@JohnRennie ^ please have a look at this whenever you are free.

@AshishAhuja If d is the distance from the optic axis and R is the radius of curvature of the lens then the spherical aberration scales approximately as d/R.

The objective has a very short focal length i.e. a very small radius R, so to keep the level of aberration constant we need to keep d small as well.

@JohnRennie "If d is the distance from the optic axis" distance of what from the optical axis?

Of a light ray entering the lens

@JohnRennie ok, so I still don't understand why it wouldn't be beneficial to make the aperture of the eyepiece smaller, since that should reduce the spherical aberrations produced by the eyepiece?

The eyepiece has a much longer focal length so it has a much larger radius of curvature. That means spherical aberration is less of a problem. It's true that reducing the aperture makes all lenses better, but it also reduces the light intensity so there is a tradeoff.

To get the brightest image you want the aperture to be as big as possible but without causing excessive spherical aberration.

ah yes okay, that makes sense.

also, one more thing:

I can't remember exactly how a compound microscope works. I think the objective forms a real image near the eyepiece and the eyepiece then uses it as a virtual object.

Give me a moment and I'll Google it.

sorry I'll just post the deleted message again: I have used a compound microscope in my school biology lab where we had to focus to see the cell wall on a plant leaf or something like that. In that you could see the clearest image in only one position. Am I correct in saying that this position is when the object is placed at a distance of just over the focal length of the objective away from the objective?

For the objective v >> u, so yes u will be only slightly greater than f i.e. the point of sharpest focus would be slightly greater than f.

ok, thanks.

That diagram is slightly misleading as I think they've made u only slightly less than v for clarity. In fact u << v.

@JohnRennie Hi !, Can you give an example where mass is converted into energy ??

Nuclear bomb! :-)

For a simpler example how about decay of a neutron to a proton and electron and neutrino.

This is where my doubt arises, So after nuclear fission Still there is same mass ryt ? Like Uranium is bombarded and it transforms into other atoms.. still the mass are conserved, So the mass is not directly converted into energy instead the energy is released when we break the bonds between them ?

Before the decay the KE is zero because we just have a stationary neutron.

@KavinIshwaran No, the mass decreases in an atomic bomb.

But let's look at the neutron decay as that's a nice simple example.

@JohnRennie ahh yes, this is what i need

@JohnRennie yes

But after the decay the proton, electron and neutrino are all flying apart away from the position of the neutron. So now the KE > 0 i.e. kinetic energy has appeared from nowhere. OK so far?

yes

Now, if you add up the masses of the proton and electron (the neutrino mass is so small we can ignore it) you find the sum is less than the mass of the original neutron.

What has happened is that some of the mass has been converted into the KE of the decay products.

Got it !

So an example for Energy to mass ?

Discovering the Higgs boson at the Large Hadron Collider.

at CERN ?

Yes. The LHC collides two protons. Each proton has a mass of about 1GeV so the total mass is 2GeV. But the collision can create a Higgs boson with a mass of 125GeV.

The extra 123GeV of mass comes from the kinetic energy of the colliding protons.

I heard that the protons collide at a speed 99% of C

@JohnRennie Ohh

It's more like 99.999% c :-)

If you're interested I can explain how this works ...

@JohnRennie Yes pls

I need to make an analogy. This is going to sound a bit odd at first, but you need to go with me.

Suppose you pluck a guitar string so it vibrates.

If you pluck it gently it vibrates a little, while if you pluck it hard it vibrates a lot. OK so far?

yes

Now suppose this is a quantum string. What this means is that the amplitude of the vibration cannot vary continuously.

If you pluck very gently nothing happens, then as you pluck harder and harder suddenly the string starts vibrating with some amplitude A.

Now as you keep plucking harder the string carries on at amplitude A until suddenly it starts vibrating with amplitude 2A.

The idea is that while a regular string can have any amplitude a quantum string can only have amplitudes A, 2A, 3A and so on.

Kind of like the way a hydrogen atom can only have energy levels 1s, 2s, 2p, etc.

Does this make sense so far?

Yes

Now the bit that's a bit hard to understand is that the energy of the vibrating string behaves a bit like a particle.

