A good intro blurb, comparing the gravitational interaction,

$$\vec{F}\left(r\right)=-\frac{G m_1 m_2}{r^2}\hat{r}$$

to the Coulomb interaction,

$$\vec{F}\left(r\right)=\frac{k q_1 q_2}{r^2}\hat{r}$$

in which \(\hat{r}\) points from object #1 toward object #2.

Reminiscent of the "Tokamak" structure of ITER.

Be sure to read through this section on torque.

Not quite.

If \(\theta=0\), then the field point \(P\) is directly ahead of the wire element \(d\vec{\ell}\).

That is, \(d\vec{\ell}\parallel\hat{r}\), because \(\hat{r}\) points from \(d\vec{\ell}\) to \(P\).

There is a set of locations \({P_i}\) in spacetime where the field \(\vec{B}\) is absent. For the electric field, you could say that for MANY sets of locations, like the midpoint between two like charges, a ++ array or a -- array.

Decent description of using right hand rule for the central \(\vec{B}\) field direction

Angular velocity \(\omega=\frac{d\theta}{dt}\) is like the rate of revolutions per minute, RPM, except in physics we generally do not work in units of whole revolutions and minutes. but in radians of arclength per second. I.e., \(\left[\omega\right]=\text{seconds}\)

An angular velocity like this will make sense for any pixel of an extended object, at any distance away from the center or rotation or spin.

An example of this is a driver making a turn onto the interstate (so a bit of centripetal acc.) and speeding up at the same time (a few \(m/s^2\) of tangential acc.), speeding up to merge with the flow of traffic.

E.g., a spring laid out horizontally on a smooth tabletop, so smooth we can neglect friction for small amplitude experiments. The other forces are downward weight force,

$$\vec{W}=\left(mg\right)\hat{j},\quad g=-9.8\frac{m}{s^2}$$,

and the upward normal force \(\vec{N}\) from the rigidity of the surface, which exactly balances the weight force.

An example of a "more complicates" force is the restoring force of a spring on a object free to move along some axis, like an x-axis on a frictionless tabletop.

In that case, \(\vec{F}\left(x\right)=\left(-kx\right)\hat{i}\)... an interesting work integral.

Good. "rapidly propagating change in the electrical field" means $$\frac{d\vec{E}\left(x,\,t\right)}{dt}\gg 0$$ which depends on density of the conduction electrons in the conductor, the temperature, the voltage gradient and a few other factors.

Drift velocity is quite slow.

Although, it may seem that the lights come on when you click the switch, the "signal" does not move instantaneously!

Idealized view, but useful for practice.

A good example to look at, because we will be studying the electrostatic field between two parallel plates, one positively charge and the other negatively charged. This is the famous device known as a capacitor.

Equilibrium, no accelerations, is the key concept in this section.

Mislabeled in Fig. 6.39. The figure's cylinder has height marked as \(\sigma\) but should be \(\delta\).

This is a nice synopsis.

Integration at its most basic is adding up a bunch of tiny areas shaped like trapezoids, flat bottom and parallel vertical sides \(\perp\) bottom, with slanted top determined by the derivative of the integrand, i.e., of \(f^{\prime}\left(x\right)\) when integrating \(\int\, f\left(x\right)dx\).

But Gauss' Law simplifies that tedious, difficult process by taking advantage of symmetry in the source array.

Anytime you use the word "dynamics," you imply that forces are under consideration in some manner.

This is a good mini-workout, and the solution is given. Note the introduction of unit vector \(\hat{u}\).

It is so much more powerful than the energy we can easily get from oil, coal or natural gas!

Author means \(t_2\)

Unfortunately, the radians column is messed up, too tiny. $$30^{\circ}\longleftrightarrow\frac{\pi}{6}$$ $$45^{\circ}\longleftrightarrow\frac{\pi}{4}$$ $$60^{\circ}\longleftrightarrow\frac{\pi}{3}$$ $$90^{\circ}\longleftrightarrow\frac{\pi}{2}$$ $$120^{\circ}\longleftrightarrow\frac{2\pi}{3}$$ $$135^{\circ}\longleftrightarrow\frac{3\pi}{4}$$

Skim in this chapter for lecture on Friday, Feb. 20th...

