Did you know that physics involves the study of crystals? From radios, diodes, metals, medical technology, smart phones, and more; crystals have been vital to the development of our modern world.

Well before quantum mechanics had first been discovered people had already began to understand some of the fascinating effects of electricity and magnetism through the use of crystals. In 1861 Maxwell had formulated a series of equations that defined how electricity and magnetism could be linked to one another using Faraday’s observations, whom in 1845 had already discovered some of the first evidence that light was itself an electro-magnetic effect. All of this had already happened when Hall found himself busy in 1879 measuring magnetic forces on charges in wires and attempting to describe how these effects changed the resistance of wires. These observations and early electrical experiments that scientists completed in an attempt to describe the electro-magnetic phenomena of the world around them were both astounding and eye opening to a world never before seen by humans. And yet almost none of their observations had even begun to construct a theory of how the materials themselves could possibly allow these phenomena.

A particular discovery that led to one of the first working theories behind why metals were able to conduct was spurred off by J.J. Thompson’s discovery of the electron in 1897. Metals are themselves a kind of crystal with many free electrons that when allowed to coalesce into a larger piece of metal, also known as a bulk material, are able to have these electrons flow through them as is the case for copper wire. Paul Drude first used that discovery in 1900 in conjunction with the highly successful kinetic theory of gases to formulate a working model of conduction of heat and electricity in metals. He formulated four basic assumptions that gave an insight into some of the quantitative and qualitative properties behind bulk metals. Those assumptions are:

1. The electrons experience no interactions between collisions*
2. Collisions are instantaneous
3. Any electron picked at random will experience a relaxation time such that this will be the probabilistic time between electron-ion collisions.
4. Electrons only achieve thermal equilibrium with their surroundings through collisions

*(external fields will cause effects on the electron gas, but no resulting forces cause any interactions i.e. an external electric field would interact with the charged particles, but the movement of the electrons within the system does not cause interactions for other electrons)

Image Source: Wikimedia Commons

Description: The above picture represents the collisions that electrons (tiny blue dots) would be undergoing with the ions (larger red dots). It should be noted that this is absolutely NOT how electrons actually travel in metals. This is only a model that was relatively successful in a pre-quantum mechanical physics world view.

However, Drude’s model is absolutely flawed. In his time, how light was emitted and absorbed by atoms had not yet been understood. The quantization of energy levels for electron orbitals and the inability for electrons to exist between these energy barriers had not been discovered. However, even without this knowledge of the world he was able to apply previously successful ideas and gather some accurate information about the magnificent world of crystals.

To me, this is important. Physics is much more a way of critical analysis and educated guesses about the world than it is applying a kind of absolute knowledge of what we already know. It is important to understand that in the face of the unknown physicists and many other professions do not simply stop and find themselves lost. The art of discovering the physical properties of the universe that is about us is as much an applied philosophical quest as it is one that stands within the art and creativity of the minds that pursue it. Developing a wide array of models and testing these ideas is vital to discovering new and never before seen properties about the unknown.

Over the next few days we shall explore the details of this model and how its early successes helped give early scientists a means by which we could begin to describe some of the different effects going on within the different metals. The assumptions by Drude are wildly inaccurate, however they have a wide array of successes that come with their overall failure.

Also, I am going to make so many more puns.

Confused about what’s going on here? Great! Send your questions and comments to (JoshingWithPhysics@gmail.com) or respond in the comments section below! I want to make sure that you are able to get a detailed and clear understanding of this fascinating world! Your feedback is absolutely vital.

Best wishes,

Josh Lofy

Have you already read the recent Joshing With Physics post “Hooke’d On Physics“? I left an open ended question that asked how could the mass in Hooke’s law be negative, as shown here, ? is the spring constant, is the distance traveled, and is the acceleration experienced (in this case, gravity).

This question arose from the presentation of Hooke’s law as . Hooke’s law was represented this way because if you were to pull the spring in any direction it would pull in a negative direction .

We set Hooke’s law equal to Newton’s force of . This representation can be shown as and simplified to the previous expression . However, the mass cannot be negative! So what is our mysterious negative entity here?

Image Source: Wikimedia Commons

More puns coming your way soon!