@JohnRennie Duality ?

When we pluck the string hard enough to create a vibration A it is equivalent to creating a particle.

@JohnRennie interesting

@KavinIshwaran It is kind of related to the wave particle duality, yes.

If pluck it even harder to get an amplitude 2A this is like creating two particles. 3A is three particles, and so on.

But there is more to this!

Do you know about harmonics on a string?

This:

@JohnRennie Rhythm ?

See the link I posted.

@JohnRennie oh yes I am going through it

You don't need to read the whole thing. You just need to know that the string can have the different vibrational modes shown in this diagram:

Each harmonic behaves like a different particle.

ohhh

Or more precisely each harmonic behaves like a particle with a different momentum and energy.

These things also apply to light ?

Now the point of all this is that these "particles" can easily be created an destroyed by adding energy to the string by plucking it or taking energy away from the string e.g. by touching the string to damp it. Yes?

yes

Now, in physics we describe particles using a theory called quantum field theory. This says all of space is filled with a field called a quantum field that behaves kind of like a 3D vibrating string.

For example there is an electron field, and the electrons are just vibrations in this electron quantum field, in the same way that we talked about vibrations of a string being particles.

sounds like all particles vibrate in that field ?

Likewise photons are vibrations in a photon quantum field, quarks are vibrations in a quark quantum field and so on for every type of particle.

@KavinIshwaran No, there is a different field for each type of particle.

@JohnRennie ohh

But now you can see how we can collide two protons and create a Higgs boson. Protons are made up from quarks, so we are actually colliding quarks.

yes

What happens is we start with two quarks that are vibrations in the quark field. These have a lot of energy because they are moving at 99.999% c.

The collision transfers the vibrational energy from the quark field to the Higgs field.

So what happens is the two quarks disappear because their energy was removed, then the Higgs boson appears because that energy has gone into the Higgs field.

Energy is conserved - it has just been moved around between different quantum fields.

interesting

Thank you Sir for sparing some time ! :)

The theory is muuuuuuuuuuuuuuch more complicated than I made it sound. It's about the hardest type of theoretical physics there is.

@JohnRennie I could hear :)

And... can "theory of everything" exist ?

As universe has a symmetry.. it is most likely to exist ?

since everything you explained now has a link to everything ?

Phone's ringing ...

@KavinIshwaran Hi, sorry about. I'm back now.

no problem !

The idea of a "theory of everything" is that all these different fields - electron field, quark field, photon field - are actually the same field and it just looks like it is built up of different fields.

So there is just one field and all the different kinds of particles are different types of vibrations in this single field.

like different magnitude of behavior ?

No-one knows how this would work. There are lots of ideas how it might work, but so far none of these ideas have worked out.

oh

You've probably heard of string theory, and this is yet another idea of how all the different types of particles could be combined into a single field. But again no-one has been able to make the idea work.

So right now we simply don't know if a unified theory exists.

@JohnRennie I heard that the idea of "String theory" only enables to include a particle graviton ?

@JohnRennie oh....

If string theory could be made to work it would include all the known particles like the electron, quark, photon, etc plus it would also include the graviton.

But I must emphasise that so far no-one has been able to make it work this way.

ohoh

That doesn't mean string theory is wrong because no-one has been able to prove it wrong either. At the moment we don't know if string theory could be made to work or if it's plain wrong.

if someone made string theory work.. will it affect any laws of physics ?

We simply don't know ...

Thank you for all the explanations !

You're welcome. This sort of thing is what got me interested in physics.

catching me too !

@JohnRennie This is the current-voltage characteristic for a zener diode; for voltage regulation a zener diode in reverse-bias is used, but why can't it simply be used in forward bias? The forward bias graph is just as steep as the reverse bias graph.

Weight of pio

Forgive the author's handwriting. But in solving this problem. They have drawn a free body diagram of the forces acting on the boy. They've labelled the angle the gravity force makes with the direction parallel to the surface as $\theta$

As you can see in the second image, the author has also labeled $\theta$ as the angle the boys position makes with the vertical in the circle. How does the author deduce that these two angles are equal? I.e. the gravity angle and the arc measure from the highest point on the circle