, extremely strong magnetic fields (typically, [math]B_0\stackrel {_{\normalsize >}}{_{\normalsize\sim}}1[/math] tesla) are hardly shielded at all by Mumetal, or similar magnetic materials.

That is, just like I said, mu-metal quickly fails to shield anything when you are trying to stop the close proximity field of an ordinary magnet.

 

Your first Link (1) was cute by the way. The effort (and cost!) that goes into shielding a computer monitor inside an NMR medical scanning room is a great example of why it is probably more sensible now to replace a CRT with a liquid crystal display! (another obselete product...) But can only imagine the engineering efforts that went into that shielding project, that caused them to turn to mu-metal as the best 'kludge'. And here the magnetic fields are well below safety standards for human beings for long-term exposure. NOT the close-proximity fields of permanent magnets! The same goes for the other examples there: Shielding the earth's magnetic field? Come on: It's so weak you have to use a magnetic compass to detect it! All the examples given where mu-metal shielding is effective happen to be with fields so weak you need special equipment just to measure them accurately!

 

The other examples on the website page are equally complex and incredibly expensive and difficult shielding projects, and while 'mu-metal' is inevitably involved, I'd suspect a far more important factor would be the creative engineering skills of the scientists overseeing the projects! (as I pointed out in my previous posts describing the problem of shielding.)

 

You know that most shielding materials have \\mu _s = 100,000 ~to~1,000,000 or more, and you have the gall to make it look like having \\mu _s > 1 is a rarity among shielding metals ?

Whoa. Let's take it back a few notches: The point wasn't to speak of permeability per se, but to show that a second factor, thickness of the shield was far more significant relatively speaking for shield effectiveness.

 

This is clearly indicated right in the bit you quoted: "...and the shield thickness is small...only very thick shields...", but is even more clearly indicated in the equations themselves.

 

Mumetal® can have a relative permeability of 230,000 or higher. For now, I'll use a conservative number \\mu _s = 2 \\cdot 10^5. Now if the radius of your spherical shell (or whatever the heck it is you wrote that equation for) is 10cm (about the size of a bowling ball) and has a wall thickness of 10mm, this gives you S = 1/10000 !!! Even if you had giant sphere of radius 2m (a 13 ft dia enclosure, bigger than my kitchen) with a wall thickness of only 4mm (less than a quarter inch, hardly enough for just structural strength) you have an impressive shielding ratio of S = 1/200 !

This is good work: But look at what you are doing as well, over and over again. You are talking about distances ranging from a bowling ball radius to a 20 foot square room. In these cases, supposing the placement of a strong permanent magnet the size of a golf-ball (no lightweight item), you are dealing with shielding field-strengths [math] \frac{1}{10^{10}} [/math] of the strength at the source.

 

Of course if you are either shielding incredibly weak fields (like the earth's) or working at gargantuan distances from the source (10 feet from magnet) mu-metal is your man!

 

Now look at the original post: He wants to completely shield a permanent magnet less than a mm from its core on the axis of the field lines, from end to end.

 

In a word, "impossible" (at least with a mu-metal shield).

Mu-metal cannot block a permanent or D.C. magnet. All that happens is the Mu-metal becomes magnetized and hence becomes part of the magnet!

I was going to say "hogwash" !!! But I'll be polite and so, I'll merely ask you to back up your claim.

I may have exaggerated a little, but anyone who wants to play with mu-metal cages and magnets can confirm the basic observation. (As a hi-fi audio designer I have done so for many decades, and speak firsthand.) Only incredibly thick pieces of mu-metal could hope to be effective, and its a law of diminishing returns, due to skin-effects, hysteresis, magnetostriction and reduction of useful area near the magnet.

It's true that you can't use this material to shield one magnet from another, but you can shield a region of space from magnetic fields

Well, that's another one for me. In what sense can you shield 'nothing'? (a region of space)? Obviously, to talk about effective shielding you have to put something in there, and show it was shielded. Yet if we put a magnet in there, the house of cards collapses.

 

If I just admit I exaggerated a little, will you let me go now? I'm just a bit quirky in my old age.

