Introduction
In this video I am going to explain the dot product, the cross product, the divergence, the curl and how it all works for Maxwell’s equations AND what Maxwell’s equations mean about magnets and light and a tiny bit about why that is important. But before I do that, I want to take a moment to say why I am making this video.
See, as many of you know, I am a science historian who makes videos and writes books where I use the personal stories of the scientists to teach the science. I am currently in the middle of making a series of videos on the minute details of how Maxwell’s equations developed over time. I am covering the people whose experiments inspired Maxwell (like Coulomb, Oersted, Ampère, and Faraday), how Maxwell used their ideas in his three long sets of papers on his laws in 1855-6, 1861-2 and 1864, and am working on telling the story of people like Hamilton, Tait, Hertz, Gibbs, and, of course Oliver Heaviside whose work was vital to converting Maxwell’s equations to their modern forms.
It was making these detailed historical videos that I realized two things: 1) there seems to be a need to have a video that starts with the very basics and shows not just what the math does but what it means and 2) because I have studied the history in such depth, I feel like I have a unique angle on how to give an overview of these laws by using a little backstory of where they came from, which I think (or at least hope) helps it all make sense.
OK, Ready? Let’s go
Table of Contents
Introduction
Part 1: Some Basics: Vectors and Vector Multiplication: the Dot Product and the Cross Product
Part 2: The Del Function, Divergence and “Gauss’s Laws”
Part 3: Maxwell’s Equations with Curl
Part 4: Maxwell’s Laws and Magnets
Part 5: Maxwell’s Laws and Light
References
Part 1: Some Basics: Vectors and Vector Multiplication: the Dot Product and the Cross Product
I would like to start with the basics: scalars and vectors. Scalars are things with value but without direction and vectors both have value and direction. For example, your mass is a scalar, as it has no direction. Your weight, on the other hand, is how much you are pulled by gravity, so it is a vector. That is why you say you feel weightless when you feel like gravity is not pulling on you but you never feel massless as you are never massless even in zero gravity.
As vectors have a magnitude and direction, you can represent them by a single arrow pointing in some direction, where the size of the vector is represented by the size of the arrow. Or you could represent it by how much it points in the x, y, and z directions like this:
\overrightarrow{A} = A_x\hat{x} + A_{\text{y}}\hat{\text{y}} + A_z\hat{z}
Now Maxwell knew about vectors, but he didn’t know about vector multiplication, so he couldn’t use them. Currently, there are two common ways to multiply vectors:
- the dot product which involves multiplying the components in the same direction
- the cross product, which involves multiplying the components that are perpendicular
The best way to understand this concept, I think, is by giving an example of multiplying two vectors with some physical meaning, I like thinking about multiplying force and distance.
If you take the dot product between the force and the distance (multiplying how much they are pointing in the same direction), you get the work done on an object or the amount of energy that you gave to the object. Like those high school physics problems where you push a lawnmower and the problem wants you to calculate the work done you get:
Work =\overrightarrow{F}\cdot\overrightarrow{d} = F_xx + F_{\text{y}}\text{y} + F_z z
Or, how much you push the object in any direction times how far you push it in that direction. Note that the dot product produces a scalar, with a value but no direction. Just as Force and distance are vectors but the work done on an object (how much energy you give it) has no direction.
The cross product, on the other hand, results in another vector, or a variable with both magnitude and direction. Let’s look at the cross product of the force and the distance, or how much the force is perpendicular to the distance you get the torque, or spinning force, which has a direction depending on what direction the spin goes (like with turning a screw, which direction the screw moves). Now with spinning, the order of operations matters. Let’s take the cross product of distance cross force to get the torque:
\overrightarrow{\tau} = \overrightarrow{d}\: x \: \overrightarrow{F}
If you take the cross product of a distance in the x direction with a force in the y, then your object twists out of the page, or in the positive z direction, if you take a cross product with distance in the y direction with a force in the x direction, then your object twists into the page, or in the negative z direction. Since the force and the distance could have components in the x, y and z directions, you have to determine all the possible twists. You end up with three equations, one for each direction.
\tau_x = \text{y}F_z - zF_\text{y}
\tau_\text{y} = zF_x - xF_z
\tau_z = xF_\text{y} - \text{y}F_x
Or, a single, very long equation:
\overrightarrow{\tau} = \overrightarrow{d}\:x\:\overrightarrow{F} =(\text{y}F_z - zF_{\text{y}})\hat{x} + (zF_x - xF_z)\hat{\text{y}} + (xF_{\text{y}} - \text{y}F_x)\hat{z}
Now figuring out all the directions can be hard to visualize, or at least I find it hard to visualize. Luckily, I was taught this little trick to figure this out where you draw an x, an y and an z in a triangle where if you take a cross product in the clockwise direction you get a positive result and if you take it counterclockwise it is negative. For example, y cross z is clockwise so it results in the positive x direction while z cross y is counterclockwise, so it is in the negative x direction. You can pause the video and check it out, it works every time!
