Early Quantum Theory

Prerequisite: physics, algebra2

"Quantum" is a scary word, but is really only means "quantity". The most significant discreet quantity is in Planck's quantum hypothesis for photon energy:

E = nhƒ
in which the integer n quantizes light such that you cannot have fractional photons.



The graph shows the wavelength of light bombarding blackbodies of certain temperatures, and the intensity of light radiated from the blackbody. Wien's Law tells what wavelength makes a blackbody radiate the most intense light, at a certain temperature T.

When light shines onto the curved metal plate, it emits electrons and causes a current to run through the setup. The voltage supplied to oppose this current is the stopping potential. This voltage and the electron's charge would be the electron's kinetic energy.



The classical understanding of radiation is that when sufficient energy builds up, a particle breaks free of the object. Intuitively, a metal would eventually radiate if a light is shined on it long enough. Experiments on this photoelectric effect show that the only way to get a metal to radiate is to shine a light of high enough frequency, no matter how intense or prolonged the light.

The work function is the energy required to remove an electron. In accordance with experimental results, there is a certain light frequency that must be achieved before an electron can escape.








There are four ways in which particles can interact:

1) photoelectric effect
2) electron excitation
3) Compton effect
4) pair production

This is pair production:



Light hits a nucleus and forms a positron and electron. Charge and momentum are conserved. Beautiful.

Electromagnetic Waves

Prerequisite: physics, algebra 2

Maxwell came up with a couple equations pertaining to light:

1) some kind of charge-field relation that involves calculus.. not in the scope of AP
2) magnetic fields have no beginnings or ends, whereas electric fields do
3) change in magnetic field generates an electric field
4) a current or change in electric field generates a magnetic field

An electromagnetic wave is essentially energy transmitted through fields. Change in magnetic field causes change in electric field by Faraday's Law, then the change in electric field causes a magnetic field by Ampere's Law, that may change the existing magnetic field or not.

I forgot to label the third picture; the vertical waves are electric fields and the horizontal waves are magnetic fields.

Since light rides on electromagnetic fields, it does not rely on a medium, thus explaining how it can travel through a vacuum. The constancy of the speed of light also relies on this fact; light is not something that you actively move, but rather it is something you send through a field.

If the speed of light is your cake, the derivation of Lorentz Transformation might be of interest.

Early Atomic Models

Prerequisite: physics, algebra2

First two equations are for the cathode ray experiment, which sets magnetic force equal to the centripetal force. The cathode rays then became known as electrons. The third is the oil drop experiment, in which the electron's electric force is equal to its weight. Simple and elegant.

Scientists then became interested in why elements only emit certain wavelengths. They came up with various mathematical models for the hydrogen atom, in which the Balmer series describes visible wavelengths, the Lyman series UV, and the Paschen series IR. All three combined shows the emission spectrum for hydrogen. Rydberg's constant R is 1.0974 E7 /m.

The orbit radius then became of interest. The second step of the derivation is electric and centripetal force. The Bohr radius is the inner orbit of a hydrogen atom.

The energy of light emitted is the energy it takes for an electron to jump levels. The quantum condition is interesting where the angular momentum L is.. quantized. The integer n refers to the orbit level, or principal quantum number. The most amazing thing is probably how the angular momentum reduces to an integer and some constants.

The total energy is the kinetic energy minus the potential energy. Substitute v from angular momentum and r from radius. The ground state of a hydrogen atom is 13.6eV. The second set of equations describe the wavelength associated to an electron skipping energy levels, as Balmer, Lyman, and Paschen had previously attempted.

To explain the quantized energy levels, the Broglie wavelength of an electron must fit the circumference of its orbit by an integer n. The equation can then derive angular momentum as in the quantum condition, thus verifying its legitimacy.

Except that physics likes to be counter intuitive. It continues to boggle our minds and turn our brains inside out..

Convergence

Prerequisite: calculus

Following Infinite Series, it is of interest to note the values of x that make the power series converge. This range of values is the interval of convergence. The radius of convergence is the value measuring from the center to either boundary. Usually the radius is one of three possibilities: center ± remainder, all real numbers, or center. Remainders are listed accordingly.

There are many ways to test for convergence, the simplest of which is the nth term test. The function a(n) must approach zero in order for the series to stop growing. The direct comparison test compares p(n) with another function of known result. It is almost a form of sandwich theorem. Note that p(n) must not have negative terms.



The ratio test sets two consecutive terms of p(n) as a ratio to show whether p(n) increases of decreases, which can then be used to conclude whether the series converges or not. Note again that p(n) must not have negative terms.



The absolute convergence test. Think about it.



The integral test sets the function p(n) as an improper integral. If the improper integral diverges then the series diverges (and p(n) must not have negative terms). The p series test is a strange one to conceptualize (has nothing to do with p(n)). A value of p = 1 makes the series grow incredibly slowly, and anything larger than one even by a bit makes the series decrease.



The limit comparison test is a combination of direct comparison test and ratio test.



