- So, I'd like to start
with a little story. Back in the late 1950s, the faculty at Cal Tech became concerned that their undergraduate
science curriculum wasn't keeping up with the exciting,
new developments in physics, including the discoveries developed by one of their star professors,
named Richard Feynman. So, as Feynman was an excellent teacher and also a common complainer
about that very issue, they asked if he would
teach their two year undergraduate program in
physics that they just revamped. And he agreed, but only for one time, which is why every lecture was recorded and every drawing photographed. These series of lectures
were so beloved that they were immediately
compiled to make a book. And the Feynman Lectures in physics are considered the gold
standard of physics education, where they continue to
inspire and influence physics education to this very day. The reason I mention
Richard Feynman is that, like many theoretical physicists, Feynman was deeply
enamored of Maxwell's laws. In fact, he started his
lecture on Maxwell's laws by stating, "There can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of
the laws of electrodynamics. The American civil war will pale into provincial
insignificance in comparison." Woo, Feynman also concluded
his talks by saying that Maxwell, "Brought together all the laws of electricity and magnetism and made one complete
and beautiful theory." However, despite Feynman's
well deserved reputations for the excellence in his lectures, Feynman himself admitted that
there was something missing from his lectures on
electricity and magnetism. I'll let him say it, roll the tape. - And it's necessary to learn all this physics in a limited time. I'm sorry, but we have to lose something. And one thing that we
have a tendency to lose in these lectures is the historical
experimental development. It is hoped that in the laboratory, some of this error is removed. Another possibility, if you're interested, is to read the "Encyclopedia Botanica," which has excellent
articles on many subjects of this kind or other histories or other textbooks and
electricity magnetism. Anyway, if you wanna know some more, you can always find it out. - So, I thought, I have the time, why don't I take a crack at it? So, with a mixture of hubris and a deep love of the history of science, I've decided to do a series of videos, a deep dive on the history of what inspired modern Maxwell's equations. This is my first video on
the history of the physics, experiments, and theories
behind Gauss's law. This is a story of
Charles-Augustin de Coulomb and how he inspired Michael Faraday and how Michael Faraday
inspired James Clerk Maxwell and why Maxwell's
equations as he wrote them in 1856, in 1862, and
in 1864 are equivalent to our modern version of
his Gauss's equation law, albeit with different letters,
names, and math conventions. Ready? Let's go. ♪ Electricity, electricity ♪ - [Instructor] Part one, Coulomb's law and experiments, 1784 to 1786. I would like to start in the early 1780s, when an engineer named
Charles-Augustin de Coulomb was told there was something wrong with his sensitive magnetic compass. See, it turned out that
the compass that Coulomb had made was too
sensitive as it would move whenever an operator would walk to the machine to take measurement. After viewing it himself, Coulomb realized that
the machine was twitching because the assistant was getting a bit of static charge as he
shuffled up to the machine and he could solve the problem by putting a grounding wire near the
machine for the assistant, and also possibly removing a carpet, but it got Coulomb thinking maybe this twisting force was so sensitive, it could be used to measure the minute electrical forces. Inspired, Coulomb studied
the physics of twisting and in 1784, presented a paper on what is called atorsional
balance or twisting scale, which was the world's most precise measuring device at the time. Heck, a degree on the
scale was so sensitive, it was equivalent to
the weight of 100000th, the weight of a grain of sand. The following year, 1785, Coulomb used that scale to
study electric repulsion. He did that by charging
up a tube by rubbing and then having it touch
a ball in the scale, he then moved an identical
metal ball until they touched and then studied the repelling forces. In this way, Coulomb
experimentally determined that electrical repulsion force is proportional to one
over the distance squared, which he confidently wrote was a fundamental law of electricity. For his second paper on electricity, Coulomb studied the attractive
force by charging one object by rubbing a glass tube
and touching one sphere while the second sphere was
electrified by being touched by a wax or resin tube that was rubbed. In this way, Coulomb determined
that the attractive force of opposite charges follows
the same relationship. In addition, in the same paper, Coulomb undetermined
that the electric force also depends linearly on the charges. Of course, he couldn't
directly measure the charge, which is currently measured in Coulomb's. So, instead he charged
up a ball in the machine and then had to touch a neutral ball of the same size and then