Now, what Maxwell was trying to do in Maxwell’s equations was put Faraday’s idea of electric and magnetic lines of force into mathematical forms. For that reason, he needed a way to describe how they changed over space or he needed to take the derivative of a vector. For that he needed a function that had a derivative and a direction. Which brings me to…
Part 2: The Del Function, Divergence and “Gauss’s Laws”
See, this triangle function,\nabla it has three common names, alted, nabla, and del. I am going to use the name “del” because I am American (I will explain what my birthplace has to do with the name in later videos). Anyway, the del function is a vector that has the derivative of each direction pointing in each direction, ie.
\overrightarrow{\nabla} = \frac{d}{dx}\hat{x} + \frac{d}{d\text{y}}\hat{\text{y}} + \frac{d}{dz}\hat{z}
Let’s start with the divergence (\overrightarrow{\nabla}\cdot), as it is the easiest. Remember with the dot product you multiply both x components plus both y components plus both z components.
Work = \overrightarrow{F}\:\cdot\:\overrightarrow{d}= F_x x +F_{\text{y}}\text{y}+F_z z
Therefore, if you take the divergence of a vector \overrightarrow{A}, you get:
\overrightarrow{\nabla}\:\cdot\overrightarrow{A} = \frac{dA_x}{dx} + \frac{dA_{\text{y}}}{d\text{y}} + \frac{dA_z}{dz}
Or how much the vector \overrightarrow{A} changes in all directions as you move in space away from a point.
Let’s look at the Maxwell’s equations that use divergence, both of which are called “Gauss’s Law” as they are usually given:
\overrightarrow{\nabla}\:\cdot\overrightarrow{E} = \rho/\epsilon_0 \:\:\text{ and }\:\: \overrightarrow{\nabla}\:\cdot\overrightarrow{B} = 0
Now before I go into what they mean, I feel like I need to give a tiny backstory. In 1784 a French engineer named Charles Augustin Coulomb published a description of his very sensitive torsional scale that was described as an, “all too unsteady twisting machine.”[1] The following year Coulomb used his device to experimentally determine a “Fundamental Law of Electricity” that the electrical force is dependent on the size of the two charges and inversely dependent on the distance squared.[2] Then, the year after that, 1786, Coulomb used his machine to determine that charges only reside on the outside of a conductive material and do “not penetrate into its interior parts.”[3]
Then 51 years later, in 1837, Michael Faraday was inspired by Coulomb to study conductive containers. That is why Faraday built a giant conductive cage, big enough to live in, which is how he discovered that not only did the inner surface not have any excess charge, but also, if electrical experiments were conducted outside the cage Faraday found that the conductive cage shielded him from these forces.[4]
Faraday immediately realized that what was going on was due to static electric induction – or charged objects moving charges without touching. As he wrote, “the effects are clearly inductive effects produced by electricity.”[5] Because of this experiment Faraday imagined that all charged objects emanated what he called “lines of inductive force”[6] or as he later called it “lines of electric force,”[7] similar to how he felt that magnets emanate “lines of magnetic force.”
That is why, in late 1855, and early 1856, a young mathematical genius named James Clerk Maxwell decided to make a mathematical model of Faraday’s ideas of electric and magnetic lines of force. Using fluid mechanics (tubes of incompressible fluids) as an analogy, Maxwell imagined how these lines of force would “be supplied by a source” (positive charge) and “afterwards be swallowed up by a sink” (negative charge).[8]
Maxwell then used a method that he learned from a mathematician named Carl Fredrick Gauss where he made an imaginary closed surface around the charges and measured the amount of the lines \overrightarrow{D} that pointed out of the surface or:
\oint \overrightarrow{D}\:\cdot\:d\overrightarrow{S}
Where Maxwell decided that the number of lines that emerge from a surface have to be dependent on the charge in that surface:
\oint \overrightarrow{D}\:\cdot\:d\overrightarrow{S} = Q _ {inclosed}
Maxwell then used integration by parts to show that this integral was the same as the integral of the sum of derivatives of \overrightarrow{D}over the volume inclosed, ie:
\oint \overrightarrow{D}\:\cdot\:d\overrightarrow{S} =\int\int\int\left(\frac{dD_x}{dx} + \frac{dD_{\text{y}}}{d\text{y}} + \frac{dD_z}{dz}\right)dxd\text{y}dz
Note that Maxwell didn’t know about the divergence (\overrightarrow{\nabla}\:\cdot\overrightarrow{D}) but we can recognize that he had just proven that:
\oint \overrightarrow{D}\:\cdot\:d\overrightarrow{S} =\int\int\int\left(\overrightarrow{\nabla}\:\cdot\overrightarrow{D} \right)dxd\text{y}dz
In addition, Maxwell knew the total excess charge enclosed is the same as the sum (or the integral) of the excess charge per volume(\rho) integrated over that volume \rho , ie.:
Q _ {inclosed} =\int\int\int \rho dx d\textit{y} dz
Which means that:
\oint \overrightarrow{D} \cdot d\overrightarrow{S} = Q_{inclosed}
Is the same as stating that
\int\int\int\left(\overrightarrow{\nabla}\:\cdot\overrightarrow{D} \right)dxd\text{y}dz = \int\int\int \rho dx d\textit{y} dz
Which means that:
\overrightarrow{\nabla} \cdot \overrightarrow{D} = \rho
Which is sometimes how “Gauss’s Law” is written. Note that this \overrightarrow{D} field is not the electric field, Maxwell called it the displacement field as it displaces the charges in the medium as these lines of force emerge from charged objects.[9] However, Maxwell stated that if the medium is uniform then the displacement field \overrightarrow{D}, and the electric field \overrightarrow{E} are related by a simple constant, k:
\overrightarrow{E} = k\overrightarrow{D}
which Maxwell called the “electric elasticity.”[10]
Years later, Oliver Heaviside decided that he would prefer to use a constant that he called the permittivity that was defined as the inverse of Maxwell’s “electric elasticity”. [11] If we use the Greek letter \epsilon = 1/k for the permittivity then \overrightarrow{E}=\overrightarrow{D}/\epsilon, which makes Gauss’s law to be:
\overrightarrow{\nabla}\:\cdot\:\overrightarrow{E} =\rho/\epsilon
Note that \epsilon is a constant that depends on the electric properties of the medium.