Spontaneous Magnetostriction

 

When a material becomes ferromagnetic at the Curie point, spontaneous magnetization appears within the domains and with it an associated spontaneous strain e or magnetostriction [math]\lambda_O,[/math] along a particular direction. The amplitudes of such are independant of crystallographic direction. Within each domain the strain varies with angle from the direction of spontaneous magnetization, according to this: [math] e(\theta) = e (\cos \theta )^2 [/math]

 

The average deformation throughout due to magnetostriction can be gotten from integration (assuming randomly oriented domains):

 

[math] \lambda_O = \int^{\pi/2}_{-\pi/2} e( \cos\theta)^2\sin\thetad\theta = e/3 [/math]

 

This is caused by the ordering of the magnetic moments at onset of ferromagnetism. Because the strain is in all directions, the sample will change dimensions but stay the same shape. For details on the quantum mechanical aspects of this, see:

Introduction to Magnetism and Magnetic Materials D. Jiles 1991 Chapman & Hall. Jiles is from the Ames Lab at the U.S. Department of Energy)

 

As a practical matter, the best technology for both d.c. and low frequency magnetic shielding is still relatively simple steel alloy casings. (refs. Magnetic Circuits and Transformers ,1943 (nothing has changed) Dept EE, M.I.T.)

 

For anyone who wants to properly understand the electromagnetic field at a deeper level than the almost idiotic 'Maxwell' equations of Heaviside from the last century, A difficult to locate but valuable book is:

Antennas in Matter RWP King, GS Smith, with M Owens & Tai Tsun Wu 1981 - M.I.T. press This 850 page tome deservedly commands the title 'Advanced'.

"Tranformers are notorious avenues for introducing noise on powerlines. ... The magnetic field cannot be confined and can be a source of troublesome hum. Since the field is highly directional, the problem can sometimes be minimized by orienting the transformer in the direction that produces the least interference, or it can be located further away. Magnetic shielding is an expensive last resort. More exotic constructions use high Ni alloy (mu-metal) as a casing but the benefit obtained from high theoretical permeability is partially lost in fabrication and assembly techniques. A greater attenuation (not elimination) of the stray field is achieved if a double-case construction is used: an inner case of magnetic nickel steel surrounded by a 2nd case of mild steel. " Handbook of Transformer Design and Applications W. Flanagan McGrawHill 2nd ed. 1993 (sec 3.7)

 

Here again my points are re-affirmed.

Solving jointly all of the equations with regard to the integration constants, we finally obtain:

 

[math]S = \frac{H^I}{H_O} = \frac{4\mu_s r^2_e}{r^2_e(1 + \mu_s)^2 - r^2_i (1 - \mu_s)^2} = \frac{4}{\mu_s} \frac{1}{1-\frac{r^2_i}{r^2_e}} [/math]

 

This equation indicates that magnetostatic shielding effectiveness depends upon magnetic permeability of the shield and its radius and thickness. Even if the shield's magnetic permeability [math]\mu_s > 1[/math] and the shield thickness is small, [math] ( r_e = r_i ) [/math], the shield effectiveness is not very good. Only very thick shields with large permeability provide protection from static and low-frequency magnetic fields.

 

Shielding effectiveness [math]A_m = 20 log |1/s| in dB [/math].

 

While the Equation above is simple, the magnetostatic problem is not! The following complicating factors make the design and evaluation of magnetostatic shielding a very tricky business:

 

(1) High magnetic permeability materials are non-linear: with change of field intensity the permeability varies, reaching saturation quickly at key induction values. Shielding effectiveness is also affected by some parameters that are specific to magnetic materials ( i.e. mu-metals), including hysteresis, magnetostriction, and core loss. All these factors must be accounted for not only in shield design but also in performance measurements.

 

(2) Material properties vary broadly with respect to manufacture, handling conditions, and especially temperature, flexing, and impact shock. As a rule the higher the permeability of the material, the larger the instability in shielding effectiveness, with experimental variances observed as large as TENS of decibels.