have the neutral ball removed so that the charge in the
machine would be reduced by two. And then, he found that the
force was also reduced by two. Then, after he wrote a paper about how electricity dissipates over time, for his fourth paper, which was published the following year in 1786, Coulomb decide to use
his electrometer to study how charges are distributed
in conducting spheres. Coulomb then found that quote,
"For conductive objects, the electric fluid is only
diffused over its surface and does not penetrate
into its interior parts." Fast forward 50 years to 1836, when a 45 year old Michael Faraday read an English translation of Coulomb's
work and was fascinated. In fact, Coulomb's fourth
paper in particular with a material about how charges reside on a conductor directly
influenced Michael Faraday to discover electric
lines and dielectrics, which brings us to part two, Faraday's "Electric Lines
and Dielectrics," 1837. When Faraday read Coulomb's paper, it was just five years after Faraday had discovered magneto-electric induction or how changing a magnetic field in a coil of wire could induce a
current in that wire. And Faraday was on a tear of
innovation about electricity. That is why the following year, in 1837, for his 11th paper in six years, Faraday decided to study
Coulomb's law about how all the charges were on
the surface in more depth. This is why Faraday ended up
upscaling Coulomb's experiment and building a giant cube that he lived in while electrical experiments
were conducted outside and how he found he was
shielded from these forces when he was inside the cube or cage. Faraday instantly realized that this was an effect of static
electric induction, i.e., moving charges
electrically, without touching. As he wrote, "The effects are clearly inductive effects
produced by electricity." And it was because of this experiment that Faraday started to imagine
that all charged objects emanate what he called
lines of inductive force, or as he later called it
lines of electric force. What Faraday was saying here is that all charged particles
emit these lines of force. And if you have multiple charges, then all the lines from the
opposite charges cancel out. And the total surplus of
charges emit a force field, depending on the amount of extra charge. This force field will apply a force on any extra charge you put in that field where the amount of force
depends on the amount of charge you add times the
strength of the electric field or F=qE, in modern language. As everything that you can see is composed of an unfathomable amount of charges, both positive and negative. It turns out that the electric field has an effect on all matter. Note, electric fields
do not affect neutron, but Faraday didn't know about that because the neutron wasn't discovered for almost a hundred years
after he did this experiment. If an electric field
goes through a conductor, the charges are free to move. So, they move until the electric field inside the conductor is zero. That is why if you are surrounded by a metal cage, called a Faraday cage, you are protected from the electric field. But if you pop your head out of the cage, the electric field can
make a visible effect. I have a whole video about the
physics of the Faraday cage. You can check it out here. Anyway, if the electric field goes through an insulator, on the other hand, Faraday theorized that the
molecules become polarized, but do not fully move
to cancel out the field. For this reason, Faraday gave
a new name for insulators. He called them dielectrics
to express that substance through which the electric
forces are acting. The next immediate question
is, do different materials affect the electric
field more than others? Or as he put it, "I now proceed to examine the great question of
specific inductive capacity, i.e., whether different
dielectric bodies actually do possess any influence on the degree of induction which takes
place through them." At the time, they knew for over 90 years that if you had two conductors and they were separated by an insulator, that object could store a lot of charge and what they called a Leyden jar, but we now call a capacitor or condenser, therefore Faraday examined
how differently shaped Leyden jars worked when
you inserted a dielectric and found that it would
increase the charge stored on those Leyden jars or
stored in the capacitor. He took a few samples this
way and found for example, that the specific inductive capacity, what we now call the dielectric constant of Shellac is about two,
or twice that of air. To recap, in late 1837, Faraday built the Faraday cage, created the idea of electric field lines, which he first called
inductive lines of force, and then electric lines of force, theorized the conductor's
let charges move to delete the electric lines of force within them to explain the Faraday cage, coined the name dielectrics
to describe inductors as he thought they could twist
due these lines of force. So, that one side was more positive and one side was more negative. Thus, they had die two electrics. Created the idea that