In the case of a vacuum or air, with either no or minimal electric properties you still need a constant \epsilon_0 (called the permittivity of free space) to make the units work out. This situation is so common that often we are given Maxwell’s laws for that case as if it was universal:
\overrightarrow{\nabla}\:\cdot\:\overrightarrow{E} =\rho/\epsilon_0
Even though it only works in a vacuum or air at room temperature and pressure. Heck even in air this equation isn’t totally accurate as the real equation is actually:
\overrightarrow{\nabla}\:\cdot\:\overrightarrow{E} =\rho/(1.0006\epsilon_0)
But most scientists figure that is close enough to \epsilon_0to not bother with the factor of 1.0006.
However, don’t get distracted by the constant, what is more important is what this equation is saying, which is if you have a volume with more positive charges than negative, then the divergence of the Electric field will be positive (\overrightarrow{\nabla}\cdot\overrightarrow{E}>0), and more lines of force will emerge from a surface around that volume than enter it. If you have more negative than positive then the divergence is negative (\overrightarrow{\nabla}\cdot\overrightarrow{E}<0) and more lines of force will enter the surface, and if the volume has equal amounts of negative and positive, then the total number of lines entering equals the lines exiting and the divergence is zero (\overrightarrow{\nabla}\cdot\overrightarrow{E} =0).
In addition, Maxwell also discovered from manipulating other equations that independent of the medium a magnetic field, \overrightarrow{B}:
\overrightarrow{\nabla}\:\cdot\:\overrightarrow{B} = 0
Which means that although you can have more positive electric charge than negative or more negative than positive in a volume, for magnets you cannot have more Norths than Souths or more Souths than Norths. So, when you follow the Magnetic field lines, they cannot converge on a point.
Even on the North part of a bar magnet, where the magnetic field lines look like they are emerging from the North like a point but the lines sneak back to the North inside the bar magnet leaving the divergence of the magnetic field equal to zero. All of which means that magnetic fields don’t diverge or converge on a point, all they can do is curl around in circles which brings us to…
Part 3: Maxwell’s Equations with Curl
First, let us start with the cross product with del or the curl, curl\overrightarrow{A}=\overrightarrow{\nabla}\times\overrightarrow{A}. Let me refresh your memory for how the cross-product works [between a displacement \overrightarrow{d} and a force,\overrightarrow{F}, to get the torque, \tauand you end up with three equations, one for each direction.
\tau_x = \text{y}F_z - z F_\text{y}
\tau_{\text{y}} = zF_x - x F_{z}
\tau_z = xF_{\text{y}} -\text{y} F_{x}
Or, a single, very long equation:
\tau = \overrightarrow{d}\:x\:\overrightarrow{F} = (\text{y}F_z - z F_{\text{y}})\hat{x} + (zF_x - x F_{z})\hat{\text{y}} + xF_{\text{y}} -\text{y} F_{x})\hat{z}
By analogy, if you take the curl of a vector \overrightarrow{A} , you get three equations for the three directions:
(\overrightarrow {\nabla} \times \overrightarrow{A})_x = \frac{dA_{z}}{d\text{y}} - \frac{dA_{\text{y}}}{dz}
(\overrightarrow {\nabla} \times \overrightarrow{A})_y = \frac{dA_{x}}{dz} - \frac{dA_{z}}{dx}
(\overrightarrow {\nabla} \times \overrightarrow{A})_z = \frac{dA_{\text{y}}}{dx} - \frac{dA_{x}}{d\text{y}}
Or, a single, very long equation:
curl \overrightarrow{A} = \overrightarrow{\nabla}x \overrightarrow{A} = \left(\frac{dA_{z}}{d\text{y}} - \frac{dA_{\text{y}}}{dz}\right)\hat{x} + \left(\frac{dA_{x}}{dz} - \frac{dA_{z}}{dx}\right)\hat{\text{y}} + \left(\frac{dA_{\text{y}}}{dx} - \frac{dA_{x}}{d\text{y}}\right)\hat{z}
If you think of the divergence as demonstrating how the fields change as they diverge from a point, you can think of the curl as demonstrating how much they change as they curl around in a circle around a point.
