Thursday, September 16, 2021

The Electric Potential - Is It Really Unphysical?


          Illustration of magnetic and electrical Ahranov-Bohm Effects.
          (From Physics Today,  August, p. 63,  cf.  Vavagiakis et al)

In a previous (June 16) post we saw that the electrical potential is defined as the potential at a point A when work is done per unit positive charge in bringing a charge from infinity  (¥ ) to A, expressed:

V = W/ q =  Q / 4p eo 

Although Maxwell’s equations e.g.

i)  Ñ X (E) J    + D / t

ii)             Ñ X E  - B / t       

iii)         Ñ ·0    

iv)          Ñ ·r            


Contain only electric  (E ) and magnetic (B, H)  fields, they can also be expressed in terms of electric and magnetic potentials.  In this post we focus on these two potentials and attempt to discern which is real and physical – and which isn’t.  This was originally examined by Vavagiakis et al in the Quick Study section of Physics Today (Physics Today August, p. 62,  by Vavagiakis et al

Technically, as the authors note these potentials have no real significance.  The basic physics is contained in the forces on charged particles e.g. for the E-field

F = Q q/ 4p eo r2

And the spatial and temporal variations of the potential determine their field strength. For example the electric field strength:

E = F/q

Or: E = 1/q { Q q/ 4p eo r2   

The main point is the forces are directly proportional to the field strengths and vanish where the field strengths vanish. Physicists are at liberty to add terms to the potentials that leave the fields invariant.

Few undergrads know that Maxwell’s classical field equations regarded the potentials as non physical.  This view was revised with the development of quantum mechanics. For example, as previously noted (see my post of Sept. 14th) the Heisenberg uncertainty principle is incompatible with the notion of a point particle. It is impossible to precisely fix the position of an electron in a hydrogen atom, for example, without losing all information about its momentum.

 For this reason, QM replaced point particles with wave functions, e.g.

y(x)=  | y(x) |  exp i  qx

 Defined by an amplitude | y(x) |   and a phase qx.(Note the phase is irrelevant in classical mechanics, demonstrated when one works out the Heisenberg principle for a classical object, say bullet).   The squared amplitude | y(x) | 2   meanwhile is the probability density of finding the particle at position  x, more completely written:

P ab  =     ò b a   | y(x) | 2  dx

 Where the location must be between points a and b.

Why the need to include phases for charged particles?  Well, because in the modern format we include quantum mechanics when examining electric and magnetic potentials and in QM the phase of a particle is fundamentally connected to these  - specifically the magnetic potential A and the electrical potential  f.

The phase of a particle of charge q  over traversing a spatial trajectory  g  over a time interval t  changes by (op. cit.):

D q = q/ h  ( ò g ds · A  - ò  t    dt’ f)

Relative to the phase of a particle that traverses the same trajectory with vanishing potentials.  Because differences in phases can be observed in quantum physics experiments the equation has consequences for the physical significance of the potentials,  A  and f . In other words they are supposed to be real, physical quantities.  But are they?   The Quick Study essay in  Physics Today  (August, p. 62)  by Vavagiakis et al questions that assumption.   

Along with the preceding they ask us to consider the top left graphic shown, which depicts a copper solenoid which generates a magnetic field B inside it.   That field is related to the magnetic potential through its curl, i.e.

B  =   Ñ X A

At the same time it prevents the field from existing outside it. And yet, when an electron beam (blue)  splits and passes around the solenoid, the respective electrons in each path interfere and exhibit a phase difference.   This is evidence of the magnetic potential's influence, i.e. that it is a physical existent.  When the wavefunctions interfere on the far (right) side of the solenoid they still have a total phase difference of:

q = q f(B)/ h

Where f(B) is the magnetic flux in the solenoid.  This result is called the Ahronov -Bohm effect.

But is there an electric Ahranov-Bohm effect? In this case consider two identically-charged particle passing through charged metallic tubes (Faraday "cages") as depicted in the right top graphic.   The object here is to induce an electric potential difference between the two particles but without encountering any E-fields. To obtain this effect a voltage  D is applied between the tubes/cages for a time Dwith the particles inside the tubes.  What do we find? Well, the E-field ≠   0  only outside the tubes. Also, limited to the time the particles are inside them. One finds the following phase- voltage relationship whereby the particles acquire a relative phase shift (ibid.):

 D q = q/ c   x    DV/ m V   x  DT/ h s  

This equation describes the electric version of the Ahranov -Bohm effect, i.e. in which the electric potential determines the evolution of the phase - say as a charged particle traverses a time DT.  (The units of microvolts, m V and nanosconds are included to give an appreciation of the scale in changing the electron's phase.)