When the series flips between addition and subtraction, it is an alternating series. The alternating series test looks for all three conditions in order for a series to be considered convergent. It is worth noting that if the alternating series does converge, the truncation error applies here as well.

There are two types of convergences. If you were to rearrange the order of the terms in an absolutely convergent series, it will not make a difference. However, the rearranging the terms in a conditionally convergent series will result in anything. You can make any sum with it~

Here is a nice systematic guide for testing convergence:

Testing convergence is like using a magical telescope to see infinity~

Infinite Series

Prerequisite: calculus

This is an infinite series:

[k, ∞] ∑a(k) = a1 + a2 + a3 .. + a∞

Sorry for the type. Here:

Here is a basic review of sequences and series. The sum of infinite geometric series will be incredibly important~



The bulk of investigating infinite series is about the power series. This is merely the sum of infinite geometric series, previously notated as [1, ∞] ∑ar^(n-1). Evaluating this sum gives c/(1-x), previously known as a/(1-r). You can have "x centered at a", notated as "x = a". The term "a" is essentially your x shift from the origin. We will come to understand the purpose later.



There are some fancy tricks you can do with the power series.. differentiate and integrate~ You can apply it to the notation itself, or expand a finite sum into its additions and differentiate / integrate term by term as you would a very long function.

The power series is all very well for polynomials and geometric sums, but for others we need to use the Taylor series:



This is merely the power series, having cn replaced with "the k order derivative of f(x) divided by k factorial". The purpose of all this can be better understood with a graph:

Say f(x) = e^x like in this graph. Pn(x) denotes an n degree Taylor polynomial, or a partial sum of f(x), or an approximation of f(x). An infinite sum gives you f(x), so the greater the n of Pn(x), the closer it approximates f(x). The series represented in this graph is centered at zero so that Pn(x) hugs f(x) at x = 0 (any series centered at zero is a Maclaurin series). It can be centered elsewhere to approximate some value far from the y axis. You might want to revisit power series with this in mind~

Since the polynomial Pn(x) is only a finite segment of the infinite function f(x), the excluded part makes the truncation error. The remainder Rn(x) of Pn(x) is the next order term. To put a number to it, the remainder estimation replaces f^(n+1)(c) with a larger version Mr^(n+1) that covers the interval [x, a].



Series and functions clash - funeries~

Magnetism

Prerequisite: physics

Magnets exert fields in a way that resembles electric fields:

One piece of magnet has little magnetic domains within itself, domains with more or less the same electron spin. A ferromagnetic metal such as iron, cobalt, and nickel have magnetic domains that do not cancel each other out, causing an uneven distribution of charge, and so it is a magnet. Hard magnets have more consistent magnetic domains, so they can be used to stroke other metals to make more magnets.

The qvB equation applies to a charge moving in a magnetic field. The BIL equation applies to a conductor carrying current, placed perpendicular to a magnetic field. We will only consider uniform magnetic fields.

In a qvB situation, the magnetic force can become a centripetal force when the magnetic field area is at least the size of the circle. In this example, the magnetic field goes into the page (and imagine that it is uniform all over the page).

This is the conventional notation for magnetic fields perpendicular to the page (me as in me reading this blog, and you as in you awkward blog).

A lot of perpendicularity is involved in magnetism. The first right hand rule applies to the BIL equation and the third to the qvB equation.

A magnetic field produced by a current involves the vacuum permeability constant µ0. The force exerted by one current carrying conductor on another depends on length, current and the other conductor's magnetic field. To know the direction of the force, use the second right hand rule to figure the direction of B between the conductors, then use the BIL right hand rule to figure the force.

One ampere is defined as the current in each wire that is one meter apart to produce 2E-7N for every meter of wire. Should not have used the equal sign there.

A solenoid is a coil of wire that makes a strong magnetic field inside when a current is applied. This makes an electromagnet, and an even stronger one with an iron rod inside.

The magnetic field of a solenoid can be calculated with this, where N is the number of loops.

Magnetic flux is the amount of theoretical magnetic field lines. The unit is weber (Wb), or (kgm^2)/(As^2). I am not sure what one Wb is supposed to mean, but it looks like kinetic energy over current to me.

If you are still wondering, magnetic field is in a sense the density of those lines. The unit is weber per meter square (Wb/m^2), or kg/(As^2), best left as Wb/m^2 for conceptualization.

Michael Faraday discovered that one can induce a current by changing the magnetic field on a conductor. This EMF can be described as ∆ϕ/∆t, yielding J/C, or volts. According to the magnetic flux equation there are several ways to induce EMF with magnetism:

1) change magnitude of magnetic field
2) change effective area of magnetic field
3) change angle of effective area
4) change effective magnetic flux on conductor (move)

Lenz's law is almost Newton's law for magnetic flux. It can be used to determine which way an induced current will run.

Below is just a nice thing to know: since changing magnetic flux induces a current, it induces an electric field to produce the current.

Huff, there. Strange stuff. Still getting my head around.