This is essentially a traffic light problem.

I.e., there's no such thing as horizontal gravity.

Be careful. This is true for vectors

$$\vec{H}-\vec{P}=\left(-\vec{P}\right)+\vec{H}=\vec{H}+\left(-\vec{P}\right)$$

but not exactly the same for numbers: $$7-2.2\ne2.2-7$$

A good example is the solar system's comets. We took centuries of observation to get a rough idea of what they were. But in the 20th century and this century we have observed and seen the mathematical patterns of a few large groups of comets. Each group has orbital or compositional properties that vary significantly from other comet groups.

...headlined by Copernicus, Kepler and Galileo!

Those extra circles are known as epicycles. Cf., Ch. 2.4

It is interesting that Galileo, our hero from the origin of modern science, figured out that Earth definitely orbits the Sun, and, through observation, many other astronomical facts. But he did make a few errors. For instance, he initially thought that comets were objects high up in our atmosphere!

This is the most abused phrase in all of science. It is sometimes known as "waving your hands," and it is usually a subsitute for doing a big pile of calculus, in this case, three-dimensional calculus. So for PHY2054, we will let discretion be the proverbial better part of valor and accept this important result. Why is it important? It is important because we now have an oscillator system!

As I have emphasized!

Flux is a fancy way of counting field lines cutting through a given area.

Very important, especially when we get to the topic of flux.

Not electromagnetic radiation, per se, but charged particles, the protons, electrons and other ions that get lofted from the surface of the Sun, sometimes explosively so.

We will tackle this calculus problem using a spreadsheet method, as we have before!

A handy analogy. Think of resistance like a load of gravel in a water pipe. Water flows, but not so plentifully as in an empty pipe.

Here is the answer to the riddle, drift speed very small, but lights go on almost instantly.

Here is the link,

https://phet.colorado.edu/sims/html/capacitor-lab-basics/latest/capacitor-lab-basics_all.html

I,=.e., a good solvent

A nice workout.

There seem to be some typos in the equations below this point. I will try to mend them.

Yes, we discussed the Feynman diagram for this process in lecture on 6/23/25.

Similar to $$y\left(t\right)=y\left(0\right)+v_y\left(0\right)t+\tfrac{1}{2}gt^2$$

This is basically a "sideways" gravitational system, \(U_1\left(x\right)=\kappa\,x\) as compared to regular gravitational potential energy, \(U\left(y\right)=\gamma\,y\).

$$\kappa=-E/x_0$$ and $$\gamma=-mg$$

If horizontal gravity DID exist, it would be something like \(U_1\)

Similar to my notation, \(d\vec{s}\).

ambiguous :(

That is a method of pain.

It is better just to know that 1. if \(F\left(x\right)=m\ddot{x}\left(t\right)=-kx\), then \(x\left(t\right)\) is sinusoidal in time, \(x\left(t\right)=A\sin\left(\omega t\right)\) or \(A\cos\left(\omega t\right)\) 2. alternatively, if \(U\left(x\right)=\tfrac{1}{2}kx^2\text{ then }F\left(x\right)=-\frac{d}{dx}U\left(x\right)=-kx\), so \(x\left(t\right)\) is sinusoidal etc.

Then go from there.

One of the most important examples in the entire book.

Even though it is not exactly a \(\tfrac{1}{2}kx^2\) potential.

Sometimes known as a "sombrero potential" in quantum field theory.

Especially when thinking about 1. oscillators and 2. various field like the electromagnetic field.

A famous type of example. Einstein wrote his first papers on relativity with explanations of street cars moving at various speeds relative to an observer on the sidewalk. 🙂

Frequently, we have the freedom to assign the origin to that equilibrium position \(\vec{x}_{eq}\) so that \(\Delta\vec{x}\longrightarrow x\) and \(\vec{F}=\left(-kx\right)\,\hat{i}\).