Here's the kicker:  an observation of a phase shift would be a direct confirmation that the electric potential is physical.  However, up to now no such observation has ever been made. The further ramifications of this and especially the experimental impediments will be explored in a future post.

The E-field is non-zero only outside the cages and only during the time the particles are inside them. According to the phase- voltage relationship the particles acquire a relative phase shift. 

To Clarify: Gen.Mark Milley Is The Patriot, Trump Is The One Who Needs To Be Tried For Treason


                                           Gen. Mark Milley - The True Patriot

   Trump - imagining how he'd blow up the world if Milley wasn't around

As soon as the new anti-Trump book 'Peril' hit the market it was inevitable Trump would squawk.  Never mind this psycho bum grifter loser ought to be in orange prison garb by now. And the fact he's free and still walking around is mocking any sense of justice after his insurrection. That this deranged imp now has the nerve to call for Gen. Mark Milley to be "tried for treason" is therefore laughable.  Especially after Gen. Milley acted the part of a true patriot by protecting the nation during Trump's lame duck period marked by his insanity and overreach.  

What got Trump's dander up? Well, learning from the new book by Bob Woodward and Bob Costas that Gen. Milley had contacted China without Trump's knowledge and reassured its Gen. Li that there would be no sudden attacks despite Trump's bluster and bombast linking China to the virus.  In addition, Gen. Milley had to give assurances to Nancy Pelosi he would run interference to keep Trump from using the nuclear codes to mount an attack - just to avoid being a one term wonder.

Recall back in early January House Speaker Pelosi took the extraordinary step of publicly revealing that she had talked with Gen. Mark A. Milley, chairman of the Joint Chiefs of Staff, to discuss “available precautions for preventing an unstable president from initiating military hostilities or accessing the launch codes and ordering a nuclear strike.” 

Having witnessed first hand what the combustible traitor Trump was capable of-  with the incitement of his mob terror takeover of the Capitol on Jan. 6th -  Pelosi was quite justified in pursuing inquiries.  Unlike Dotard she had the national interest in mind not her own.  Moreover,  anyone with at least two brain cells would know the traitorous ape was off kilter mentally and capable of anything.   As I noted in early January that included the ultimate potential terror: unleashing a full scale nuclear war.  One could therefore conceive of a scenario much like that depicted in the movie, The Dead Zone, e.g.

 Where another insane president ("Greg Stilson") engaged the nuclear codes "to fulfill my destiny".  Obliterating the destiny of the nation in the process.  Well, there was no General Milley there to prevent that outcome.

  In the case of reaching out to China - because Milley believed Trump's erratic behavior could lead to war-   Milley also was thinking of the national interest not loyalty to an insane traitor rat.  Hence, I do not use the fraught term "undermine Trump's authority" like many in the breathless media.  The reason is that an insane man has no authority because he has no rationality.   

Meanwhile we have learned a spokesperson for Gen. Milley has confirmed many of the revelations in the book, e.g.

"His calls with the Chinese and others in October and January were in keeping with his duties and responsibilities - conveying reassurance in order to maintain strategic stability."   

However, Chris Miller, who served as Trump's acting Secretary of Defense over his final two months - and was the one who refused to send the National Guard in to quell the insurrection when first asked (to allow Trump more time to watch it on TV) had this to say:

"I never authorized Milley's actions with China. His actions were a disgraceful and unprecedented act of insubordination and he must resign immediately."  

 Give me a break, you Trumpie traitor rat.  You ought to be licking Milley's boots for protecting this country.  Which you didn't do when you failed to call in the Guard after numerous pleas as his terrorists ravaged the Capitol.

In fact, as reported last night ABC News learned Milley was not the first Pentagon official to reach out to China.  Former Defense Secretary Mark Esper reached out shortly before the November election - already aware of Trump's increasingly unhinged behavior. His purpose was to assure them the U.S. would not launch a military attack. 