Now a tiny bit of history. In the early 1800s a Danish philosopher named Hans Christian Oersted spent almost two decades searching of ways to make physics not a “collection of fragments on motion, on heat, on air, on light, on electricity, on magnetism and who knows what else”[12] but instead he wanted to determine a way to, “include the whole universe in one system”[13] Oersted was thus delighted when in May of 1820 he found a link between electricity and magnetism when he discovered that current in a wire would move a magnetic compass needle to point around the wire. Oersted published his results in July of 1820.[14]
As soon as a French polymorph named André-Marie Ampère heard about it, Ampère immediately decided that maybe all magnets are electric. To validate his claim, Ampère decided that he wanted to model magnetism with electricity as he put it in 1820, “This action that M. Oersted discovered led me to look for … how electricity might produce all the phenomena presented by magnets.”[15]
However, Oersted’s original experiment with straight current carrying wires didn’t act anything like a bar magnet, but wires aren’t required to be in straight lines, so maybe some other orientation of wires would act like a bar magnet. Ampère thus created a pancake of wires (in what is called a planar spiral). To his delight he found that this “pancake” worked like a magnet, where one side of the pancake was a “North” and the other a “South,” depending on the direction of the current. As one pancake spiral worked like a magnet, Ampère wondered if two spirals would work like magnets with each other, which they did. Ampère therefore realized in 1820 that, “In replacing the magnet by another spiral with its current in the same direction, the same attractions and repulsions occur. It is in this way that I discovered that two electric currents attract each other when they flow in the same direction and repel each other in the other case.”[16]
But that is not all, in that same paper, Ampère then wrapped his wire into a helix around a glass tube, and to his shock, the “joint actions” of the coils made this helix (or as he called it a few years later a “solenoid”)[17] act in a way that was, “perfectly similar to the action of a magnet.”[18]
In 1826 Ampère published a collection of his work in a book titled “Theory of Electrodynamics, Uniquely Deducted from Experiment” where he found that, for an infinitely long straight wire with current i, if another straight wire is placed parallel to it of length l, and current i’, then the force between the two wires is proportional to the two currents times the length divided by the distance between them, a:[19]
F \propto \frac{ii^{'}}{a} l
In modern units the force per length is written as:
F/L = \mu \frac{l_1 l_2}{2\pi r}
Where \mu is a constant called the permeability, which is dependent on the magnetic properties of the medium.
That is why in late 1855, and early 1856, Maxwell stated that if a magnetic field in a closed curve is constant over time, it can be made to be “a measure of the quantity of the current which passes through it.”[20]
Maxwell then imagined a flat or open surface,\sum, with a perimeter \partial\sum, surrounding a current. Under Maxwell’s logic, [the integral of a magnetic field (\overrightarrow{H}) around a current is equal to the current enclosed by that surface \overrightarrow{I}:
\oint_{\partial\sum} \overrightarrow{H} \cdot d\overrightarrow{l} = \overrightarrow{I}_{enclosed}
(Note that for a line \overrightarrow{l}, d\overrightarrow{l}points along the length of the circumference, so as you are taking the dot product with \overrightarrow{H} you are measuring how much \overrightarrow{H} points around the circumference.)