different dielectrics have different electrical effects and created a method to measure the dielectric constant
that he just devised. Wow, right? Faraday was
very excited about this. And just two months
later on January, 1838, he published another paper with a subtitle "On Induction, continued," in February another "On
Induction, continued." And in June, 1838, a paper on the nature of electric force, which was
all about lines of induction. Although Faraday was convinced that these curved induction lines of force or electric lines of force
explained everything, other scientists basically hated it. The big problem was they felt like forces should be in a direct line. Nothing should be curved, but the other problem was they wanted to have some math in their equations and Faraday didn't know any math. So, he didn't include any math. Then, Faraday had a bigger problem. On November 29th, 1839, Queen Victoria's personal physician, Dr. Peter Latham, made an emergency visit as Faraday was so dizzy
he could barely walk and was having horrible memory issues. Dr. Latham wrote that Faraday said that his mind for a long time had been dreadfully overworked from not only his own abstruse speculations
upon electricity, but also all of his lectures
and constant consultations. Faraday tried to keep on teaching, but his doctor said that if he didn't take at least a year off, he will suddenly break down and Faraday reluctantly
stepped away from science. Faraday then spent
years away from science, struggling to even read scientific texts. But, whenever he had a moment
of scientific lucidity, he would rush back to the laboratory and still manage to make
astonishing discoveries. For example, in 1845,
Faraday found a relationship between the polarization
of light and magnets, which was partially what inspired
him to propose that light, i.e., radiation, was a wave of electric and or magnetic lines of force in 1846. I will go into this much more in further videos, don't worry. Now, I wanna jump forward
to February 20th, 1854. This would be almost
16 years after Faraday proposed the idea of
electric lines of force. And over six years after Faraday proposed the idea that light was
a wave of electric lines of force or magnetic lines of force, or maybe both because
that is when a 22 year old Scottish mathematician and physicist named James Clerk
Maxwell wrote his mentor, William Thompson, AKA Lord Kelvin. The reason that Maxwell
was writing Thompson is because Maxwell said
that several of us here wished to attack electricity
and Maxwell asked Thompson if he should start with
Faraday, who has no mathematics, or Ampere, who had quite a lot. Thompson recommended that
Maxwell start with Faraday. And as Maxwell recalled years later, "As I proceeded with the study of Faraday, I perceived that his methods of conceiving the phenomena was also a mathematical one, though, not exhibited in the conventional form of mathematical symbols." Moreover, Maxwell, who
was very comfortable with mathematics felt
that he had the ability to convert Faraday's math into the ordinary mathematical forms, which brings us to part three, Maxwell on Faraday's "Lines
of Force," 1855 to 1856. In December of 1855, 24
year old James Clerk Maxwell gave a talk on Faraday's "Lines of Force." In the introduction, Maxwell said that he wasn't, "Attempting to establish any physical theory of a science which I have hardly made
a single experiment." Instead, Maxwell wanted to show how by a strict application of the ideas and methods of Faraday the connection of the very different
orders of phenomenon, which he has discovered
may be clearly placed before the mathematical mind. In this paper, Maxwell had a challenge because not only had Faraday
come up with the idea of charged lines of force
emanating from charged particles, but also according to
Maxwell's biographer, Faraday went further than this. He conceived the notion of
causing the lines of force to represent the intensity
of the force at every point, so that when the force is great, the lines might be close together, and far apart when the force is small. Maxwell, thus had to find a way to model not only the direction of the force, but also some method of indicating the intensity of the force at any point. Maxwell's initial solution was to model these lines of force, not as mere lines, but as fine tubes of variable section carrying an incompressible fluid. His logic was that he
could use the physics of fluid dynamics to model
how the electric force changes as one over the distance squared, by changing the size of the tubes since the velocity of
a fluid is inversely, as the section of the tube. Note that Maxwell wasn't
thinking these tubes were real, but he was using what he called a purely imaginary
substance as a metaphor. Maxwell then imagined
that the positive charges were sources of these tubes
and the negative charges were sinks of these tubes
where the electrical lines of force flowed in these
tubes continuously. Maxwell then began with determining how the pressure in these tubes varied as you moved from sources to sinks. And he declared that for any arrangement of sources and sinks,
there must be a surface with equal pressure as