Milley's first Chinese contact came after that and had been fully authorized by Secretary Mark Esper - the last real Defense chief - i.e. before Trump appointed his toady Miller.  Thus Miller's barking about resignation has no foundation, even less because Miller is a traitor too.  Both he and Trump out to be tried for treason and hung - preferably in the Capitol where their white terror apes ran amuck on January 6th. 

 As a person whose heritage goes back to the American Revolution, e.g.  

 I have little tolerance or truck for traitors, especially those in high office.  So I make no apologies for wanting to see Trump either hung by his scruffy orange neck or put away for keeps,

 but certainly unable to ever attain national power again. He isn't even fit to be a dog catcher.

Meanwhile the idiot editors at the WSJ are as dense and clueless as ever, spouting off today that: "it would be disturbing to find out that military leaders responded to the President's norm breaking by betraying their institutional obligations."  

But Milley and Esper's sole obligation was to the Constitution and protecting the nation from an obvious lunatic and sore loser.  This they did as authors Philip Rucker and Carol Leonnig also made clear in their book, 'I Alone Can Fix It'.  In that case ensuring the "plane" (electoral transition) "would land safely" with no military interference on behalf of Trump. In this case it was essential to out maneuver the lawless (not merely 'norm breaking') Trump to ensure the Chinese would not launch a protective pre-emptive nuclear attack. 

The WSJ editors are also daft to insist that if Milley was "really so outraged" at Trump's reckless behavior he "ought to have resigned".  No, schmoes, that would be the coward's way out. Milley understood - as Leonnig and Rucker noted - he had to remain in place to protect the nation from a Hitler -adulating psychopath and he did.  Case closed.  

Gen. Milley doesn't owe anyone an explanation and he has the full backing of President Biden and Gen. Lloyd Austin.  'Nuff said.

See Also:

Milley’s revelation about Trump leaves us with five questions

Wednesday, September 15, 2021

A Newsome Blowout? California & Nation Barely Escape Trump Clown....Errr Clone, Larry Elder


                                      Trump Clown Larry Elder and Gavin Newsome

The  mainstream media this morning will mostly report that California governor Gavin Newsome won in a "blowout" in respect to the recall vote.  Let's remember this was a moron invention wherein voters who get PO'd at their choice (or vigilante- inclined repukes) try to "recall" an already elected official.  The worst aspect is that if voters vote 'yes' they then pick some Schmoe to take the sitting guv's place. The candidate with the highest percentage  of all then gets to be the next governor - even if it's only 14%.

Up until last month, as noted by Steve Kornacki last night, the pro-recall and no recall forces were almost even at 47% vs 51%.  Then the Dem politico establishment suddenly woke up and realized if it didn't fire up Golden State voters they could be looking at a national nightmare.  

This time it was mainly the Right's pro -Trump, 'Big Lie'  loons who drove the recall effort.  

Their main mistake, given too many Dem voters were content to sit on their butts going "ho hum", was to push Trumpie sycophant Larry Elder - a conservo talk show host- into the spotlight.  That one self-inflicted error was all the help Newsom needed to light a fire under the Dem voters (especially Latinos) and get them to cast their ballots.  What did Newsom use? Well, fear of course!  And fully justified.  Just imagine a black Greg Abbott running Sacramento - taking away women's a abortion rights using vigilantes like Abbott did in Texas-  and wiping out the minimum wage as well. 

 But that wasn't the worst of it.  This stupid Uncle Tom pawn of Trump also  suggested Women's March participants are too ugly to rape and argued that it's the descendants of slave owners and not slaves that deserve reparations.  When some in the left media then painted him as a de facto white supremacist the WSJ's Willam McGurn went off like a stink bomb in an op-ed today ('Gavin Newsome's White Privilege').  He insisted that never ever again can the left or Dems use the white supremacist label to tar anyone in the opposition.  In Willie's own words:

"If white privilege is a thing, Mr. Newsom is drenched in it. Mr. Newsom’s father was a well-connected state judge who once managed one of the trusts for the family of oil magnate J. Paul Getty. ....In sharp contrast, Mr. Elder is a talk-radio host whose father, a former Marine, took two jobs as a janitor to support his family before opening a small cafe. Mr. Elder grew up in South Central L.A., and his former high school was featured in the 1991 film “Boyz n the Hood.” If elected, Mr. Elder would be California’s first African-American governor.