But, the current enclosed is the same as the current per area (\overrightarrow{J}) integrated over the area:
\overrightarrow{I}_{enclosed} = \int \int_{\sum} \overrightarrow{J} \cdot \overrightarrow {dS}
So:
\oint_{\partial\sum} \overrightarrow{H} \cdot d\overrightarrow{l} = \int \int_{\sum} \overrightarrow{J} \cdot \overrightarrow {dS}
Now Maxwell didn’t have vector notation, so he solved the particular case where the current is only in the x direction so the area is in the y-z direction or
\oint_{\partial\sum} \overrightarrow{H} \cdot d\overrightarrow{l} = \int \int_{\sum} J_x{dydz}
Assuming that the flat surface is a tiny square of sides {dy} and {dz}, Maxwell proved that:
\oint_{\partial\sum} \overrightarrow{H} \cdot d\overrightarrow{l} = \int\int_{\sum} \left(\frac{dH_z}{d\text{y}}-\frac{dH_{\text{y}}}{dz}\right) d\text{y}dz
This is how Maxwell derived that:
J_x = \frac{dH_z}{d\text{y}}-\frac{dH_{\text{y}}}{dz}
And, by analogy how he got that:
J_y = \frac{dH_z}{dx}-\frac{dH_{x}}{dz}
J_z = \frac{dH_x}{d\text{y}}-\frac{dH_{\text{y}}}{dx}
Which, as we know vector notations, is clearly just:
\overrightarrow{J} = \overrightarrow{\nabla} \times \overrightarrow{H}
Note that Maxwell liked to use the \overrightarrow{H}-field instead of the magnetic field \overrightarrow{B} where they are related by the simple ratio for a uniform medium: \overrightarrow{B} = \mu\overrightarrow{H}, where \mu is the permeability, which is the same constant that relates the force and the currents divided by the distance, so Maxwell was really saying that:
\overrightarrow{\nabla} \times \overrightarrow{B} = \mu \overrightarrow{J}
Then, starting in 1862, Maxwell decided that there is another reason that a magnetic field could have a curl aside from the motion of charges, it could come from a changing electric field. His logic was that the displacement field \overrightarrow{D} polarizes the molecules and the “ether” so he boldly decided that “a variation of displacement is equivalent to a current.” That is why Maxwell derived “Ampere’s Law with Maxwell’s Addition” to be:
\overrightarrow{\nabla} \times \overrightarrow{H} = \left(\overrightarrow{J} + \frac{d\overrightarrow{D}}{dt} \right)
(note that he had messy constants in 1862, he fixed them in 1864, but it doesn’t matter as it just changes the value of the constant). Anyway, we can multiply both sides to by \mu to get:
\overrightarrow{\nabla} \times \mu\overrightarrow{H} = \mu\left(\overrightarrow{J} + \frac{d\overrightarrow{D}}{dt} \right)
In a uniform medium, by replace \mu\overrightarrow{H} with \overrightarrow{B}, and \overrightarrow{D} with \epsilon\overrightarrow{E} with to get:
\overrightarrow{\nabla} \times \overrightarrow{B} = \mu \overrightarrow{J} + \mu \epsilon \frac{d\overrightarrow{E}}{dt}
Where \mu (the permeability) is the constant that is dependent on the magnetic properties of the medium, and \epsilon (the permittivity) is the constant dependent on the electric properties of the medium. And, once again, in the case of a vacuum or air, these constants are still needed for the units and we use the magnetic permeability of free space, \mu_0, and the electric permittivity of free space \epsilon_0 to get:
\overrightarrow{\nabla} \times \overrightarrow{B} = \mu_0 \overrightarrow{J} + \mu_0 \epsilon_0 \frac{d\overrightarrow{E}}{dt}
Which is, once again, how this law is often taught.
Now don’t get bogged down in the constants, what is important is this equation is saying in a fundamental way is that the [curl of the magnetic field is either due to a current that the magnetic field is circling, or from a changing electric field that is inducing a perpendicular magnetic field around it.
One more equation, and one more short backstory. In 1825, an English shoemaker named William Sturgeon accidentally discovered that Ampere’s coils of wire with current in them worked as a much stronger magnet if they were wrapped around iron, where they got the new name of electromagnets. Then, in 1831, Michael Faraday was playing with electromagnets trying to get the iron to move the electricity from one coil to another remotely.
He then discovered that he could move electricity without touching but only when he changed the strength of the electromagnet. In other words, when he changed the current of an electromagnet a neighboring coil would have current induced in it, something he called “electromagnetic induction.”[21] After Faraday realized that the trick to inducing current in the wire was changing a magnet, he found that if he moved a bar magnet into or out of a coil, he could also induce current in the wire.
After this Faraday had the insight to create the idea of magnetic fields (or what he called “magnetic lines of force”) and to describe that the rule of induction is that current is induced in a circle when these “magnetic lines of force” are broken or “cut” by the wire.]
Fascinatingly, Maxwell wasn’t initially particularly interested in modeling this phenomenon. Instead, in 1862, he was modeling how to determine the “work done on the vortex [ie. the molecule] in unit of time,”[22] for magnetic and electric fields and bumped into it almost by accident, and did not give it a name. However, by his famous paper of 1864, Maxwell acknowledged that it was a law, “deduced from the laws of induction.”[28] Anyway, what Maxwell derived in 1862 and 1864 is:
\overrightarrow{\nabla} \times \overrightarrow{E} = -\mu\frac{d\overrightarrow{H}}{dt}
But since \overrightarrow{B} = \mu\overrightarrow{H}, Maxwell was also deriving:
\overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{d\overrightarrow{B}}{dt}
This equation is saying [that a changing magnetic field will create an electric field that curls around the changing magnetic field, just as Faraday experimentally discovered over 30 years earlier.
