Problem Solving Strategies

John Rennie

06:01

@TheCuriousOne To do nodal analysis we look at each node in the circuit and the total current flowing into that node must be the same as the total current flowing out of the node.

If we do this with your circuit we get this:

If it isn't obvious how I got this we can go through it when you're around.

Ronald Becker

@JohnRennie hello sir

John Rennie

Hi :-)

Ronald Becker

Are you free right now?

John Rennie

Yes :-)

Ronald Becker

A rectangular block of glass is placed on a printed page laying on a horizontal surface. Find the minimum value of the refractive index of glass for which the letters on the page are not visible from any of the vertical faces of the block.

This question

John Rennie

06:17

I'm guessing it means something like this.

The red is the printed page and there is a block of glass on top of it,

and the light from the page gets totally internally reflected off the sides as shown by the blue line.

But I don't understand the question as there will always be some angle at which the light isn't reflected so the print will be visible.

Ronald Becker

06:35

@JohnRennie hmm me too i did not understand how can it reflect every light coming striking it's surface does minimum word help?

John Rennie

Unless there is more to the question than you have typed I don't see how it can be answered.

Ronald Becker

There only this much to the question I share ans if you want?

John Rennie

06:51

Yes, post the answer so I can see how it does the calculation.

Ronald Becker

07:20

sarthaks.com/490068/…

John Rennie

It isn't clear to me what the diagram used in that answer is showing ...

Ronald Becker

infinitylearn.com/surge/question/physics/…

John Rennie

Aha!

I think I get it.

Ronald Becker

@JohnRennie Sorry sir be I'll be back later are you going to be online at 9 IST ?

John Rennie

About 10 p.m.

Ronald Becker

07:33

OK I'll ask that time :)

Bumblebee

08:16

@JohnRennie Hello :)

I was hoping I could talk to you about cold fusion

John Rennie

Hi :-)

Cold fusion?

Bumblebee

Yes, the idea that fusion can be achieved at room temperature.

John Rennie

OK, what do you want to discuss?

Bumblebee

I wanted to know more about it in the context of the scientific method.

I suppose they hypothesised that fusion could be achieved at room temperature, and they set out to design some experiment to prove this was the case.

John Rennie

The problem is well understood.

The easiest way to do fusion is to fuse a deuterium and a tritium nucleus.

This forms a helium nucleus and a spare neutron, and the energy is released as the kinetic energy of the neutron.

The second easiest way is to fuse two deuterium nuclei to form ³He and again a spare neutron that carries away the energy.

Bumblebee

08:25

Wait, so has cold fusion been achieved?

John Rennie

Cold fusion hasn't been achieved. So far I'm just discussing the general requirements for fusion (of any type).

Bumblebee

Ah ok.

John Rennie

Where the temperature comes in is that both nuclei are positively charged and they repel each other, so it's hard to get them close enough to fuse.

To get the nuclei close together we have to fire them at each other with a very high velocity.

@Bumblebee OK so far?

Bumblebee

Ok, but how did Stanley Pons and Martin Fleischmann obtain the data which suggested that they achieved cold fusion? I mean, fusion usually occurs in the sun, and they are trying to achieve it within a test tube at room temperature which seems impossible

John Rennie

Their idea was that the metal they were using absorbs deuterium very well, and they though maybe the electrons in the metal would screen out some of the positive charge of the nuclei and reduce the repulsion between them.

If you can reduce the repulsion in this way maybe you can get the deuterium nuclei close enough to fuse even at low temperatures where the velocity of the nuclei is quite small.

They did observe energy coming from their system, but it looks as if what they measured was just chemical energy and not energy from fusion.

I suspect what happened was they had this great idea, and when they observed energy coming from their system they assumed their idea worked without checking it carefully.

I remember the excitement when they announced their results. I was still working as a scientist back then and lots of people I knew got very excited about it.

It's always a danger for scientists that you get so excited about your idea that you just assume it's true and don't do your experiments carefully. That's what happened to Pons and Fleischmann.