So guess who’s the white supremacist in this race?"

 Well, McGurn, Elder is -  given his pro-slave owner stance, which also proves he's the classic 'Uncle Tom'.  This is not even close to being rocket science, more like the proverbial no brainer.  I don't give a royal crap if Elder grew up along side a sewer, if he howls for slave owner descendants to get "reparations" he's no better than David Duke.

The lesson about this result is also to process that any Trumpie ass clown or clone will try to use the "Big Lie" if it suits his interests.  In Elder's case he was prepared to claim vote fraud even before all the votes had been counted.  The only thing that spared us from another ungodly electoral battle was the fact he ultimately got blown out.  But this is a lesson for the mid terms next year.  All the Repuke House candidates need to be blown out in their respective elections.  It can't be  close or they will claim voter fraud like Trump did in 2020.

Let the media talk blowout as they will - the same bunch who got hysterical and pilloried Biden for doing the Afghan pullout wrong. The sobering reality is we narrowly dodged a bloody disaster, and the moral of the story is we will likely have to use vast gigabytes of fear in the midterms as well. To remind short memory voters what these Reep insurrectionists and seditionists tried to pull last year - complete with graphic videos - and why they don't deserve political power.

See Also:

by Amanda Marcotte | September 15, 2021 - 6:57am | permalink

— from Salon


Republicans have given up entirely on the basic premise of democracy. They no longer think that politicians should try to appeal to a majority of citizens to win power. Instead, their focus is entirely on finding ways to circumvent public will and gain power anyway. In some states, this manifests as gerrymanderingoutright voter suppression and even a new slate of laws meant to allow Republican officials to simply vacate the results of elections they don't like. In California, it's about forcing an off-year election in the most confusing way possible, hoping to hoodwink voters into accidentally giving themselves a Republican governor.

Managerial "Bezos-ism": The Basis For How Amazon Wrings Productivity From Workers - And The Template Is Spreading

 Heike Geissler's book (Seasonal Associate) laid out the brutal conditions she had to endure working in an Amazon fulfillment center in Leipzig, Germany.   In one particular harsh day she writes (p. 100):

"You replace the Band Aids on your hands. Your thumbs, forefingers, middle fingers  on both hands now have long hangnails from all the reaching into totes and boxes and from cutting and folding cardboard. Harmless irritations but they make every movement harder."

Heike didn't elaborate on the need (or any use) for 'Depends' -but the logical inference is she likely did use these adult diapers, as many Amazon workers have to.    This is because the automated  timing system used by Amazon  (to ensure workers "made rate") never allows enough time to use the rest room and get back to all the tote and consignment areas without incurring demerits.   

Highly educated and a freelance writer, the last thing she wanted to do was join the Amazon drone brigade, but had few options.  There simply was no other way to get money and pay off her debts at the time, so she decided to become a seasonal associate.

As she related the situation: (p. 5) "You do get child benefits for the two boys, you can pay your bills, but unfortunately they don't get paid on have to take the first job that comes up and get money in the bank."

She had the option of  applying for  welfare, of course, but like here in the U.S. there were way too many bureaucratic hoops to jump through and besides - as a proud German- she had her pride. So it was "Amazon or bust". 

 In the Afterword to Ms. Geissler's eye-opening book,  Kenneth Vennemann writes:

"Heike plays with these insidious euphemisms, the barefaced lies of 'flat hierarchies' and 'special handling'.  Hence the ugly title, SEASONAL ASSOCIATE, the word associate here so far removed from the idea of partnership and sharing that it makes me snort with cynical laughter."

Of course,  there are hundreds of variations on the theme of 'working for Amazon' which anyone can find by Googling. Sob stories galore, and many of which do elicit sympathy for the those who found themselves in this capitalist behemoth's grip. But as Vennemann observes -  "there have been a few undercover pieces on Amazon by journalists, but they went into far less detail."

Indeed, reading Heike's account you are right there with her as she has to work at the receiving end, unpacking boxes and entering products into the system.  Also, consuming half of her lunch break time just getting to and from the company cafeteria. Sounds like a piece of cake? Well it wasn't. It was sheer numbing hell for the 8- odd weeks she spent during one holiday season.   But what of workers who are locked into such algorithmic-driven  drudgery for years?