 

(3) You have probably noticed that the Equation above doesn't include frequency, (rightfully so for magnetostatics) So strictly speaking it is only applicable at d.c. or low frequencies where equivalent penetration depth [math] \delta = (2/(\omega\mu\sigma))^{1/2} [/math] is large. But because of high magnetic permeability, the skin effect becomes significant at VERY low frequencies, leading to a reduction in the effective thickness of the shield. This in turn leads to a deterioration of the magnetostatic shielding effectiveness. The only good news is that with frequency rise other mechanisms can enter the picture.

 

To give a practical grip on the picture, Here are some measurements done with Helmholtz coils:

 

A 1/16th in. iron pipe provided around 20-25 dB shielding effectiveness at 50 - 5KHz. By comparison, an RG59 cable shielded by relatively thin but high permeability double-layer conetic braids yielded only 14-20dB and 16-23dB. As a rule it isn't simple to obtain large cable shielding effectiveness in magnetostatic fields period.

(For refs see Sellers J et al, "Flexible Braids for Improved Magnetic Shielding of Cables" 1978 EMC symposium Alta. & Rikitake, T. "Magnetic and Electromagnetic Shielding" 1987 226 p etc.)

OMG. The Electrical Engineer inside me forces me to speak out after reading the beginning of this thread:

 

Mu-metal does indeed 'shield' items or 'block' electromagnetic waves.

However, to think that you will 'block' the lines of force and hence the 'pull' of a magnet is a complete misunderstanding of what Mu-metal shielding actually does, how it does it, and what it can be used for.

 

Mu-metal cages and covers are used to 'block' (by absorption of field energy of) MOVING magnetic fields (from A.C. currents!), which induce electrical currents in sensitive circuits. That is, a Mu-metal box or 'Faraday' cage captures the energy from radio waves and other changing electromagnetic fields, such as those radiated by (60 Hz) transformers or chokes in power supplies.

 

Mu-metal cannot block a permanent or D.C. magnet. All that happens is the Mu-metal becomes magnetized and hence becomes part of the magnet!

 

The theory behind Mu-metal 'blocking' is not 'true' blocking like in the sense of electrostatic 'shielding', where forces are screened out. In the case of Mu-metal, significant portions of the energy in the electromagnetic field are diverted and converted into heat, or re-radiated in a different direction /orientation than before, improving the shielding of sensitive parts, which usually have more and less sensitive orientations.

 

Mu-metal really is best understood as an 'antenna' that steals electromagnetic radiation by absorption or diverts it by electrical currents. Something like a large metal framed building would reduce or reflect your T.V. signal causing 'ghost' images and poor quality reception.

 

At best a Mu-metal cap or plate might act as a 'magnet keeper' by providing a more direct and low resistance route between the poles of your magnet. But any piece of iron or nickel would do the same thing.

I'll have a go at providing you with the information you need.

 

In the meantime I am posting this tantalizing graph of the gap between the equation for a solid sphere near the surface and Newton's 1/d^2 formula, for a quantization involving 4000 slices of a sphere (the number of particles would be approaching the millions).

 

 

(text from one of my articles analyzing various models of quantization of mass:)

 

A Closer Look at the Solid Sphere Formula

 

It is even more remarkable when we enhance the formula for accuracy and plot the results. Note the n-1 summation endpoint and +.5 centre-point:

This doubles the accuracy of the finite approximation. The convergence with Newton’s 1/d2 formula is actually tighter than current capabilities for measuring the gravitational constant. Nonetheless we cannot be pleased with the formula from a theoretical standpoint. The reason is that for the Sphere Theorem to be self-consistent, which is a mandatory requirement, it must hold for both hollow and solid spheres. It clearly does not. This means that some other mechanism must be sought to explain the apparent validity of the solid sphere portion.

 

(note: here the jiggling of the plotted lines reflect the rounding errors due to long floating point computational variations/rounding errors. Interestingly, this actually mimics the Brownian-like motion expected from small fluctuations of force. The error in the equation at 4000 slices is still way above the noise floor of the computer algorithm calculating the forces. )

Can you tell me, if there are \\approx 10^{23} charges in the shell, what distance (as a fraction of the radius) you must travel inwards before the field drops off to below, say 0.0001% of the field at the surface ?