It happened again a few years back when scientists thought they had observed neutrinos travelling faster than light. But it turned out to be a fault with their equipment.

2011 OPERA faster-than-light neutrino anomaly

In 2011, the OPERA experiment mistakenly observed neutrinos appearing to travel faster than light. Even before the source of the error was discovered, the result was considered anomalous because speeds higher than that of light in vacuum are generally thought to violate special relativity, a cornerstone of the modern understanding of physics for over a century.OPERA scientists announced the results of the experiment in September 2011 with the stated intent of promoting further inquiry and debate. Later the team reported two flaws in their equipment set-up that had caused errors far outside their...

Bumblebee

08:40

Well i'm glad it's making a comeback

Being able to harness that kind of energy at room temperature would be increadible

John Rennie

The problem is that the repulsive electrostatic force is very, very strong at distances close enough for the strong force to take over and fuse the nuclei.

So far no-one has figured out how to get round that apart from using very high temperatures to give the nuclei a very high thermal velocity.

Every now and then people come up with a new idea for getting round the repulsion problem at low temperatures, but so far none of the ideas have worked.

John Rennie

09:03

deleted

John Rennie

12:13

@TheCuriousOne Hi :-)

TheCuriousOne

@JohnRennie Sir Could you explain why such a thing happen.(Your answer to this question asked may be “due to kirchkoff’s law”). But what actually happens. Also is it possible to make AB and BD lie in the same straight line. So The structure would be like this

John Rennie

We can go through how to do the problem if you want ...

TheCuriousOne

Sure Sir

John Rennie

We know there is some current I coming in to node A, and that current has to split into the two branches I₁ and I₂ as I've drawn. Yes?

TheCuriousOne

Yes Sir

John Rennie

12:21

Now suppose we follow the I₁ branch to B. Here the current I₁ has to split into two currents going to D and C. Yes?

TheCuriousOne

Yes

John Rennie

Let's call the BC current I₃. We don't know what I₃ is yet but we'll be able to find it later.

TheCuriousOne

Yes Sir

John Rennie

The BC and BD currents have to add up to I₁ so that means the BD current has to be I₁ - I₃. Yes?

TheCuriousOne

Yes Sir

John Rennie

12:24

Then at D the current I₁-I₃ splits again. Let's call the D to C current I₄.

TheCuriousOne

Yes Sir, I understand the currents. But while writing the equations, we get 4 right. So won’t it be easier if we associate this figure to something like a cube in the diagram which I drew?

John Rennie

The trouble is I don't see any way to make that work.
Maybe there is a way and I just don't see it.

TheCuriousOne

Sir so how do we logically understand why current doesn’t flow through BC or CD.

John Rennie

The only way I know of is to go though the nodal analysis and find al the currents.

TheCuriousOne

Logically (I mean without solving equations or math). I mean like associating this diagram to a whole diagram which would fill the symmetry like a cube. And somehow the symmetry maintains for this circuit as well

John Rennie

12:30

I cannot see a symmetry argument for it.

TheCuriousOne

Ok Sir thanks for the clarification

Sir also I plan to gain knowledge in relativity

like the basics. Is there any books you recommend for high school level

John Rennie

There are hundreds of books on relativity. I don't know which is best. You should just Google for best special relativity book or something like that.

TheCuriousOne

@JohnRennie Sir. Also there is something in electrodyanmics called the reciprocity theorem. Why is the theorem true? I remember seeing some vector calculus but couldn’t understand it

Sir, last question. When we find magnetic potential is it just like electric potential but instead of charges, we find for vector fields. Does this also stem due to Gauss law which is analogous to the work done by electric field being zero.

Harjot Dhillon

15:35

@JohnRennie hello sir

John Rennie

@HarjotDhillon Hi :-)

Harjot Dhillon

I was having problem in finding Voltage of parallel combination of batteries

John Rennie

Do you want to post the problem and I'll have a look.

Harjot Dhillon

Batteries have no internal resistance. According to equation I have to divide there emf with there resistance but how could we divide something with zero

John Rennie

The batteries are all the same, i.e. 5V, so the voltage across them is just 5V.