The WSJ's Christopher Mims examines  this robotic management style in his piece:  ('How Amazon Squeezes Performance Out Of Workers') Sept. 11-12).  He writes, for example, on the travails of one recent Amazon employee:

"On the job, no one ever stood behind Mr. Morreale and barked at him to work faster. They didn’t have to. Twice a day at a stand-up meeting, his shift managers told the group how everyone was doing. They knew because Amazon’s software, and an assortment of sensors in the warehouse, tracked workers’ every move. “Those numbers are always in the back of your head somewhere,” he said.

Mr. Morreale’s story represented pretty much the median experience of the Amazon fulfillment center workers I’ve interviewed. On one end of the spectrum, there were those who found the work intolerable, and lasted less than two weeks. At the other end were those with an appetite for the work."

But the latter are doped up on energy drinks and caffeine.  The only "generous" concessions the managers make to the working stiffs.  You can't sit down while you're on the job, but you can gulp more coffee and have a few aspirin if you're aching.  Impossible to imagine infecting other working venues, corporations?  Don't be so sure.

 At this very moment, Mims writes, Bezosism is "diffusing through the world of work, rewriting the source code of the global industrial machine. If it proves as popular and durable as the systems of organization on which it builds—from Fordism to the Toyota Production System—it could be, along with the e-commerce and space companies he built, Mr. Bezos’ most important legacy."

Don't laugh or reject this as hyperbole. We  already know (WSJ, Sept. 11-12, p. C3)  corporate bosses across the nation have insisted on remote employees installing "bossware" or "tattle ware".  This allows distant supervisors to monitor home workers every move: logging webcam screenshots, keystrokes and browsing history. 

Meanwhile, in Amazon’s fulfillment centers, human productivity is measured by an overall "stow rate",  calculated for each worker at a robot-fed pick-and-stow station. Amazon's egregious managerial error is in applying aggregate 'stow rates' and 'flow rates' to all employees.

The overall rate at which workers must complete a task in an Amazon warehouse, whether it’s putting items on shelves, taking them off, or putting them in boxes, is calculated based on the aggregate performance of everyone doing that task in a given facility.  This according to an Amazon spokeswoman who spoke to Mims. This floating rate, Amazon argues, shows that "none of its employees is being pushed beyond what’s reasonable, because that rate is something like an average of what everyone in a warehouse is already doing."

Yeah, right, except that if every other worker is trying to outdo peers to 'get ahead' the floating rate increases, to the detriment of those workers (maybe older) who are unable to keep up. And Mims points out "anyone can have a bad week" or maybe they're developing one of the repetitive injuries so common in the workforce (which is why aspirin dispensers abound along with coffee stations. )  In Heike Geissler's case it was getting sick and exhausted to the point of getting flu - along with back problems.  But why expect otherwise when workers have to perform the same task for an entire 10-hour shift, with only a half-hour for lunch and two 15-minute rest breaks?  

And yet Bezos wrote in an April letter to shareholders:

We don’t set unreasonable performance goals,”  

If you don’t make rate you’ll get a warning, triggered by an algorithm, and if it happens often enough your job is in danger, can be a powerful psychological spur to work harder, and possibly to exceed your physical limits, as Mr. Morreale discovered.

One day at the fulfillment center, he pushed himself too hard. Lightheaded and clammy, he sank to his knees, a no-no that Amazon’s performance algorithm treats as “time off task.” Associates aren’t allowed to sit down while on the job, unless it’s lunchtime or one of their 15-minute breaks.  

Amazon's solution to the relentless pace is to provide free aspirin vending machines, as well as free coffee machines. Caffeine and NSAID pain killer are supposed to be the solution for the bulk of associates, but they aren't.  Many, like Morreale, still break down and ultimately must leave.  

For the associates like Heike Geissler the only solution was to cut corners, a technique she described on page 105 of her book:

"You receive inbound units and enter them into the system.  You could hide products. Perhaps not forever, but at least for a few days you could hide products from others and thus remove them from the commodities cycle.  You could damage products and pretend they arrived already damaged. …"

And so on.  Anything  to slow down or halt the relentless, soul killing "commodities cycle" - which like the assembly line of old is remorseless in its mechanical demand for human compliance and attention.