Important. If initial point A is above final point B, then (yB - yA) is negative and the product -mg(yB-yA), is positive....it uses the author's g value, 9.8 m/s^2. This means the object gains energy as it falls, kinetic energy.

You occasionally see commodities and machines rated in foot-pounds.

In our case, the component of the force acting parallel to the path, \(F_{\parallel}\)

Momentum and energy are part of the four-dimensional vector that completely encodes an object's dynamics: $$\mathbb{P}= \begin{bmatrix} E/c \ p_x \p_y \ p_z

\end{bmatrix}$$ c = speed of light. This vector is known as the four-momentum and the energy-momentum four-vector.

Useful non-calculus shorthand: \(\vec{F}_{net}\Delta t =\Delta\vec{p}\)

I.e., \(\vec{F}_{net}=\frac{d\vec{p}}{dt}\)

The grip is wrong in this diagram, by about \(90^{\circ}\). Do you verify?

We will lecture on this on 2/7/25

The famous small angle approximation. See, for instance, "Trig function series."

I like this way of diagramming the set of x- and y-components. : )

A pattern

I never use the term "normal force" in this manner, because there is no surface in this example. But the textbook author uses the term this way, so take it with a grain of force.

Slightly different strategy than what I use: * calculate rise time to apogee, using \(v_y\left(0\right)\); * calculate y-coordinate of apogee; * figure out the drop distance on the downhill side of the arc; * figure out the drop time from apogee to landing point;

• calculate final value of \(v_y\) from the drop time.

This method avoids the big nasty quadratic formula, but you can surely use quadratic formula. : )

Note: in this problem, the rise time (from volcano to apogee) is shorter than the drop time, because the total arc is not symmetric.

Good exercise

Reverse of the "baseball from the top of a cliff" example. I..e., it STARTS at the bottom of the cliff. Key time = 4.0 s.

WHOA. Very intriguing!

Hmmmm. I am intrigued by this one. :D

home run problem

home run problem

This makes it a "home run" type of problem, i.e., using the symmetries of the ballistic arc that launches and lands at the same y-coordinate.

I never use the equation involving \(v_{0y}^2 \) and \(v_y^2\), because they are best to study in relation to energy and momentum concepts in later chapter. You can do all projectile problems without it, so THINK carefully.

We will discuss this in lecture on Monday, Aug. 19.

This distinction -- internal vs. external -- will be important when we take a peek at atoms.

...an acceleration of the professor and the cart of equipment at the same rate.

This is an alternate description of the idea of dynamic lift. The usual description of dynamic lift involves a pressure differential in the surrounding fluid, air, due to the Bernoulli Effect.

In the energy-momentum method for studying the dynamics of a system, later in this textbook, we will find a perfect quantity that encodes the interaction equally, so that "action-reaction" is no longer a loose usage.

An understatement.

Similar to a Ferrari going off a cliff, a problem that has been on midterms in previous semesters.

This kind of variation is how petroleum geologists discovered the Chicxulub impact crater in the Yucatan and Gulf of Mexico. This asteroid/comet impact is what scientists now believe triggered the extinction of most dinosaurs.

Be aware.

The textbook uses \(g=9.80\,m/s^2\) for the acceleration due to gravity at the surface of Earth. I, however, prefer to encode the direction with a - sign, $$g=-9.8\frac{m}{s^2}$$

The definitive vector is the velocity vector, also defined as a tangent vector especially when working in four dimensional spacetime.

D for \(\Delta\), D for difference, i.e., subtraction.

Labeled as "5 lbs," so that is like the notional value "45000 miles" in the previous example.

If you were at a rifle range, sighting in your hunting gear, then you'd say, "Yes, I am inside the first ring, but that would not be good enough unless I am shooting the side of a barn." So maybe you adjust the magnification of your optics or buy a new sight with better optics.... or keep practicing!