the pressure is high near the source and zero
infinitely far away. And it continuously changes. He then declared that,
"It is easy to see that these surfaces of equal
pressure must be perpendicular to the lines of fluid motion." By lines of fluid motion, Maxwell meant the flow
of electric field lines. If that sounds suspiciously like Maxwell was saying that there's
surfaces with equal potential and that the electric field is dependent on the change of potential over space, or what's called the gradient of field. Well, then you're a
hundred percent correct. And Maxwell knew it. In fact, he explicitly
said that this pressure was the same as the
voltage from a battery, then called the electric tension, and even more impressively
Maxwell connected this electric tension to the potential in static electricity, writing, "This pressure, which is
commonly called electric tension, is found to be physically identical with the potential in static electricity. Thus, we have the means of connecting the two set of phenomena." Maxwell even went as far as
deriving the pressure, P, from a single source,
S, in a uniform medium and got P equals kS over 4, pi, R, which is exactly the same as the modern definition of potential from a point charge
i.e., V equals kQ over R, it only looks a bit different, but that is because Maxwell's
constant K in this paper is four pi times larger than
modern Coulomb's constant K, it seems likely, but I
don't have proof for this, that the reason we use
K in Coulomb's constant is because of Maxwell. Anyway, don't let that
four pi disturb you, as Maxwell would play with adding and removing four pi's all over the place. As you can either have four
pi in Gauss's equation, or in your equation for electric potential and field from a point
charge or use two constants. On his second talk on
Faraday's lines of force, which were given in February, 1856, Maxwell continued his
analogy of a flow in pipes and connected the electromotive force or the intensity of electric action, i.e., the electric
field, to what he called the quantity of electrical
current times a resistance, K. Note that Maxwell didn't
have vector multiplication. So, he wrote three equations
for the three directions, which he labeled equation B. But if we define the
intensity of electric action as the vector E and the
quantity of electric current as a vector D, then
equation B is simply E=kD. Note that this quantity
of electric current is actually the quantity of electric lines emanated from or to a
charged source or sink. As I feel this is confusing, I'm gonna give this variable the temporary name of
quantity of electric lines, just so you don't think I'm talking about an actual current. Anyway, Maxwell needed
a way of determining how these quantity of electric current or quantity of electric lines changed as it emanated from a source. So, he went to a method
devised by the great German mathematician,
Carl Frederick Gauss. Therefore Maxwell started
with a surface in space and stated that the total
quantity of conduction through any surface,
S, depends on how many of these lines, D, point
out of the surface. Maxwell then used calculus to
show that if you integrated how much of the D pointed
out the surface, S, it was equivalent to the
integral of the derivative of each component of the
vector D over the total volume. Note that the sum of
derivatives is now called the divergence of D and
can be written as Dell.D. Then, Maxwell made one of the most astonishing statements in physics. He just basically said, "Let's
make all those derivatives, the divergence of D equal to four pi times the charge per volume rho that made the lines of force in the first place." Now, don't worry about the four pi. He just added it to remove the four pi in the pressure equation, like I said, Maxwell played around with four pi a lot, and it just changes value of constant. So, it doesn't really matter, what matters is that he
just stated Gauss's law without really spelling
out how he got there, probably 'cause it wasn't
clear to Maxwell yet, irrespective of how he derived it. These equations B or C
are just Gauss's law. First, as I said before,
in modern language, equation C can be written as a divergence of D equals four pi rho. Then, if you multiply both sides by K and then use the equation B that E=kD, you get that the divergence
of E equals four pi K rho. If you define a new constant Epsilon, so, that Epsilon equals
one over four pi K, you get that the
divergence of E equals rho over Epsilon, which is Gauss's law. Maxwell was very pleased with the results and even the now frail
and kind of still sickly, 65 year old, Michael Faraday
was excited about it. Despite all the math, writing Maxwell, "It gives me much
encouragement to think on. I was at first, almost frightened when I saw such
mathematical force bear upon the subject and then wondered to see that the subject stood it so well." However, Maxwell needed
an actual electric model of the atom rather than imaginary tubes of incompressible fluids, which brings me to part four, Maxwell's "On Physical Lines of
Force," 1861 to 1862. 5 years before Maxwell's
first paper in 1850, another Scottish scientist
