Some Notes on Feynman Vol I

This is not meant to be a summary of The Feynman Lectures on Physics Vol I but rather a collection of ideas that are useful to “know cold” as well as some cool, non-obvious ideas.

On Atoms:

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.

Some Fundamentals

Energy

$$J=N\cdot m=kg\cdot \frac{m}{s^2}\cdot m=kg\cdot \frac{m^2}{s^2}$$ $$J=\mathrm{Force}\cdot \mathrm{distance}$$

Energy is some quantity that we can’t quite see but only measure and that when we are operating under conditions of energy conservation, the scalar (aka number) we have for energy will always be the same.

Voltage

$$\mathrm{Voltage}=\frac{\mathrm{Energy}}{\mathrm{Charge}}=\frac{J}{C}$$

For energy then (of moving one charge $q$ from one end of the wire to another one).

To get the energy from voltage we multiply voltage times charge.

$$\mathrm{Energy}=qV=\cancel{C}\cdot \frac{J}{\cancel{C}}=J$$

• The higher the $V$ difference, the more work has done when the charge “falls” from the high potential end of the terminal to the low potential end

Current

$$I=A(\mathrm{ampere})=\frac{C}{s}=\frac{\mathrm{Charge}}{\mathrm{s}}$$

Current is analogous to velocity.

$$I=\frac{dq}{dt}\leftrightsquigarrow V_{velocity}=\frac{dx}{dt}$$

This is handy for making analogous computers which are computers that allow us to simulate problems in classical mechanics with circuits.

General Laws and Fields

Most laws like Newton’s law of graviation are given via a force:

$$F=G\frac{m_1{m}_2}{r^2}$$

This is very similar to the law for electrostatic force:

$$F=k_e\frac{q_1{q}_2}{r^2}$$

Now something useful for later is that a more elegant way of viewing these laws is through fields. At first, it was not immediately obvious to me why we should resort to a more mathematically abstract formulation to solve problems but the key is this:

We introduce the field so that we only need to look at one particle at a time. The field handles the bookkeeping of what the rest of the universe is doing. (paraphrased)

For the law of gravitation and electrostatic force, the fields would be described then as

$$\overrightarrow{g}(\overrightarrow{r})=-G\frac{M}{r^2}\hat{r}$$ $$\overrightarrow{E}(\overrightarrow{r})=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}\hat{r}$$

Which then leads the laws to take the form of

$$\overrightarrow{F}=m\overrightarrow{g}(\overrightarrow{r})$$ $$\overrightarrow{F}=q\overrightarrow{E}(\overrightarrow{r})$$

On Relativity

The minimum you need to know to go ahead and solve problems related to relativity is this:

For over 200 years the equations of motion enunciated by Newton were believed to describe nature correctly, and the first time that an error in these laws was discovered, the way to correct it was also discovered. Both the error and its correction were discovered by Einstein in 1905.

Newton’s Second Law, which we have expressed by the equation

$$F=d(mv)/dt,$$

was stated with the tacit assumption that $m$ is a constant, but we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula $m$ has the value

$$m=\frac{m_0}{\sqrt{1-v^2/c^2}},$$

where the “rest mass” $m_0$ represents the mass of a body that is not moving and $c$ is the speed of light, which is about $3\times {10}^5\mathrm{km}\cdot {\sec }^{-1}$ or about $186,000\mathrm{mi}\cdot {\sec }^{-1}$.