So, if you were at a rifle range, you'd want to adjust your sights slightly, to get that group up and to the right, into the center.

The metric system is based on cubic centimeters (cc) of liquid \(H_2O\). Balance the object you are measuring with a number of cc's of liquid water, that will tell you the number of grams in your object. That is the theory. In practice, yes, you'd use various carefully machined metal cylinders etc.

A famous dictum, ascribed to Galileo, is

"Measure what is measurable, and make measurable what is not so."

Good to review and double-check from time to time during the semester.

Where Planck's constant, \(\hbar=6.58\times 10^{-16}\,eV\,s\) rules all, in which the wave properties of matter are significant.

Systematic explication of how four-dimensional spacetime behaves under various energy conditions, how four dimensional spacetime curves and affects various forms of energy, and all because of the nature of LIGHT. \(E=mc^2\) is just one equation from the most basic relativity ideas.

"Drama" :D

Electrical structures. You will learn this in PHY2054.

An enormous statement. It emphasizes what scientists do: figure out new things, by putting together a framework from first principles. That is problem-solving, which is what we want to be able to do on PHY2053/4.

And because GPS, of necessity, uses Einstein's general theory of relativity, even though the spacetime curvature in the neighborhood of Earth is so tiny, the civilian GPS in your vehicle can locate your vehicle to within a few feet almost anywhere on Earth. Without relativity, GPS would not be nearly as precise.

I.e., learn to THINK.

This is why studying for a physics class is different from a class where memorization is of paramount importance, like an anatomy class.

...as well as the theory of currved four-dimensional spacetime, Einstein's general theory of relativity.

Sub-nuclear particles that bind together to form protons and neutrons plus other exotic subatomic particles like the \(\pi^+\) meson.

A nice simulator for RLC and RC and LC circuits.

Worth fooling around with this one.

Derived in lecture on 11/15

Here is the precise combination in terms of \(R\), \(L\), and \(C\).

Discharge curve, a combination of real exponential \(e^{-\alpha t}\times\) complex exponentials \(e^{\pm i\omega t}\)

Phase \(\phi=kx-\omega t\)

Phase \(\phi=0\) corresponds to the first peak of the \(E_y\) field; \(\phi=2\pi\) corresponds to second peak, and so on. This means that the equation \(kx-\omega t=0\) is the equation of motion for the first peak of the wave, the "wave top" of the wave. Another way to view this is that

$$x=\frac{\omega}{k}t$$

That factor \(\frac{\omega}{k}\) must be the same as \(c\), the speed of the light wave.

Fourier transforms, another huge tool

This concludes the derivation of the wave equation for electromagnetic waves, specifically for \(E_y\left(t,x\right)\). The constants \(\epsilon_0\text{ and }\mu_0\) are related therefore to the speed of light, \(c\).

a.k.a. the vacuum solutions

Understatement. It is a lifetime's study, and that would only be scratching the surface.

i.e., only a vacuum is needed.

Even through a vacuum

Here is a nice blurb in YouTube about interference patterns. It is a HUGE tool for physicists and engineers. https://youtu.be/D7aftTF--5w?si=hABIsE089aUZpVT0

A nice diagram

Einstein's great quest, but he died before he reached it, and, in fact, no one has reached it yet.

Initial condition: Charged up, i.e., \(q\left(0\right)=q_0\)

Nice exercise

I.e., initial load in the capacitor

plug-in exercise

Good exercise

It is interesting to normalize the total energy equation to 1 by dividing both sides by U -- should result in a version of the Pythagorean Identity \(\sin^2\left(x\right)+\cos^2\left(x\right)=1\)

Joule heating, for which the energy dissipation rate is \(P=I^2 R\), and which is also known as "\(I^2 R\) heating."

idealization

Similar to a spring system at equilibrium, \(x=0\), where kinetic energy

$$\frac{1}{2}m\dot{x}^2$$

is maximum.