Harjot Dhillon

15:41

So current would not increase

John Rennie

Correct.

Harjot Dhillon

But I was unable to digest it

I have another problem with this equation of batteries in parallel

John Rennie

OK ... ?

Harjot Dhillon

I have to divide emf of each 5 V battery with there resistance that is 0 so I am unable to do that

Although I intuitively know that voltage would remain 5 volt no matter how many batteries we connect

TheCuriousOne

@JohnRennie Sir, Could you kindly address the questions?

John Rennie

15:48

@HarjotDhillon I can't think of an intuitive way to explain this.

Suppose you do a KVL loop round any pair of batteries.

No, wait, that isn't a good example.

Suppose you go from A to B. You could choose to go through any of the five batteries.

And since there are no resistances in between A and B the voltage must increase by 5V whichever way you go.
Yes?

Harjot Dhillon

I tried this way to get rid of zero in denominator

John Rennie

Well cancelling the zeros isn't mathematically well defined since 0/0 is undefined.

Suppose you put in some value of R, then look at the limit as R ⟶ 0

Kavin Ishwaran

@HarjotDhillon Are the batteries identical ?

Harjot Dhillon

Yes

Kavin Ishwaran

If yes, you could easily see that when n identical batteries are connected in parallel, the voltage is E (emf of one cell) regardless of what the resistance is.

17 mins ago, by Harjot Dhillon

Let's say the internal resistance is r (it may be zero but let's assume it is r), so E_eq/r_eq = nE/r (as batteries are identical) so E_eq = nE(r_eq)/r

but in parallel cells, internal resistance are also in parallel, that means r_eq = r/n, and substitute in the above equation.

Harjot Dhillon

16:06

@KavinIshwaran thanks

Kavin Ishwaran

you will see the r in numerator and denominator always cancels out, so when you take limit R ⟶ 0 the r and r cancels hence the limit value is just E

@JohnRennie Hi !

TheCuriousOne

So technically and unknowingly Harjot was correct

But that was the limiting case

John Rennie

@KavinIshwaran Hi :-)

I'm just starting lunch so I'll be half n hour or so.

Kavin Ishwaran

Oh Ok :-) Please ping me when you are free

TheCuriousOne

16:23

Sir, could you please clarify. When we find magnetic potential is it just like electric potential but instead of charges, we find for vector fields. Does this also stem due to Gauss law which is analogous to the work done by electric field being zero.

John Rennie

16:40

@KavinIshwaran I'm back! :-)

@TheCuriousOne I don't know much about electrodynamics I'm afraid. Sorry :-(

Kavin Ishwaran

@JohnRennie Hi !

In Heisenberg's uncertainty principle. what do we exactly mean by uncertainty ?

John Rennie

That's a good question but it's a complicated one.

Kavin Ishwaran

Oh

John Rennie

If we consider a classical particle then it has a precisely defined momentum i.e. p = mv.

OK so far?

Kavin Ishwaran

Yes

We have already discussed about HUP earlier (sometime ago) till how it comes from Matrix mechanics

John Rennie

16:45

But suppose we have two particles with slightly different momenta, or three particles, or more, and we mix up the particles in some vaguely defined way that means the mixture looks like a single particle.

Now we ask, what is the momentum of the mixture?

And the answer is that we can't say what the momentum is because it's a mixture of particles with different momenta.

Kavin Ishwaran

We would arrive at calculating average

John Rennie

Yes, we could calculate an average, and a standard deviation. Then we'd say the momentum is <p> ± σₚ where σₚ is the standard deviation.

OK so far?

Kavin Ishwaran

Yes

John Rennie

Well this is exactly what happens with quantum particles. They exist as a mixture of different momenta called a superposition.

That mixture has an average momentum <p> and a standard deviation σₚ.

And by now you've probably guessed what this has to do with the uncertainty principle ... yes?

Kavin Ishwaran

so the standard deviation is the uncertainty ?

John Rennie

16:49

Exactly!