named William Rankine was inspired by Faraday to model the atom with what he called molecular vortices. This strangely named model
is astonishingly modern as Rankine just stated that quote, "Each atom of matter consists
of a nucleus or a central physical point enveloped
by an elastic atmosphere, which is retained in
its position by forces attracted toward the nucleus or center." Soon, Maxwell's friend
William Thompson started incorporating some of Rankine's
ideas in his own paper. And by 1861, Maxwell adjusted
his views of electric and magnetic fields using this
model of molecular vortices, rather than the model of
tubes of incompressible fluids in a series of papers titled,
"On Physical Lines of Force." This paper caused Maxwell to
make major changes in his laws. However, in terms of Gauss's law, it's pretty similar with two major and multiple minor differences. The first major difference is that, back in 1855 and 1856,
Maxwell just stated outright that the sum of derivatives was dependent on the charge density
because it made sense to him. This time, Maxwell started
by deriving another equation, specifically his equation that currently has the unwieldy name of Ampere's Circuital law
with Maxwell's addition. Maxwell then used that
equation to establish how the divergence of the electric field is related to the charge per volume. Now, given the letter lowercase "e." I will show how Maxwell
did this in future videos, but for now just put a pin
in it and save it for later. The second major change
he did is that he gave a physical meaning to
what he initially called the quantity of electric
current that I called the quantity of electric
lines from sinks and sources. If you remember when
Maxwell first created this, he used an analogy to
current in a tube where the electric field is
analogous to the voltage. And the quantity of electric current is how much these fields flow
from sources to sinks, which depends on a resistance. These equations work, but they're kind of
physically meaningless. But then in 1861 and 1862, with the help of these molecular vortices, Maxwell could finally model them through what the atoms were doing in a dielectric. Writing, "In a dielectric under induction, we may conceive that the electricity in each molecule is so displaced. That one side is rendered positively and the other negatively electrical, but that the electricity does not pass from one molecule to another." As the molecules in the
dielectric are displaced, Maxwell renamed his
quantity of electric current to be the displacement, a
name it retains to this day. Just as before the displacement is related to the electric field by a constant. This time, however, the
constant is not a resistance, but instead of constant that is dependent on the nature of the
dielectric as I said before, this doesn't change the
nature of the equation at all. It just changes the meaning
behind all the variables. If you look at the equation
as Maxwell wrote it in 1862, however, it looks very
different than 1855, but that has to do with the superficial changes that Maxwell made. First, Maxwell added a four
pi to this relationship, instead of to the Gauss's law part, which makes no difference
aside from changing the magnitude of the displacement field. Second, he squared the constant, which he gave a new letter capital "E." That sounds significant, but it only changes the
value of the constant. Third, he made the
displacement to be negative with the electric field. The reason he did this is Maxwell decided it'd be more interesting to talk about how the lines converge
on sources and sinks instead of how the lines diverge away from sources and sinks. For that reason, Maxwell
made the displacement vector negative to the function we use currently, or he took the convergence of
a negative displacement field, negative Dell.D prime
instead of the divergence of a positive displacement field, Dell.D. This only changes the value
of the displacement field and not the end result for
force or electric potential. By the way, in 1883, Oliver Heaviside switched the notation when
he wrote about Gauss's law and said that the
expression on the right side of this equation with a
negative sign prefixed, Maxwell called the convergence, but what he may as well
use the term divergence for the same quantity with
the plus sign prefixed. We have basically used
Heaviside's notation and sign convention and the
term divergence ever since. Back in 1862, Maxwell wrote Gauss's law as one equation instead of two. So, the charge per volume equals a divergence electric
field times a constant. If we use Maxwell's old term, rho for E, use the Greek letter Epsilon
for one over four pi E squared and vector notation
for the electric field. Maxwell's equation transforms from this to rho equals Epsilon
times the divergence of E or the divergence of E
equals rho over Epsilon. So, once again, Gauss's law now with some physical meaning
for the displacement, way back in 1862, however, there's one concept in this equation that wasn't very well developed and that had to do with the constant one over four pi E squared, because all Maxwell said was that it was related to the