Let me explain the negative sign, if the curling electric field is in a conductor it will create its own circling current which, according to “Ampère’s law” will make its own curling magnetic field. The negative sign means that the magnetic field from the current will work against the changing magnetic field to push against the changing external magnetic field. If the negative sign wasn’t there, you could not only build a perpetual motion machine, you could build one that constantly accelerates. Although that might be convenient, that is not physically possible as it violates conservation of energy. Ergo, we need a negative sign so that the induced current works against the change in magnetic field and.. you know… the laws of thermodynamics still work. (Which always reminds me of my favorite “Simpson’s” clip:[23])
Now that I have gotten to all four equations, I want to take a moment to say part of why they are important. Specifically, I want to talk about what they mean in regards to magnets and to light. Which brings me to
Part 4: Maxwell’s Laws and Magnets
Let us look at 2 of Maxwell’s laws again in “free space” as they say:
\overrightarrow{\nabla} \cdot \overrightarrow{E} = \rho/\epsilon_0
\quad\overrightarrow{\nabla} \times \overrightarrow{B} = \mu_0 \overrightarrow{J} + \mu_0\epsilon_0 \frac{d\overrightarrow{E}}{dt}
The first equation (Gauss’s law) is showing that if you have excess of charged particles (\rho) then an Electric field will diverge from those charges. The second equation (Ampère’s Law with Maxwell’s addition) states that you will have a curling magnetic field induced if either you have charges moving (\overrightarrow{J}) or you have a changing Electric field (\frac{d\overrightarrow{E}}{dt}).
Now compare that to the other 2 Maxwell’s laws:
\overrightarrow{\nabla} \cdot \overrightarrow{B} = 0
\quad \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{d\overrightarrow{B}}{dt}
Where the first equation (Gauss’s Law for magnets) is showing that there are no magnetic particles (\rho_{magnet}=0), and the second equation (Faraday’s Law) states that a curling electric field can be induced by a changing magnetic field but not a magnetic current (\overrightarrow{J}_{magnet}=0) because, as there are no magnetic particles, there is also no such thing as magnetic current.
In this way, what Maxwell’s laws are doing is mathematically supporting Ampère’s theory. According to Maxwell’s equations, there are no magnetic particles, all magnets and magnetic fields are due to the action of electric particles. Take a moment to stop and think how profound a statement that is.
Electromagnetic forces are considered one type of force for a reason, it is because both electric AND magnetic forces are at their core electric.
Not that it has stopped people from trying to find a magnetic monopole, after all, who wouldn’t want to be the person who found an addendum to a fundamental equation?
OK, if you feel like your brain is exploding I have more. Ready?
Part 5: Maxwell’s Laws and Light
Part of the reason that Maxwell added that changing displacement current to his equation for curling magnetic fields in 1862 is because 5 years earlier in 1857 a German team named Dr. Weber and Kohlrausch experimentally measured the ratio of the electric constant k where (k =1/\epsilon_0) to the magnetic constant \mu_0 and found that, for the right set of units[24]:
\sqrt{\frac{k}{\mu_0}} = \frac{1}{\sqrt{\mu_0\epsilon_0}} = 3.1 \times 10^8 m/s
Which seemed close enough to the speed of light to be considered… the speed of light.
This is why in “free space” “Ampère’s Law with Maxwell’s addition” is sometimes written as
\overrightarrow{\nabla}x \overrightarrow{B} = \mu_0 \overrightarrow{J} +\frac{1}{c^2} \frac{d\overrightarrow{E}}{dt}
where c is the speed of light. Now, let’s look at Maxwell’s curling equations again in a vacuum or air:
Faraday’s Law: \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{d\overrightarrow{B}}{dt}
Ampere’s Law with Maxwell’s Addition: \overrightarrow{\nabla}x \overrightarrow{B} = \mu_0 \overrightarrow{J} +\frac{1}{c^2} \frac{d\overrightarrow{E}}{dt}
In the specific case where there are no with no electric currents then \overrightarrow{J} =0 you end up with:
\overrightarrow{\nabla}x \overrightarrow{E} = \frac{d\overrightarrow{B}}{dt}
\overrightarrow{\nabla}x \overrightarrow{B} = \frac{1}{c^2}\frac{d\overrightarrow{E}}{dt}
In this case, a changing magnetic field induces a curling electric field and a changing electric field induces a curling magnetic field times the sp1eed of light squared. With this, Maxwell felt that he had mathematical proof that Faraday’s crazy idea from 1845 was correct, light is a vibration of electric and magnetic lines of force!
But also, Maxwell proved that Faraday was incorrect, as Faraday thought that light might be a longitudinal wave and Maxwell’s wave only worked for electromagnetic waves where the electric field is perpendicular to the magnetic field and both are perpendicular to the direction of wave propagation. Or, as Maxwell put it in 1862, “the velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of Mr. Kohlrausch and Weber, agrees so exactly with the velocity of light… that we can scarcely avoid the inference that light consist in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena”[25]
Note that Faraday didn’t believe in the aether and Maxwell did, but Maxwell’s equations work either way.
With these laws, Maxwell took Faraday’s ideas of curving electric and magnetic lines of force and put them in mathematical form so that you could get all of the previous equations of force and induction from Faraday’s fields. Also, in the case of a vacuum, Maxwell demonstrated that these laws would describe an undulating electric and magnetic wave that moves at the speed of light, which seemed pretty conclusive to him that light WAS an electromagnetic vibration in space.
This is why people have T-shirts that start with “And God said” then they display Maxwell’s equations and end with the line “…and there was light.” (A shirt that might have offended the devout Maxwell if he could see it).