Amazon, we learn, is trying to change its ways to lessen the injury rate - currently at 5.6 injuries per 100 workers.  To that end it's introduced a "Working Well" initiative.  But perhaps a more effective solution might be to be less demanding in terms of the physical demands.  An Amazon worker, for example, could be forgiven for being dubious after reading that "Current and former Amazon executives describe the management philosophy as performance drive and hard charging - built on the idea that everyone should be pushed to the limit.  And underperformer cut."

But workers, associates are not machines. They are not designed to work relentlessly at an inhuman rate for 10 hours - nonstop with only a 1/2 hr. and two 15 minute breaks.  And NO worker - male or female -  should have to wear Depends to get through a shift!

Thinking of working at Amazon? Be sure you have a solid health insurance policy as backup after you're cut for not making rate. With the relative higher risk of injury you will surely need it!

Tuesday, September 14, 2021

Solutions To Physics Problems (Set 4)

 1) Given we effectively have a flat rotating cylinder than it behooves us to make use of cylindrical coordinates.  The divergence is:

Ñ = ^h  / r + ^mr  ( / ¶f)   + ^k   / z

And we have:  r =   r^h  + z^k

(z = constant)

The velocity is:

v = dr/ dt =  r’  ^h + r  f’  ^m =  r w ^m

Ñ x  v =  (^h x ^mw  +   ^mr    ( / f) [r w ^m]

^k x ^m   (r w)/ z

But:  ( / f) ^m =   - ^h

So that:

Ñ x  v =    2 (^h x ^mw   =   2 w   (B)

2) Based on the presentation in (1) it must be in the +z direction (D)

3)  curl ix2 + jy2 + k (x2 – y2)

=   Ñ x x2 + j y2 + k (x2 – y2)

= (i x j) ( / x) y2  + (i x k)  / x (x2   +   y)

+ (j x i)   / y (x2 )     +  (j x k / y (x2   -   y)

+ (k x i / z (x2 )     +  (k x j / z (x2   -   y)

=   -2 ( iy + j x)  

4)    First, construct a sketch of the string, e.g. in u(= y) , x  coordinates as shown in order to do a brief analysis as shown:

We then have in terms of the string tension T:

 T u  = T sin q

As usual per these approaches, assume q is very small, in which limit, sin q  » tan q


T sin q = T tan q =  T  (  u/ x)

Take the force difference:  

[T u ] x + dx -   [T u ] x  =    / x (T  (  u/ x) dx

To ensure no net horizontal forces due to tension,  we limit the situation to small slopes. Also, neglect the possibility of a vertical force per unit length, so:

r dx ( u/  t2) =   / x (T  (  u/ x) dx

Which simplifies to:

r  ( u/  t2) = T  (  2 u/ x2 )

Now, transform to standard wave equation in 1-dimension:

2 u/ x2  -  1/ c2 ( u/  t2) = 0

From this we can solve for the velocity  c:

c= (T/ r ) ½

The boundary conditions can be written:

u(x, t) =  u(L, t) = 0


u(x,t) =  X(x) Y(t)

Then, on forming the particular 2nd order differential equation:

c2/ X  (d X/ dt 2  ) =   1/ Y  (d Y/ dt 2 )

From which:

1/ Y  (d Y/ dt 2 )  =  - w2

And:    c2/ X  (d X/ dt 2  ) =   - w2

The solutions are of the form:

X  =  C cos  wx/ c  + D sin   wx/ c 

Check boundary conditions:  X(0) = C = 0

And X(L) =  D sin   wL/ c  = 0

Then w will assume values:

w  =   n p c/ L   =   n p / L   [T/ r ] ½

The frequency is f =  w/  2p  =    n p / L   [T/ r ] ½ w/  2p 

Or f =  n/ 2L [T/ r ] ½

Note: Though in the actual GRE there will be multiple choice options, this is generally the type of problem you want to avoid because – in the absence of memory (say already knowing the wave equation for a string) and the useful DEs, the derivation is excessively long.  Bear in mind this is a timed test. So you have 3 hours (180 mins.) to get through 100 problems. That works out to roughly 1 m 48s per problem. That means if you need to send something like 8 minutes deriving the answer for one problem, you will lose valuable time for 4-5 others (even given 2 may be easy) .   If, however, you can make an educated guess, say based on a knowledge of dimensional analysis, then that might be worth a try!