dielectric constant somehow. He then clarified it in
his next paper of 1864, which brings me to part five, Maxwell's "Electric Elasticity" versus Heaviside's "Permittivity." On December 8th, 1864, a now 33 year old James Clerk Maxwell wrote his famous paper where
he used his new equations to describe how his laws could be used to mathematically explain light as a wave in electromagnetic field, a phrase he coined for that very purpose. I know he looks much
older in this picture, but I think that has to do
with his big bushy beard, which he grew to make
him look more mature. Anyway, this paper was to
have major implications for the progress of science,
but in terms of Gauss's law, it only had two minor changes, one superficial, and one profound, both about the relationship
between the electric field, represented by the letters P, Q, R, and the displacement field, represented by the letters F, G, H. The superficial change
is that Maxwell decided that if the displacement is negative, the electric field
should be negative, too. So, this relation no
longer has a negative sign. Maxwell then changed the sign for the force and potential equations. So, the results are all the same. More interestingly, Maxwell
decided to drop the four pi in the E squared and go
back to a simple constant K. So, the equation between
the electric field and the displacement field is
the exact same relationship that he found in 1856, namely E=kD. What makes this different is that in 1856, Maxwell did not give K a
name besides a resistance. In 1864, however, Maxwell
called "kD" electric elasticity, as he imagined the
molecules in the dielectric being pulled by the electric displacement. And then bouncing back when the displacement field was removed, the more the dialectic was
pulled by the displacement field, the smaller the electric elasticity is. Using this logic, the elasticity was inversely proportional to
the dielectric constant. Although, depending on the units you use, you might need another constant as well to make all the units work out. Finally, we have Gauss's equation where every letter has a physical meaning. Once again, it was Oliver
Heaviside in the 1880s and early 1890s who
decided that it was often more logical to define the constant, which Heaviside called the permittivity. If you defined it as
the ratio of D over E, rather than E over D, so that the constant is linearly proportional
to the dielectric constant. In other words, define a new constant, the permittivity that Heaviside labeled C, but we currently use
a Greek letter Epsilon that is UN over the elasticity, where Epsilon equals one over K or K equals one over Epsilon. That way, instead of E=kD, you get E equals D over Epsilon. The advantage of using
Heaviside's "Permittivity" rather than Maxwell's
"Electric Elasticity," is that you can see visually
that the electric field is reduced by the value of
the dielectric constant. Note with the units that we use for charge and distance and force, and the like, even if the field is
going through a vacuum where there are no molecules to be pulled or air at room temperature and pressure, which reduces the electric
field by a factor of 1.0006, you still often need a constant
to make the units work out. This constant is currently called the permittivity of free space, or this symbol pronounced,
Epsilon zero or Epsilon naught. Naught is an old fashioned
way of saying nothing, as in "Naught but a fool." This is such a common
situation that we are often taught Gauss's equation
only for that case, which is exactly what Richard Feynman wrote in 1962, roll the tape. - [Feynman] The Maxwell
equations of Maxwell, which we're going to study
all during this year, but you see that they are
of the form advertised, that there's a divergence,
there's a formula that tells you what
the divergence of E is. It's equal to rho over Epsilon zero. Rho is the charge for unit volume. - So, that is the historical
origin of Gauss's law, but I didn't really give very many details of how we got for Maxwell's equations to the modern form of Maxwell's equations. And what does that have to do with math functions called quaternions? Why did Oliver Heaviside get involved? Why was he listened to? And what about the influences of Hamilton, Tate, Gives, and even Hertz? Well, you better subscribe
and hit that bell button because all of that is next
time on "The Lightning Tamers." Woo. That was a pretty
hardcore physics video for me. I hope you enjoyed it,
and it wasn't too much. If you want the full
script for this video, including citations
that you can just click so you can read it yourself, hop on over to my website,
www.kathylovesphysics.com. Also, you can learn about my new book, "The Lightning Tamers," which
is available for pre-order. Heck, you could even download the first, over a hundred pages for
free and check it out. I'm also hosting a GoFundMe
for an audio book version. So, if you feel like
donating to me on there, it's all on my website
and a whole lot more. Thank you to my patrons and remember, stay safe and curious my friends, bye. As everything is composed
of an unfathomable, (Instructor stuttering)
Unfathomable it's hard to say.