In fact you'll sometimes see the HUP written as σₓ σₚ ≥ ℏ/2

where σₓ and σₚ are the standard deviations in the position and momentum respectively.

TheCuriousOne

Is this the same logic we use when determining the amount of charge when a current is moving, I.e using gauss law?

John Rennie

No, it is unrelated to classical electrodynamics. It is a purely quantum effect.

TheCuriousOne

No, I mean the average thing?

We consider the average of all coulombic forces of the moving charge on a sphere of charges surrounding it

John Rennie

No, the "average" we get in a superposition is unrelated to averaging any property of classical particles. The quantum average is called the expectation value if you want to Google it.

Kavin Ishwaran

@JohnRennie HUP is a result of treating particles as waves (wave nature of particles) ?

John Rennie

16:54

@KavinIshwaran It is a result of treating particles as a wave function but they do not have to be a wave as we usually understand it.

Kavin Ishwaran

I see

John Rennie

e.g. an electron in the hydrogen atom is not obviously a wave, but the HUP still applies to it.

Kavin Ishwaran

@JohnRennie But still an electron in H atom has some wave function associated with it which we call as orbitals right. So it is obvious that HUP applies to it ?

John Rennie

Yes

TheCuriousOne

Sir is it because of uncertainty principle that diffraction of light occurs more if we increase the magnification.

John Rennie

16:57

@TheCuriousOne That is a purely classical effect, though it turns out that the maths describing diffraction and the maths involved in the HUP are related.

TheCuriousOne

Could you explain?

John Rennie

No, it would take hours :-)

TheCuriousOne

any links?

John Rennie

There isn't a short cut to this. The only way you'll really understand it is to learn quantum mechanics. The problem is that it's based on ideas that are completely different from classical mechanics and those ideas seem very strange.

Kavin Ishwaran

@JohnRennie What made scientists to treat classical particles as waves

John Rennie

17:02

I think it was basically a guess. When quantum mechanics was first being discovered no-one knew anything about it and all sorts of ideas were tried out.

Treating particles as waves worked, so that's what was accepted.

Kavin Ishwaran

@JohnRennie I remember you saying particles are like wave-packets. Are those wavepackets solutions to schrodinger's equation ?

John Rennie

Yes

Kavin Ishwaran

So since at different points, like tangents, the wavelength differs from point to point and hence by de-broglie's idea p = h/lambda, there will be infinite wavelengths associated with a particle and hence we could find the average wavelength and the standard deviation of all those wavelengths is uncertainty ?

John Rennie

Yes.

The wavelength is related to the momentum by p = h/λ. Yes?

Kavin Ishwaran

Yes

John Rennie

17:08

So from the average wavelength we get <p> and from the standard deviation of the wavelength we get σₚ.

Kavin Ishwaran

@JohnRennie So HUP is a result of different ideas ?

John Rennie

It depends on what you mean by "different ideas".

The basic principle is that all properties of the particles are contained in the wave function.

So from the wave function we can calculate <p> and σₚ and we can also calculate <x> and σₓ.

Kavin Ishwaran

Like the concept of wave nature comes from de Broglie and concept of wavefunction comes from SE

John Rennie

de Broglie's idea came early on in the development of QM, and it's limited to infinite plane waves, which is a special case of the wave function for a free particle.

Only for free particles do we get p = h/λ.

Kavin Ishwaran

Yes

John Rennie

17:12

e.g. you cannot apply it to the electron in a hydrogen atom.

So it's a special case that only applies to free particles, while the SE applies to all quantum systems including the free particle.

Kavin Ishwaran

Yes

John Rennie

I'm going to drop out now. I'll be back tomorrow as usual.

Kavin Ishwaran

@JohnRennie A particle has a definite dimensions, but the wavefunction says the particle would be spread out over an area which gives uncertainty in position. But when we make an observation of particle, why we could never find the position with no uncertianty ?

@JohnRennie Oh Ok :-)

John Rennie

Bye :-)

Kavin Ishwaran

Bye :-)

Transcript for

Problem Solving Strategies