This is a backup of an article on Wired,’Get to know Maxwell’s Equations – You’re Using Them Right Now,” by Rhett Allain , 8/6/19
Maxwell’s equations are sort of a big deal in physics. They’re how we can model an electromagnetic wave—also known as light. Oh, it’s also how most electric generators work and even electric motors. Essentially, you are using Maxwell’s equations right now, even if you don’t know it. Why are they called “Maxwell’s equations”? That’s after James Clark Maxwell. He was the 19th-century scientist who sort of put them together, even though many others contributed.
There are four of these equations, and I’ll go over each one and give a conceptual explanation. Don’t worry, you won’t need to refresh your calculus skills. If you do want to follow the math, let me point out that there are two different ways to write these equations, either as integrals or as spatial derivatives. I’ll give both versions—but again, if the math looks uninviting, just ignore it.
The short version is that Gauss’ law describes the electric field pattern due to electric charges. What is a field? I like this description – “It’s an energy field created by all living things. It surrounds us, penetrates us, and binds the galaxy together.”
Oh wait. That was Obi Wan’s description of the Force in Star Wars Episode IV. But it’s not a terrible description of an electric field. Here is another definition (by me):
If you take two electric charges, there is an interaction force between them. The electric field is the force per unit charge on one of those charges. So, it’s sort of like a region that describes how an electric charge would feel a force. But is it even real? Well, a field can have both energy and momentum—so it’s at least as real as those things.
Don’t worry about the actual equation. It’s sort of complicated, and I just want to get to the idea behind it. (If you have seen this physics equation before, you might think I am going to go into electric flux, but let’s see if I can do this with “no flux given.”) So let’s just say that Gauss’ law says that electric fields point away from positive charges and towards negative charges. We can call this a Coulomb field (named after Charles-Augustin de Coulomb).
Everyone knows that positive charges are red and negative charges are blue. Actually, I don’t know why I always make the positive red—you can’t see them anyway.
Also, you might notice that the electric field due to the negative charges looks shorter. That’s because those arrows start farther away from the charge. One of the key ideas of a Coulomb field is that the strength of the field decreases with distance from a single point charge.
But wait! Not all electric fields look like this. The electric field also follows the superposition principle. This means that the total electric field at any location is the vector sum of the electric field due to whatever point charges are nearby. This means you can make cool fields like the one below, which are the result of two equal and opposite charges (called a dipole).
And here’s the Python code I used to create it. https://trinket.io/glowscript/18196b0cf1
This dipole field is going to be important for the next equation.
Yes, this looks very similar to the other Gauss’ law. But why isn’t the previous equation called “Gauss’ law for electricism”? First, that’s because “electricism” isn’t a real word (yet). Second, the other Gauss’ law came first, so it gets the simple name. It’s like that time in third grade when a class had a student named John. Then another John joined the class and everyone called him John 2. It’s not fair—but that’s just how things go sometimes.
OK, the first thing about this equation is the B. We use this to represent the magnetic field. But you will notice that the other side of the equation is zero. The reason for this is the lack of magnetic monopoles. Take a look at this picture of iron filings around a bar magnet (surely you have seen something like this before).
This looks very similar to the electric field due to a dipole (except for the clumps of filings because I can’t spread them out). It looks similar because it is mathematically the same. The magnetic field due to a bar magnet looks like the electric field due to a dipole. But can I get a single magnetic “charge” by itself and get something that looks like the electric field due to a point charge? Nope.
Here’s what happens when you break a magnet in half. Yes, I cheated. The picture above shows two bar magnets. But trust me—if you break a magnet into two pieces, it will look like this.
It’s still a dipole. You can’t get a magnetic field to look like the electric field due to a point charge because there are no individual magnetic charges (called a magnetic monopole). That’s basically what Gauss’ law for magnetism says—that there’s no such thing as a magnetic monopole. OK, I should be clear here. We have never seen a magnetic monopole. They might exist.
The super-short version of this equation is that there is another way to make an electric field. It’s not just electric charges that make electric fields. In fact, you can also make an electric field with a changing magnetic field. This is a HUGE idea as it makes a connection between electric and magnetic fields.
Let me start with a classic demonstration. Here is a magnet, a coil of wire, and a galvanometer (it basically measures tiny electric currents). When I move the magnet in or out of the coil, I get a current.
If you just hold the magnet in the coil, there is no current. It has to be a changing magnetic field. Oh, but where is the electric field? Well, the way to make an electric current is to have an electric field in the direction of the wire. This electric field inside the wire pushes electric charges to create the current.