This was to have MAJOR implications for the development of science. Let me give two examples. In September of 1887, a young German scientist named Heinrich Hertz experimentally validated Maxwell’s view of light when he realized that something called a “spark gap” generator would produce bursts of vibrating electricity that vibrates far slower than visible light.
Even though it wasn’t visible, Hertz felt that the vibrating electricity should, according to Maxwell, also create light, just invisible light. Therefore, Hertz added an antenea to the generator and “caught” the spark in a receiver a distance away and determined that these electromagnetic waves moved at the speed of light. These “hertzian waves” were eventually called radio waves and soon people were sending signals far distances wirelessly using these waves, and the wireless revolution began. Hertz, however, didn’t see the practical applications of it, and instead in 1892 wrote that, “the object of these experiments was to test the fundamental hypotheses of the Faraday-Maxwell theory.” According to Hertz (and to most other scientists) the discovery of radio waves was a complete validation of Maxwell’s laws and, “the result of the experiments is to confirm the fundamental hypotheses of the theory.”[29] In later years, that comment has been exaggerated to be a conversation where Hertz was asked about the purpose of his new wave and responded that they were, “of no use whatsoever, this is just an experiment that proves Maestro Maxwell was right – we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there.”[26]
I can mention so much more but I would be remis if I didn’t mention how in 1905 a young patent clerk named Albert Einstein wrote a friend that he had an “amusing and seductive” about the relativity relations that he had just discovered. Einstein wrote that he had realized that the, “relativity principle, in association with Maxwell’s fundamental equations, requires that mass be a direct measure of the energy contained in a body.”[27] In other words, Einstein stated that his E = mc2 is a direct consequence of Maxwell’s laws.
So that was my description of Maxwell’s Laws, hope you found it helpful. I have already finished my script for my second video on a deep dive of Maxwell’s laws, this one on the amazing life of a mathematical scientist named William Hamilton, how he created something called quaternions, and how quaternions work to inspire vector algebra, so make sure to click on subscribe and the bell thingy because that is coming very soon next time on the lightning tamers!
References:
[1] John Munro, Pioneers of Electricity: Or, Short Lives of the Great Electricians (London: Religious Tract Society, 1890), p. 81 https://www.google.com/books/edition/Pioneers_of_Electricity_Or_Short_Lives_o/SAgJAAAAIAAJ?hl=en&gbpv=0
[2] Coulomb, C “Second Memoire” p. 611 Mémoires sur l’électricité et la magnétisme : Coulomb, C. A. (Charles Augustin), 1736-1806 : Free Download, Borrow, and Streaming : Internet Archive
[3] Coulomb “A Fourth Memoir on Electricity, in which two principal Properties of the Electric Fluid are Demonstrated” The Monthy review, or, literary journal vol. 81 (1789), 604 In French (p. 69):.Mémoires sur l’électricité et la magnétisme : Coulomb, C. A. (Charles Augustin), 1736-1806 : Free Download, Borrow, and Streaming : Internet Archive https://www.google.com/books/edition/The_monthly_review_or_literary_journal/X5tKAAAAcAAJ?hl=en&gbpv=1&dq=“A+Fourth+Memoir+on+Electricity,+in+which+two+principal+Properties+of+the+Electric+Fluid+are+Demonstrated”&pg=PA604&printsec=frontcover
[4] Michael Faraday, “Experimental Researches in Electricity – Series 11” (Dec 21, 1837) Proceedings of the Royal Society vol 128 (1838), p. 5 https://doi.org/10.1098/rstl.1838.0002
[5]Michael Faraday, “Experimental Researches in Electricity – Series 11” (Dec 21, 1837) Proceedings of the Royal Society vol 128 (1838), p. 20 https://doi.org/10.1098/rstl.1838.0002
[6] Michael Faraday, “Experimental Researches in Electricity – Series 11” (Dec 21, 1837) Proceedings of the Royal Society vol 128 (1838), p. 20 https://doi.org/10.1098/rstl.1838.0002
[7] Faraday, M (Nov, 1845) “19th Series” Experimental Researches in Electricity: Vol 3 (1855) p. 2 https://www.google.com/books/edition/_/9m-FAAAAIAAJ?gbpv=0
[8] Maxwell, James Clerk “On Faraday’s Lines of Force” (read Dec 1855, Feb 1856) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 162 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0
[9] Maxwell “On Physical Lines of Force: Part III” (read Jan, Feb, 1862) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 491 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0
[10] Maxwell “A Dynamical Theory of the Electromagnetic Field” (Read Nov, 1864) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 560 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0