If you think of the charge \(q\left(t\right)\) as a coordinate, e.g., like \(x\left(t\right)\), then \(U_C\) is like the potential energy of a spring oscillator,

$$U=\frac{1}{2}kx^2$$

but for which the spring constant is \(1/C\)

Resembles another famous system,

$$E=\frac{1}{2}kx^2+\frac{1}{2}m\dot{x}^2$$

the harmonic oscillator.

Quadratic in current \(I\), therefore current \(I\) will oscillate.

We will review this application in lecture

Note that the diagram here and Fig. 12.3, greatly exaggerate the length of the line element \(d\vec{\ell}\)

The diagram in Webcourses is much more accurate.

A big "if" with a ton of trig involved.

Since \(x\gg\ell\), the approximation as \(\Delta\ell\) is good, and so is the approximation \(r_p\approx x\)

Homework problem

We will tackle this section and 12.1 Biot Savart Law on Monday, Nov. 6 Biot-Savart is ROUGH.

Needs a diagram!!!!!! A 3-D blackboard is needed here.

heinous diagram

Because \(\vec{F}\text{ is always }\perp \vec{v}\)

Operational or experimental definition, but not a theoretical definition!

What the observer sees: the compass needle aligns to the field. Lecture, 10/23

I am not sure why this problem is here.

Good. Be sure to verify this answer.

Technically, this makes the problem insoluble, since it is not specified that \(Q_1\) and \(Q_2\) are static, fixed in place, or not. IF you consider then to be fixed in place, it is an easy problem we have already studied. So proceed on that basis: I may review this problem in lecture on Wednesday, or in Discusssions.

A good workout with the nitty gritty of EPE, KE etc.

Significant question. The potential energy curve rules the dynamical world. * Here is a potential barrier, * Here is a potential well.

There is a huge difference between them, and the implications extend from quantum electrodynamics to black holes.

Finish working this one out, and try to follow the logic of the textbook authors. It is slightly different than my description, so understanding both will help you.

"energy at infinity" is all kinetic. The charge Q effectively has no interaction with q.

As noted in lecture, 10/2

A good problem to work out.

This diagram and description will be on part 1 of the midterm, one or two multiple choice concept questions

Also a very important result in Newton's Law of Universal Gravitation.

$$k=\frac{1}{4\pi\epsilon_0}$$ so $$\epsilon_0 = \frac{1}{4\pi k}$$

Not a coincidence. The Coulomb interaction is formally an infinite range interaction, directly because it is inverse \(r^2\)

Preparing to integrate!

Nice example, very basic

$$0^{\circ}\leq\theta\leq 180^{\circ}$$ I.e., from parallel to antiparallel

Here we go. Calculus ahead

Also similar to how CGI makes Gollum from the "wire frame" generated by Andy Serkis' motion capture suit.<br />

Important

Its destiny is to become an element of area, e.g.,

$$dA=dx\, dy$$

or

$$dA=r\,dr\,d\theta$$

Unfortunate choice of symbol, because it confounds with the unit of E field, \(\frac{N}{C}\).

I.e., units of E field, \(\frac{N}{C}\,\,\times\) units of area \(m^2\).

Even better: a hula hoop held out the window by a friend as you drive up Alafaya to Publix for Chinese food. Even your hand out the window works a bit as an analogy. If your hand is orthogonal to the road and therefore to the velocity of the air rushing past, you feel its force of air resistance. But if you hold your palm flat, parallel to the road surface, you feel way less air resistance.

This statement, however, is limited, because the electric field fills ALL of space, so that is a continuous \(\infty\) of vectors, no matter what direction the element of area \(dA\) is oriented.

SO... we must agree upon a standard of field line density, somehow.

I.e., $$d\vec{E}\cdot\, \hat{n}\,dA$$, in which \(\hat{n}\) encodes the orientation of the infinitesimal tile of surface, \(dA\)

Hmmmmmmmmmmmm....... will there be time on Friday in lecture?