But there is something different about this electric field. Instead of pointing away from positive charges and pointing towards negative charges, the field pattern just makes circles. I will use the name “curly electric field” for a case like this (I adopted the term from my favorite physics textbook authors). With that, we can call the electric field made from charges a “Coulomb field” (because of Coulomb’s law).
Here is a rough diagram showing the relationship between the changing magnetic field and an induced curly electric field.
Note that I am showing the direction of the magnetic field inside of that circle, but it’s really the direction of the change in magnetic field that matters.
Do you see the similarity? This equation sort of looks like Faraday’s law, right? Well, it replaces E with B and it adds in an extra term. The basic idea here is that this equation tells us the two ways to make a magnetic field. The first way is with an electric current.
Here is a super-quick demo. I have a magnetic compass with a wire over it. When an electric current flows, it creates a magnetic field that moves the compass needle.
It’s difficult to see from this demo, but the shape of this magnetic field is a curly field. You can sort of see this if I put some iron filings on paper with an electric current running through it.
Maybe you can see the shape of this field a little better with this output from a numerical calculation. This shows a small part of a wire with electric current and the resulting magnetic field.
Actually, that image might seem complicated to create but it’s really not too terribly difficult. Here is a tutorial on using Python to calculate the magnetic field. There is another way to create a curly magnetic field—with a changing electric field. Yes, it’s the same way a changing magnetic field creates a curly electric field. Here’s what it would look like.
Notice that I even changed the vector colors to match the previous curly field picture—that’s because I care about the details. But let me just summarize the coolest part. Changing electric fields make curly magnetic fields. Changing magnetic fields make curly electric fields. AWESOME.
What About Light?
The most common topic linked to Maxwell’s Equations is that of an electromagnetic wave. How does that work? Suppose you have a region of space with nothing but an electric field and magnetic field. There are no electric charges and there isn’t an electric current. Let’s say it looks like this.
Let me explain what’s going on here. There is an electric field pointing INTO your computer screen (yes, it’s tough dealing with three dimensions with a 2D screen) and a magnetic field pointing down. This region with a field is moving to the right with some velocity v.
What about that box? That’s just an outline of some region. But here’s the deal. As the electric field moves into that box, there is a changing field that can make a magnetic field. If you draw another box perpendicular to that, you can see that there will be a changing magnetic field that can make a magnetic field. In fact, if this region of space moves at the speed of light (3 x 108 m/s), then the changing magnetic field can make a changing electric field. These fields can support each other without any charges or currents. This is an electromagnetic pulse.
An electromagnetic wave is an oscillating electric field that creates an oscillating magnetic field that creates an oscillating electric field. Most waves need some type of medium to move through. A sound wave needs air (or some other material), a wave in the ocean needs water. An EM wave does not need this. It is its own medium. It can travel through empty space—which is nice, so that we can get light from the sun here on Earth.
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“How the Oceans Came to Be” packet: How the ocean came to be
The oceans of water molecules trapped under the oceans, deep within the earth’s crust
Background vocabulary: monomer and polymer
Living things have genetic information stored in a polymer of DNA.
That info gets copied into polymers of RNA.
That is translated, with the help of mRNA, into polymers of proteins.
And each of these steps needs special enzymes.
Life in today’s cells, and even viruses, is wicked complicated.
Hard to imagine all it all evolved, all at once. But who says it had to do it all at once?
Maybe one simple kind of reaction developed, then later, other kinds of reactions, and then over a loooong period of time, even other types.
Life in the very beginning
Perhaps once upon a time, RNA was all that life had.
Pieces of RNA were both the genes and the catalyst.
e.g. RNA could do base pairing with itself, bend, and graph other molecules.
RNA sequences could be copied by other RNAs.
Only later did DNA and proteins evolve.
This is the idea of the RNA world
A hypothetical stage in the history of life on Earth
Idea – RNA developed before DNA and proteins developed.
Alexander Rich first proposed the concept in 1962
Growing amounts of evidence for this is strong enough that the hypothesis has gained wide acceptance.
How is RNA like DNA?
Both can store and replicate genetic information;
How is RNA like an enzyme?
Both can catalyze (start) chemical reactions.
Are any enzymes today made of RNA?
the ribosome is composed primarily of RNA.
Ribosomes are part of many important enzymes, such as Acetyl-CoA, NADH, etc.
So why does life depend on DNA replication nowadays?
DNA is more stable than RNA
What does RNA, and DNA, look like?