It’s been a brief amount of time since the last post of Joshing with Physics, so to bounce on back let’s talk about some of the simple physics related to a really amazing classical object, the spring!

Hooke’s law is used to determine the force that a spring will exert when pushed or pulled from equilibrium. It is a very simple relationship that surprisingly has no mass dependence!

Hooke’s Law

Here F is the force, k is the constant of the spring, and x is the distance moved from equilibrium.

Let us be careful to define the two variables mentioned above that are equal to our force. The spring constant () is how much force the spring will exert for every unit of distance () we have pulled the spring away from its equilibrium point. The equilibrium point is where the spring would sit while at rest, without any force pushing the spring together or pulling the spring apart.

Let us also consider why Hooke’s law is negative. The force is negative in this case because if we push the spring together the spring will try to return to the equilibrium point and push outward. In this case we have defined the direction inwards of the spring as being the negative direction, so the spring will push in the positive direction and give us a force in the direction! Alternatively, if we were to pull the spring apart the spring would prefer to pull itself back together to return to its equilibrium point. In this case we have set the system up to pull with a force in the direction!

But what can Hooke’s force law be used for?

Any force can be defined as a change in momentum, or a preferred change in the movement of an object. Forces can be defined in a lot of different ways such as and many more! Here m is mass, a is acceleration, p is momentum, t is time, U is potential energy, and r is distance. In our case, Hooke’s law explicitly applies to springs but it is equivalent to all of the other things that a force can be equivalent to. There are so many different things that forces can be related to! Specifically this means that it makes as much sense to say that ! But wait. Didn’t I say that Hooke’s law was not mass dependent? It isn’t! But you can use it to solve for mass if you know the acceleration of a system.

This applies in the case of a spring hanging vertically from a ceiling with a mass on the end. If we know the spring constant of a system (), how much the spring has stretched from its equilibrium point (), and the acceleration (gravity) of our local environment (), then we can solve for the mass! ()

Does it make sense that our mass is negative? I wouldn’t think so! What value do you think is negative so that our mass comes out as a positive value? Let me know in the comments below!

Image source: Wikimedia Commons

Hooke’s law is a great and simple way to show a lot of different aspects of physics and gives us some of the rudimentary concepts to begin to talk about things like the bonds between different atoms, as is the case for a 1-D infinite crystalline chain of atoms bound together! Interested in hearing more? Help me continue to write by making a donation below or sending me a happy message! You can contact me by sending an email to (JoshingWithPhysics@gmail.com).

Best wishes,

Josh Lofy

Infinity is certainly a strange mathematical concept.  It is not like many of the numbers that people are used to dealing with on a day to day basis.  However, it is extremely helpful to have as a tool for the average physicist.  Today I want to pass a math joke off to you that will help you understand some of the most basic of infinite series.

First off, what is an infinite series?

Let’s say I add 1 to 1 one time.  This is represented as 1+1 and is equal to 2.  If I add 1 to 1 two times then I can say 1+1+1=3.  If I add 1 to 1 an infinite amount of times I get 1+1+1+…= ? because this series does not converge!  There are several tests in calculus and other kinds of mathematics to help thinkers and scientists of all types discern if different kinds of series and additions converge.  Rather than get bogged down in the convergence of this moment and talking about how these things can be proven let us take a look instead at an example of a series that does converge.

If I add + I will get .  This can be shown as + = + =  = .  If I add  to that, or half of what I added the last time [* = = ], we will see that  + =  + = = .  So exciting!  Now let’s do this again, and this time see if you notice the pattern.   +  = +  = = .  If you haven’t spotted the pattern yet and want to do so on your own, try to add the next value () and see if you spot it!  Spoilers are in the next paragraph.

As we continue to do this an infinite amount of times each addition increases the denominator, the number in the bottom of the fraction, by a power of 2 ( … ).  However the numerator, or the upper number of the fraction, is always the denominator minus 1.  This means as we do this for the n = 1000000 th = time the numerator will be while the denominator is simply . Do this an infinite amount of times and .  This is as close to the value of 1 as one can truly get.

Now for a math joke that you can share yourself!