On Philosophy and Relativity

You have probably heard at some point in your life the expression “everything is relative.” You have also heard “Einstein proved everything is relative.” This is a very poisonous thought for which is a)uninsightful and b)wrong as it is related to the theory of relativity. As for a):

When this idea descended upon the world, it caused a great stir among philosophers, particularly the “cocktail-party philosophers,” who say, “Oh, it is very simple: Einstein’s theory says all is relative!” In fact, a surprisingly large number of philosophers, not only those found at cocktail parties (but rather than embarrass them, we shall just call them “cocktail-party philosophers”), will say, “That all is relative is a consequence of Einstein, and it has profound influences on our ideas.” In addition, they say “It has been demonstrated in physics that phenomena depend upon your frame of reference.” We hear that a great deal, but it is difficult to find out what it means. Probably the frames of reference that were originally referred to were the coordinate systems which we use in the analysis of the theory of relativity. So the fact that “things depend upon your frame of reference” is supposed to have had a profound effect on modern thought. One might well wonder why, because, after all, that things depend upon one’s point of view is so simple an idea that it certainly cannot have been necessary to go to all the trouble of the physical relativity theory in order to discover it. That what one sees depends upon his frame of reference is certainly known to anybody who walks around, because he sees an approaching pedestrian first from the front and then from the back; there is nothing deeper in most of the philosophy which is said to have come from the theory of relativity than the remark that “A person looks different from the front than from the back.” The old story about the elephant that several blind men describe in different ways is another example, perhaps, of the theory of relativity from the philosopher’s point of view.

Meaning we already know frame of reference matters and we did not need relativity for that. And even if frame of reference matters, there is still something such as truth.

Now as for b), the theory of relativity enables us to make definite preditions:

But certainly there must be deeper things in the theory of relativity than just this simple remark that “A person looks different from the front than from the back.” Of course relativity is deeper than this, because we can make definite predictions with it. It certainly would be rather remarkable if we could predict the behavior of nature from such a simple observation alone.

The idea of definite predictions does not match the sentiment evoked with “everything is relative.”

On Statistical Mechanics and Kinetic Theory

The key idea here is that you can’t track individual particles but you can still predict things. This is very useful then for when we are working with lumps of particles.

On Thermodynamics

Similar to statistical mechanics in that its ideas are based on thermal conditions but different in that it emphasizes information, or arrangement of things in space, rather than velocities or position of particles.

Thermodynamics was created by an engineer (Sadi Carnot) and it is very fittingly an engineer’s solution to abstracting away physical laws in order to solve problems. The beauty and elegance on Thermodynamics lies in that it allows you to make predicitions regardless of whatever is the material composition of your system. You will see here in Feynman’s summary of thermodynamics that at no point do we talk about what the system is actually composeed of.

Summary of the laws of thermodynamics

First law:

Heat put into a system + Work done on a system = Increase in internal energy of the system:

$$dQ+dW=dU$$

Second law:

A process whose only net result is to take heat from a reservoir and convert it to work is impossible.

No heat engine taking heat $Q_1$ from $T_1$ and delivering heat $Q_2$ at $T_2$ can do more work than a reversible engine, for which:

$$W=Q_1-Q_2=Q_1\left(\frac{T_1-T_2}{T_1}\right)$$

The entropy of a system is defined this way:

(a) If heat $\mathrm{\Delta }Q$ is added reversibly to a system at temperature $T$, the increase in entropy of the system is:

$$\mathrm{\Delta }S=\frac{\mathrm{\Delta }Q}{T}$$

(b) At $T=0$, $S=0$ (third law).

In a reversible change, the total entropy of all parts of the system (including reservoirs) does not change.

In irreversible change, the total entropy of the system always increases.

This is a homage to human thought.

Entropy

Based on the 2nd law we know it is impossible to extract the energy of heat at a single $T$ for an irreversible process. In layman terms, entropy alwasy increases when we do work (some energy will go toward increasing disorder).

The second law contains, deep, deep implications on technology and the universe broadly. It introudces the concept of irreversibility. According to Feynman, this has to do with the origin of the universe.

I am still in the process of articulating this idea but it is certainly clear to me that the 2nd law is a barbell that both engineers and physicts alike have to learn to “squat with” for making machines that work and figuring out the nature of the universe.

On Waves and Light

The equations here are generally useful to know and they enable us to create some very useful technology but the mathematical expressions are very speciffic to the phenomena they describe. This means that much of the “muscle” you train for these problems will not be transferable to other areas.