How would RNA monomers assemble into polymers?
How could copies be made?
So let us look at the possible in steps, in order.
At the far left is long ago… then an RNA based world of life developed… and later a DNA and protein based world of life developed.
from The Math Equation That Tried to Stump the Internet
Excerpted from the NY Times article, The Math Equation That Tried to Stump the Internet, by Steven Strogatz, 8/2/2019
… The question above has a clear and definite answer, provided we all agree to play by the same rules governing “the order of operations.” When, as in this case, we are faced with several mathematical operations to perform — to evaluate expressions in parentheses, carry out multiplications or divisions, or do additions or subtractions — the order in which we do them can make a huge difference.
When confronted with 8 ÷ 2(2+2), everyone on Twitter agreed that the 2+2 in parentheses should be evaluated first. That’s what our teachers told us: Deal with whatever is in parentheses first. Of course, 2+2 = 4. So the question boils down to 8÷2×4.
And there’s the rub. Now that we’re faced with a division and a multiplication, which one takes priority? If we carry out the division first, we get 4×4 = 16; if we carry out the multiplication first, we get 8÷8 = 1.
Which way is correct? The standard convention holds that multiplication and division have equal priority. To break the tie, we work from left to right. So the division goes first, followed by the multiplication. Thus, the right answer is 16.
More generally, the conventional order of operations is to evaluate expressions in parentheses first. Then you deal with any exponents. Next come multiplication and division, which, as I said, are considered to have equal priority, with ambiguities dispelled by working from left to right. Finally come addition and subtraction, which are also of equal priority, with ambiguities broken again by working from left to right.
Now realize… PEMDAS is arbitrary. Furthermore, in my experience as a mathematician, expressions like 8÷2×4 look absurdly contrived.
No professional mathematician would ever write something so obviously ambiguous. We would insert parentheses to indicate our meaning and to signal whether the division should be carried out first, or the multiplication.
The last time this came up on Twitter, I reacted with indignation: It seemed ridiculous that we spend so much time in our high-school curriculum on such sophistry. But now, having been enlightened by some of my computer-oriented friends on Twitter, I’ve come to appreciate that conventions are important, and lives can depend on them.
We know this whenever we take to the highway. If everyone else is driving on the right side of the road (as in the U.S.), you would be wise to follow suit. The same goes if everyone else is driving on the left, as in the United Kingdom. It doesn’t matter which convention is adopted, as long as everyone follows it.
Likewise, it’s essential that everyone writing software for computers, spreadsheets and calculators knows the rules for the order of operations and follows them. For the rest of us, the intricacies of PEMDAS are less important than the larger lesson that conventions have their place. They are the double-yellow line down the center of the road — an unending equals sign — and a joint agreement to understand one another, work together, and avoid colliding head-on.
Ultimately, 8 ÷ 2(2+2) is less a statement than a brickbat; it’s like writing the phrase “Eats shoots and leaves” and concluding that language is capricious. Well, yes, in the absence of punctuation, it is; that’s why we invented the stuff.
– Steven Strogatz is a professor of mathematics at Cornell and the author of “Infinite Powers: How Calculus Reveals the Secrets of the Universe.”_
Professor Oliver Knill addresses the same phenomenon here:
Even in mathematics, ambiguities can be hard to spot. The phenomenon seen here in arithmetic goes beyond the usual PEMDAS rule and illustrates an ambiguity which can lead to heated arguments and discussions.
What is 2x/3y-1 if x=9 and y=2 ?
Did you get 11 or 2? If you got 11, then you are in the BEMDAS camp, if you got 2, you are in the BEDMAS camp. In either case you can relax because you have passed the test. If you got something different you are in trouble although! There are arguments for both sides. But first a story….[and there is a very cool story here, click the link below. But here is the important conclusion]
The PEMDAS problem is not a “problem to be solved”. It is a matter of fact that there are different interpretations and that a human for example reads x/yz with x=3,y=4 and z=5 as 3/20 while a machine (practically all programming languages) give a different result.
There are authorities which have assigned rules (most pupils are taught PEMDAS) which is one reason why many humans asked about 3/4*5 give 3/20 which most machines asked give 15/4:
I type this in Mathematica x=3; y=4; z=5; x/y z and get 15/4
It is a linguistic problem, not a mathematical problem. In case of a linguistic problem, one can not solve it by imposing a new rule. The only way to solve the problem is to avoid it. One can avoid it to put brackets.