I'd be delighted to:

With such a large number of particles there is no real need to do any calculations. In the case of gravity the force is so weak that it is immeasurable at any distance significantly greater than a few thousand effective nuclear diameters (e.g. for a gold atom).

 

The engineer in me is tempted to frame the question in this way, but a much more productive and important approach is to view the entire field inside the sphere, and speak qualitatively about it as we consider a system evolving in time.

 

In the case of electromagnetic forces, not only is there hope of measurement of such effects, they become extremely important, as we use very small numbers of charges in controlled lab settings to measure electrostatic forces and set standards of units. Here the errors can be far from minor and it is important to set theoretical bases for interpretation of results very carefully, and physically construct our experiments with extreme accuracy.

 

I would take the view that the concept of interest is that no matter how small the imbalance in forces, as a system evolves, even particles placed at the very centre where the imbalance will be the absolute least will inevitably be forced to break symmetry sooner or later. This action for a single particle starting at the origin can be modeled as a 'biased' Brownian motion which finally is compelled to migrate away and accelerate toward the inner surface, its variations and uncertainties of velocity gradually converging to more and more efficiently directed path to the surface.

 

At first, imperceptible imbalances will be drowned out in 'Brownian noise' and so the process must at first rely upon a large enough random perturbation to allow the particle to be immersed for a significant amount of time in a given general direction of imbalance. Eventually however, the 'signal' will rise above the noise floor often enough to affect the future path of the particle.

 

From here the particle will begin a new 'life', now characterized by a 'leashed' type of Brownian motion of the most complex kind, in which its overall position is being constrained more and more along the 'normal' (perpendicular axis) to the sphere surface. In its wanderings, the particle will weave in and out of the sphere, but along with other particles will converge in a dust-like cloud enveloping both sides of the surface.

 

This second 'random walk' will be more like the particles in the Rings of Saturn: Most will remain trapped and the ring will narrow, while very rarely, some particular particle will be given 'escape velocity' through a random series of energy exchanges, and 'tunnel' out of the surface to either escape or begin again.

No Single Answer to the Question

By the way, there can be no single answer for such a question as the one you pose, with a simple mathematical result. If you thought the Three Body Problem was a tough nut to crack, this is many orders of magnitude greater. First even from a Classical point of view (simple point-masses and deterministic velocities) The field would have to be constantly recalculated using a perturbation method.

 

Once the mass is discretely localized in space, we can no longer talk about a radially symmetric field, but instead must talk about a 'scattering matrix' which reflects the exact locations of the point-particles making up the sphere.

 

Remember that all along, we used a test-mass and moved it in a straight line along the x-axis. This was possible because in a continuum-style radially symmetric shell of constant density and equal insignificant thickness, every trajectory through the Geometrical centre was identical, and our particle (in the Classical system) would naturally travel a straight line in Euclidean space.

 

Now with point-masses spread at random distances across the surface, every single trajectory of a particle will be unique and complex. No particle will be able to travel in a straight line. In the physical case (say with a sphere of gold atoms, or a charged aluminium sphere) we would have to fire millions of particles through the sphere and measure the results to build up a scattering pattern.

 

Each individual trajectory (even if theoretically deterministic) would be unique and have it's own history with an instantaneous force vector that was constantly changing in direction and magnitude.

 

Every point in space just inside the sphere would have an infinite number of actual vectors, depending upon the speed and direction of the particle passing through it. This is because the previous history of the particle in a deterministic system must also affect the lattice of charges/masses in the sphere according to Newton's Third Law. (Equal and Opposite Reactions)

 

Also of importance is the relative velocity of the particle compared to the force. For very fast or heavy particles, a small bending of trajectory would be expected most of the time, and the scattering matrix would look quite orderly and tractable. However, once we slowed the particle down and lightened it's load, trajectories would deteriorate into the Random 'Brownian Walk' we described earlier.