5)    (E)

A rigid body may move in any of three directions. It may also rotate about any of three axes. Therefore, it must have six degrees of freedom.

6)     (A)

If a rigid body is constrained to rotate about a given axis it has only one rotational degree of freedom. It cannot have any translational freedom since this would change the axis of rotation.

7)    (B)

It has one degree of rotational freedom since the axis of rotation must be parallel to the plane. It can only have one degree of translational freedom to keep the disk perpendicular to the plane. Hence the total is 1 R + 1T = 2 DF.

8)  We have:  I = K x



(0….1… ..1)

We write out the determinant with eigenvalue l:

(1 - l….0…..0)

(0….1 - l   ..1)

(0….1… ..1 - l)

Leading to the characteristic equation:

(1 - l)3 – (1 - l) = 0


(1 - l) [ ((1 - l)2 – 1] = 0


(1 - l) (l2 –  2l) = 0

Yielding eigenvalues: l= 0, l = 2


T = Kl,  so T1 = 0, T2 = K, and T3 =2K

So (B) is the answer.

9)We will  have to take:  (– T1)C

So that:

K [(1….0…..0)


  [(0….1… ..1)

K (1….0…..0)](x)

  (0….1….. 0)] (y)

  (0….0… ..1)] (z)


 (0….0…..0) (x)

  (0….0…..1) (y)

  (0….1… ..0) (z)   =   0

So that:  0  =




Therefore the ans. is (A)  or i

 (Since the x-coordinate in the column vector is absent)

10) By the analog of the parallel axis theorem:

= IG -   M(R2 I – RR)

D I =  I -  IG   =    M(R2 I – RR)

RR =   r o 2     x



(0….1/2… ..1/2)


D I =   M  r o 2    x


  [(0….0… ..1)


  (0….1/2.. 1/2)]

  (0….1/2… ..1/2)]

= M  r o 2     x



(0….-1/2… ..1/2)

11) For a circular loop: B  =  mo I /  2r at the center of the loop, where  mo is the magnetic permeability of free space.

We have: r = 0.05 m, B = (0.70) (10 -4) Wb/m2

(Remember the conversion factor: 1G = 10 -4  Wb/m2   )

Then, solving for I: I =   2 Br/  mo

where  mo  = 4p  x 10 -7  H/m

I=   2 (0.070)  10 -4  A/  (4p  x 10 -7  H/m)   =   5.6 A

12) The work done in bringing the charge is equal  to the change in the electrostatic energy of the system. Let W1 and W2 be the electrostatic energy of the system without and with the presence of the dielectric, respectively.


W1  =  1/8 p   0  ¥    ×E  ( 4p r2 )  dr  =

1/8 p  ∫ 0  b   (q/ r2)2 ( 4p r2 )  dr    +

1/8 p  ∫ a  b   (q/ r2)2 ( 4p r2 )  dr    +

1/8 p   a  ¥   (q/ r2)2 ( 4p r2 )  dr   


W2 =   1/8 p   0  b   (q/ r2)2 ( 4p r2 )  dr    +

1/8 p   b  a  (kq/ r2)2 (q/ r2) ( 4p r2 )  dr    +

1/8 p  ∫ a  ¥   (q/ r2)2 ( 4p r2 )  dr   


Work done =   W2 – W1  =

1/8 p   b  a  (kq/ r2) (q/ r2) ( 4p r2 )  dr    

-  1/8 p   b  a   (q/ r2)2 ( 4p r2 )  dr


= (k – 1) q 2   [1/b  - 1/a]


13)   We have an ideal monatomic gas so:

U (internal energy) = 3/2 [n RT/2]

The change in internal energy is:

DU =  3 nR [ 373 K  - 273K] / 2

= 3 (8.32 J/K) (100K) /2=     1250 J


14)   The change in entropy of the gas, given

dS =  dQ/ T ,   is:

D S  =   273  373    nR dT/ T

= 5/2   nR ln (373 – 273)  = 6.56 J/K


15)   The f electron has ℓ =3  so that the total angular momentum quantum number possibilities are:

j = ℓ + ½,   ℓ - ½

Then: j = 7/2,  5/2