That Vexing Math Equation? Here’s an Addition
Steven Strogatz, rofessor of Applied Mathematics, Cornell Univ, looks at a similar problem, and agrees that “questions” like these are deliberately badly written:
Recently I wrote about a math equation that had managed to stir up a debate online. The equation was this one: 8 ÷ 2(2+2) = ?
The issue was that it generated two different answers, 16 or 1, depending on the order in which the mathematical operations were carried out….
… The question was not meant to ask anything clearly. Quite the contrary, its obscurity seems almost intentional. It is certainly artfully perverse, as if constructed to cause mischief.
The expression 8 ÷ 2(2+2) uses parentheses – typically a tool for reducing confusion – in a jujitsu manner to exacerbate the murkiness. It does this by juxtaposing the numeral 2 and the expression (2+2), signifying implicitly that they are meant to be multiplied, but without placing an explicit multiplication sign between them. The viewer is left wondering whether to use the sophisticated convention for implicit multiplication
from algebra class or to fall back on the elementary PEMDAS convention from middle school.
Picks: “So the problem, as posed, mixes elementary school notation with high school notation in a way that doesn’t make sense. People who remember their elementary school math well say the answer is 16. People who remember their algebra are more likely to answer 1.”
Much as we might prefer a clear-cut answer to this question, there isn’t one. You say tomato, I say tomahto. Some spreadsheets and software systems flatly refuse to answer the question – they balk at its garbled structure. That’s my instinct, too, and that of most mathematicians I’ve spoken with. If you want a clearer answer, ask a clearer question.
That Vexing Math Equation? Here’s an Addition, The New York Times, Aug 5, 2019
“Oil” is a general name for any kind of molecule which is
that just means that its electrons are evenly distributed
liquid at room temperature
of course, it could become solid if cooled, or evaporate if heated
Molecule has one end which is hydrophobic and another end which is lipophilic
The hydrophobic end likes to stick to water molecules. But hates sticking to oils.
The lipophilic end likes to stick to oil molecules, but hates sticking to water,
Made with many C and H atoms
Oils are usually flammable. Here we see oils in an orange skin interacting with a candle.
So Petroleum is?
Petroleum is a mix of naturally forming oils, which we drill from the Earth, and use in a variety of ways. See our article on petroleum and producing power.
A dike (or dyke) is a sheet of rock that is formed in a fracture in a pre-existing rock body. Dikes can be either magmatic or sedimentary in origin.
Topic 1 – A ring dike is an intrusive igneous body. Their chemistry, petrology and field appearance precisely match those of dikes or sill, but their concentric or radial geometric distribution around a centre of volcanic activity indicates their subvolcanic origins. See here for more details: Ring dikes
Topic 2 -Giant Dikes: Patterns and Plate Tectonics
This is a photo of Shiprock (7178 ft) and southern dike, southwest of Shiprock, NM. View to northwest. Note the several small satellite volcanic necks at the base of Shiprock.
Where is this? Shiprock is a monadnock rising nearly 1,583 feet above the high-desert plain of the Navajo Nation in San Juan County, New Mexico, United States.
The following section has been excerpted from Giant Dikes: Patterns and Plate Tectonics, by J. Gregory McHone, Don L. Anderson & Yuri A. Fialko, published on Mantleplumes .org.
Giant dikes typically exceed 30 m in width and 100 km in length, with some examples over 100 m wide and 1,000 km long. Dikes are self-induced magma-filled fractures, and they are the dominant mechanism by which basaltic melts are transported through the lithosphere and the crust.
These spectacular intrusions are likely to have fed flood basalts in large igneous provinces (LIPs), including provinces where the surface basalts have been diminished or removed by erosion.
Although giant dikes can intermingle with denser swarms of smaller dikes of similar composition (and probably similar origin), others occur in sets of several to a few dozen extremely large quasi-linear or co-linear intrusions, which may gently bend and converge/diverge at low angles across many degrees of latitude.
Tectonic controls on the formation of giant dikes appear to be independent and different from structures related to smaller dike swarms. Theoretical modeling and field observations help us to understand the essential physics of magma migration from its source to its final destination in the upper lithosphere.
…in northeastern North America, huge but widespread dikes in Canada and New England diverge to the NE and ENE from a focus point east of New Jersey, but that is also not a plume center.
The dikes change their trends across the “New England Salient,” which is a bend in terrane suture zones and primary structures of this section of the Appalachian Orogen.
In addition, the giant dikes did not form together in a radial generation, but instead decrease systematically in age from the SE toward the NW (Figure 6).