Laws of Thermodynamics

Follow up of Heat

Prerequisite: chemistry, physics

Less on temperature, more on energy. Less on chemistry, more on physics~ Here is something we can all agree on:

The zeroeth law of dynamics was a bit of a "duh" statement that scientists overlooked. Thermal equilibrium can also be understood with concentration and electrostatic analogies. Molecules at molecular equilibrium are at uniform concentration. Charges at electrostatic equilibrium are at uniform voltage. Likewise, heat at thermal equilibrium are at uniform temperature.

There again, internal energy:

I forgot to include that heat = Q. And for internal energy, we will focus more on the format ∆U = (3/2)nR∆T.

Here is your old grandma in a brand new dress. The Law of Conservation of Energy combined with ∆U = (3/2)nR∆T, PV = nRT, and PV graphs can tell us many things~

An isothermal process is when the temperature is kept constant. Since ∆T = 0, it follows that ∆U = 0 because ∆U = (3/2)nR∆T. The graphs looks like 1/x.

An isobaric process is when the pressure is kept constant. Since ∆P = 0, W = P∆V. It is more apparent when written as "W = (F/A)(A*∆x)", bearing in mind that W = F∆x.

An isovolumetric (sometimes isochloric) process is when the volume is kept constant. Since there is no ∆x, W = 0.

The adiabatic process is when no heat flows in or out of the system. Recall that heat Q is "energy transfer due to ∆T" so "∆Q" is quite redundant. Since Q = 0, it follows that ∆U = -W because ∆U = Q - W. Here is a better adiabatic graph:

So why is it that there is a change in temperature if Q = 0? Recall that temperature is the average kinetic energy of molecules, and in the case of adiabatic processes the change in this kinetic energy is due to work.

Nothing mind boggling here. Molecules do not diffuse to a higher concentration. Charges do not move to a higher voltage. There is actually a section on Gibbs free energy in AP chemistry, which in a sense calculates the spontaneity of a reaction, but that is beyond the scope of AP physics.

Efficiencies are disappointing enough, but thermodynamics takes that disappointment one step further. Even efficiencies are not completely efficient, if you compare calculations with Q and with T.

The biggest annoyance with this law is the fact that 100% efficiency is impossible. When you convert heat to kinetic energy there is always a residue of heat at equilibrium, as seen in QH --> QL + W. Heat at equilibrium cannot do work! If you turn QL back into QH with yet inefficient work, you end up with less QH than you started with.

1) QH --> QL + W
2) QL + W --> QH' + QL'
in which QH' < QH
and QL' > QL

One day when all kinetic energy is spent and all QH falls to equilibrium, we will be in heat death. Dun, dun dun, there goes your dramatic finale! Moral of the story: kids, do not waste your kinetic energy~

Heat

Follow up of Temperature and Kinetic Theory

Prerequisite: chemistry

Scored me a four in AP chemistry. The first thing that tripped me up was titration (but those were good days, joking about titrating ก๋วยเตี๋ยว). Tied at second place with solubility, was calorimetry. Now I wish someone had told me this:

Sure I knew what temperature was, otherwise I would not have even gotten a three. What I did not understand was heat. It is not a difficult concept but the teacher (and the textbook) probably took it for granted and only explained temperature.

I like to think that the relation of heat and temperature is similar to that of moles and concentration, or even charge and voltage. Molecules go from high to low concentration. Charges go from high to low voltage. Likewise, heat goes from high to low temperature.

Not sure why internal energy is introduced in this chapter. Will bring this up in the next~

Latent heat is just a fancy name for heat of phase change, whether it be freezing, melting, condensing, or evapourating. It is very important because it is actually rather easy to overlook in a calculation. At least for me. And we have our old friend specific heat down there, familiar and just as important.

Physics is obsessed with rates, so there:

AP is only likely to contain conduction. The rest, again, are nice to know.

Head on to the finale of the thermal physics trilogy.. Laws of Thermodynamics~

Temperature and Kinetic Theory

Prerequisite: chemistry

My regular physics class actually did not cover any thermal physics because most were covered in chemistry. Even so, my regular chemistry teacher was *cough cough* so I learned most stuff from AP chemistry. Here are the stuff I have left to gleam, which are mostly just equations. They are not particularly important in AP, but they are good to know~

When change in temperature causes change in length or volume. Can be written in the format on the right.

Thermal stress is a form of pressure. This is not likely to appear in AP as Young's modulus is not covered. Whatever that is. It has something to do with elasticity.

Here is another form of PV = nRT. Despite this, the classic nR form is still more common in AP than this Nk form.

I came across this vaguely in AP chemistry. Now it is a lot more clear. Geez, why did the textbook not put it this way:

where k is Boltzmann's constant.

More stuff. Nice to know, but not vital.

Also nice to know. The final answer can be in any (mass / time) unit depending on what you use in your substitution.

That symbol J.. physicists are really running out of alphabets to use! They finished the Greek letters, might as well use Chinese characters next. Sure, we have plenty of Chinese characters~

Next in the thermal physics trilogy is Heat