Hmmmmmmmm.....

Flux!

A ubiqitous diagram, very famous, because many molecules, even large ones, have a polar symmetry: a more positive end and a more negative end. Example: \(H_2O\).

The guts of our Quiz 1

Covered in lecture, 8/28/23, in my comments about \(\hat{r}_{21}\)

Where we get the word "electric" and "electron."

Or to your head. :D

You can easily do this experiment at home. Kinda cool.

We normally do this demonstration in lecture on the first day of the semester.

about 103 mph

This makes the example more of a pin ball machine: give the object some speed than inject it into the "frictionless" track.

For this formula, it is also interesting to think about the circular proving track in Italy, the Nardo ring. It is banked, unlike this example, but there is still a correspondence between optimum speed \(v\) and the track's turn radius, \(r\). That is because the track surface and tires have a set coefficient of friction, and \(g\) is known, so the turn radius for the speed you want can be calculated: $$r=\frac{v^2}{\mu_s g}$$ That is, 1. Decide the top speed at which you want to test, 2. then calculate the turn radius you need, 3. and therefore the amount of land you need to build your own test track.

A good workout, and you can check your answer. If you work through Example 2 (above) then you can definitely work this problem... though they are not identical.

Think of this as similar to what we observed with the skateboarder interactions, where the smaller skateboarder got noticeably more speed.

This is like a sideways version of an old-fashioned spring scale. A modern scale uses a device called a strain gauge or a piezo-electric cell, each of which generates shanges in electrical current based on strain or pressure applied to the device.

True true true. If you have a velocity graph, you can analyze it for two other important physical quantities: 1. position and distances, which we get by calculating areas. Terms for this calculation are velocity rectangle and velocity triangle. 2. accelerations, which we get by computing slope of the graph, rise over run in geometry class but \(\frac{\Delta\vec{v}}{\Delta t}\) on a velocity graph in physics class.

Sir Isaac Newton himself figured out that position, velocity and acceleration are the main physical quantities a scientist needs to study any moving physical object, and that they all relate directly to his construct, forces.

This section, Ch. 3.4, is the target section for Midterm Exam 1, for Summer B, 2023. For that reason, these problems and exercises are good good good practice if you are preparing to crush the midterm.

A gas like \(O_2\) molecular oxygen can be converted to a liquid by compressing it. AND, a tank of LOX liquid oxygen has to be kept cold and under pressure, like on the space shuttle, one of the last things loaded before launch.

This is a good reinforcement on the lectures about tidal heating in Io and Enceladus.

NASA image of several emission spectra:

1. hydrogen
2. helium
3. oxygen
4. neon
5. iron

*

Here is a YouTube of a nice standing wave demonstration a few years ago.

https://youtu.be/n2lSvcHKtBY

This example is slightly deceptive, in that it presumes that the value of \(G\) is already known. Historically, that was not the case. The first successful measurement of the value of \(G\) in the laboratory in 1798, by Prof. Henry Cavendish of Cambridge University, led to a calculation of the density and mass of Earth.

Brain burner of the century!

Another good workout

A good workout

A good illustration.

Good workout that makes you think.

There is an error in this example.

Error. \(\left(5.0\, cm\right)=0.05\, m\)

So \(\Delta U_{elastic}\longrightarrow -0.0075\, J\)

"exact differential" is pertinent to a differential equations course and frequently seen in physical applications for the first time in a thermodynamics course.

SO... this section is the author's way of giving you a sneak preview of something almost every engineering and physics major will eventually see.

Bypass

Optional. Read or skim if you have a personal interest.

This paragraph distills an enormous amount of history, several century of thinking about quantifying motion, which led us to the idea of \(\frac{1}{2}mv^2\) kinetic energy.

Reminiscent of the impulse formula, $$d\vec{p}=\vec{F}_{net}\Delta t$$ which describes an increment of dynamical change to the momentum of the object, \(\vec{p}\)