[11] Heaviside started talking about this in 1884, named the permittivity in 1887, but his recap written in 1891 is the most clear IMHO: Heaviside (Feb 13, 1891) “Electric and Magnetic Force” found in Electrical Papers Vol. 1 (1892) p. 21 https://www.google.com/books/edition/Electromagnetic_Theory/_Zk3AAAAMAAJ?hl=en&gbpv=0 (I will go into this more in my next video)
[12] Hans Christian Oersted, Selected Scientific Works of Hans Christian Oersted (2014) p. 164 https://www.google.com/books/edition/Selected_Scientific_Works_of_Hans_Christ/kVYABAAAQBAJ?hl=en&gbpv=0
[13] Robert M. Brian et al., eds, Hans Christian Oersted and the Romantic Legacy in Science: Ideas, Disciplines, Practices (New York: Springer-Verlag, 2007), 2. https://www.google.com/books/edition/Hans_Christian_Ørsted_and_the_Romantic/yY6D_FHNiHgC?hl=en&gbpv=0
[14] Hans Oersted, The Discovery of Electromagnetism Made in the Year 1820 by H.C. Oersted (1920), p. 5 https://www.google.com/books/edition/The_Discovery_of_Electromagnetism_Made_i/1A82AQAAMAAJ?gbpv=0
[15] Ampère, “Suite du Mémoire sur l’Action mutuelle“ Annales de Chimie et de Physique vol 15 (1820) p. 210 https://babel.hathitrust.org/cgi/pt?id=ien.35556014127658&view=1up&seq=212&skin=2021 A. K. T. Assis and J. P. M. C. Chaib, Ampère’s Electrodynamics (Berlin: Apeiron, 2015), p. 55 https://archive.org/details/AmperesElectrodynamics
[16] Ampère, “Suite du Mémoire sur l’Action mutuelle“ Annales de Chimie et de Physique vol 15 (1820) p. 210 https://babel.hathitrust.org/cgi/pt?id=ien.35556014127658&view=1up&seq=212&skin=2021 A. K. T. Assis and J. P. M. C. Chaib, Ampère’s Electrodynamics (Berlin: Apeiron, 2015), p. 62 https://archive.org/details/AmperesElectrodynamics
[17] Ampère, “Extrait d’un Memoire sur les Phenomenes” Annales de Chimie et de Physique vol 26 (1824) p. 135 https://babel.hathitrust.org/cgi/pt?id=hvd.hx3dwa&view=1up&seq=171&skin=2021 also A. K. T. Assis and J. P. M. C. Chaib, Ampère’s Electrodynamics (Berlin: Apeiron, 2015), p. 173-4 https://archive.org/details/AmperesElectrodynamics
[18] “Suite du Mémoire sur l’Action mutuelle“ Annales de Chimie et de Physique vol 15 (1820) p. 208-9 https://babel.hathitrust.org/cgi/pt?id=ien.35556014127658&view=1up&seq=212&skin=2021 also A. K. T. Assis and J. P. M. C. Chaib, Ampère’s Electrodynamics (Berlin: Apeiron, 2015), p. 82 https://archive.org/details/AmperesElectrodynamics
[19] Ampère “Theory of Electrodynamics, Uniquely Deducted from Experiment” (1826) p. 68 In French: https://books.google.com/books?id=LzcVAAAAQAAJ&printsec=frontcover#v=onepage&q&f=false p. 386 translated into English: of A. K. T. Assis and J. P. M. C. Chaib, Ampère’s Electrodynamics (Berlin: Apeiron, 2015) https://archive.org/details/AmperesElectrodynamics/page/n77/mode/2up
[20] Maxwell “On Faraday’s Lines of Force” (read Dec 1855, Feb 1856) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p.194 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0
[21] Michael Faraday, Experimental Researches in Electricity, Volume 1 (London: R. and J. E. Taylor, 1839), 32.
[22] Maxwell “On Physical Lines of Force: Part III” (read Jan, Feb, 1862) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 475 (eq. 54) https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0
[23] https://www.youtube.com/watch?v=tuxbMfKO9Pg
[24] Weber and Kohlrausch “Reduction of the Measures of Intensity of Currents to Mechanical Units” found in the Appendix of Auguste de la Rive “A Treatise on Electricity” (1858) p. 747 https://www.google.com/books/edition/A_Treatise_on_Electricity/yo9PAAAAYAAJ? (Note that they got the half value of 1.5537 m/s due to the units used, but Maxwell easily changed it to twice the value with different units).
[25] Maxwell “On Physical Lines of Force: Part III” (read Jan, Feb, 1862) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 500 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0
[26] Quoted in Norton Dynamic Fields and Waves (2000) p. 83 https://www.google.com/books/edition/Dynamic_Fields_and_Waves/XRRMxjr24pwC?hl=en&gbpv=0
[27] Albert Einstein to Conrad Habicht (June/Sept 1905) Document 28 in Einstien: Correspondence Volume 5: 1902-14 (English Translation) p. 20-1 https://einsteinpapers.press.princeton.edu/vol5-trans/43
[28] Maxwell “A Dynamical Theory of the Electromagnetic Field” (Read Nov, 1864) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 534 (equation is on page 556) https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0
[29] Hertz, Heinrich Electric Waves (first published 1892, reprint 1962) p.20 https://www.google.com/books/edition/Electric_Waves/8GkOAAAAIAAJ?hl=en&gbpv=1
Thank you for including all this material! I’d like to mention another reference that provides a good intuitive grasp of the math: Div, Grad, Curl, and All That, by H. M. Schey. This has been my go-to for many decades.