A mathematician walks into a bar.  They order a beer.  They are followed by a second mathematician who orders of a beer who is followed by another mathematician who orders half of the last mathematicians order ( of a beer) who is followed by an infinite amount of mathematicians who order half of the last order before them.  The bartender gets flustered and pours two drinks.

If you are looking for me to Taylor more series for your calculus needs I would be infinitely delighted to continue this line of thinking!  Look for future posts regarding reflections, absorptions, and polarization in the upcoming weeks for how this concept can be related to different aspects of physics!

Best,

Josh

I personally love the images I see from Hubble and other deep viewing space based telescopes.  Looking that deep into space takes my breath away in the beauty of the stars, galaxies, and other cosmic objects that bless those images.  Taking images that deep of objects that are that far away shows one way in which space is strongly connected to time.  When you look at something that is hundreds, thousands, millions, or even billions of light years away you are looking through both the space and time of the universe and seeing it as it was at that time in that space.  The astronomer from the early 1800’s, William Herschel, is credited in the new Cosmos series as noting that the sky is filled with these kinds of ghosts.  Images of objects that once were and in some cases still might be.  With the new views from our modern telescopes this is easier to see than ever.

However, if it is indeed assumed that there are stars and galaxies scattered in an almost random way across the entire night sky and the universe is indeed a finite place then it would be reasonable to assume that the night sky should be filled with lights coming from all sorts of celestial objects coming from all directions!  So why then is it not the case that the night sky is bright?

Even with the spaces and distances we are considering, the accumulation of all of the light from all of the universe seems like it should in some way illuminate our night sky.  However, there is one assumption in my current line of reasoning that has managed to sneak its way in without our noticing.  We have assumed a static and non-expanding universe.

A simple explanation can be shown for yourself through the expansion of a balloon.  Start by placing several dots at random distances and positions around a partially inflated balloon.  You can place them close together, far away from one another, or equidistant from dot to dot.  Next, continue to inflate the balloon.  Notice how the distance changes between objects that are near one another.  At first the distance changes slowly, then as the inflation continues the distance changes at a faster rate.  For dots further apart the initial expansion will be much quicker and continue at an ever increasing rate!  That is until the balloon pops.

So what did we observe?

In an expanding universe some positions in space travel further apart from one another at different rates.  This expansion is related to the lack of our bright sky, although it is important to note that this is not the sole reason for the lack of our bright sky.

Looking to be more illuminated on this topic?  Look for follow up posts coming later in the week as I continue to talk on the implications of an expanding universe and even give a solution for stellar density that would result in a bright sky at all periods of all days!

Best,

Josh

Black holes are regions where there is so much mass that the gravitational forces are too powerful for even light to escape.  With the recent combined observations of gravity waves and light from the combination of two neutron stars we now have physical evidence of the likely creation of a black hole.  For it to be formed two very hot and very dense objects merged, disturbing the space about them.  That gravitational radiation has been detected by several of the LIGO instruments here on Earth.  In the case of the kilonova, the popular name for the combining of these two stars, the observation was able to be followed up from several observatories all around the planet that looked at the massive emission of light from many different spots of the spectrum.  This large amount of energy released was in part due to the absurdly high temperatures of the massive event!

So why are black holes so unbelievably cold?

A simple explanation can be seen around some of the same concepts that you run into every day in your kitchen.  When you turn on the stove and place a pan on top of it you can begin to feel the heat radiating off of the pan.  This is helpful in the context of cooking.  It allows for heat to be transferred from your heating element (natural gas, electric burners, magnetic induction, etc.) to your food!  Heat travels from hot things to colder things.

This can also be felt when you go to grab your favorite frozen treats out of the freezer.  The feeling of cold is the transferring of heat from yourself to your frozen delicacies.  In a very simplified way, this is what a black hole is doing.  It is absorbing heat from the surrounding environment because it is much colder than it, allowing the energy to be passed into black hole and preventing it to be radiated to the much hotter empty space that surrounds it.

But how do we know that the surrounding space is hotter than black holes?  Also, what evidence do we have that black holes are cold?

I will be continuing this explanation over the next week with more details relating to the microwave background radiation and thermodynamics use of entropy with references to Stephen Hawking’s work.  For now you’ll just have to find a way to chill out.

Best,

Josh