- Maxwell equations
- The Schrödinger equation
- general relativity field equation
Maxwell’s equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in a vacuum.
A introduction part 1 of electromagnetism is made here:
As Flux density = B = ϕ/A ( Wbm-2 )
so Flux = field strength per unit area ϕ= BA (unit Wb)
B (unit tesla)
Flux linkage ( =BAn ) in a coil with N coils
- where s the voltage across the device
- Maxwell came up with the right hand rule (or cork screw rule) to find out how magnetic field goes around a piece of wire with a current in it.. Khan academy presents this law here:
- Key notations:
- is a surface integral over the surface Σ,
- Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated.
- the magnetic field, B, a pseudovector field, each generally having a time and location dependence.
The sources are:
- the total electric charge density (total charge per unit volume), ρ, and
- the total electric current density (total current per unit area), J.
The universal constants appearing in the equations (the first two ones explicitly only in the SI units formulation) are:
In the differential equations,
- the nabla symbol, ∇, denotes the three-dimensional gradient operator, del,
- the ∇⋅ symbol (pronounced “del dot”) denotes the divergence operator,
- the ∇× symbol (pronounced “del cross”) denotes the curl operator.
- is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed
- is a surface integral over the surface Σ,
- electric field, E, a vector field,ss
- the magnetic field, B, a pseudovector field
- Ω is any fixed volume with closed boundary surface ∂Ω, and
- Σ is any fixed surface with closed boundary curve ∂Σ,
The schrödinger equation
Calculates the probability for a particle (e.g. electron) to be in a certain position. Is share by Brian Green in this World Science Festival video.
It is all about finding the different energies that a particle can have.
We’re looking at things that can have more than one answer! You might have studied in high school that atoms have energy levels. The Schrödinger’s Wave Equation lets us calculate what these energies are.
Let’s start with thinking what is Kinetic Energy…
It is defined as:
It turns out, talking about velocity isn’t very useful. So, we change this equation to make it depend on momentum p
We can think of particles as waves, atleast at tiny scales where we need to use Quantum Mechanics!
To help us move between these 2 ways of thinking about matter, we can use De Broglie’s Equation
This theory set the basis of wave mechanics. It was supported by Einstein, confirmed by the electron diffraction experiments of G P Thomson and Davisson and Germer, and generalized by the work of Schrödinger. ( Wiki )
Now we don’t see matter around us in our everyday lives to be behaving like waves because Plank’s Constant is absolutely tiny! (
But hold on! De Broglie’s Equation is useful when we are dealing with minute particles like protons and electrons 🙂
We also have something called It’s is related to the Plank’s Constant as
Now, it is going to be useful to talk about and the wave number instead of and …
is the wave number. We did to avoid having running around in our equations. It just turns out to be easier to work with!
We also know that is change depending upon the energy. Since depends only on , the factors that can affect can also affect . In this case, it is Energy!
We use Wave Functions to encode information about waves. These are just equations with a few special properties. They look something like this…
This is just another way to write sines and cosines. (Waves are just combinations of sines and cosines!)
Here’s the reason for writing this in this weird format…
This type of problem is called an Eigen Value Problem.
is called an Eigen Function.
is an Operator.
Eigen Function of operators are functions which return themselves and a new Eigen Value after the operator is used on them.
As you might have guessed by now, on differentiating this we get the same thing back multiplied by some constants (differential of the argument)
Let’s do this again!
Here, is the new Eigen Value returned (that’s the extra bit)
Now let’s make a Kinetic Energy Operator (we want the energy E and we are assuming a low potential)…
We know there are gonna be a bunch of answers so we are going to use our Wave Function to generate those answers and we also have a way of writing
So on putting it all together we get…
Notice it is partial derivative instead of the usual one. This just means that there is more than one variable in the function we treat as a constant when differentiating.
What we just did is valid for a one dimensional particle that is time independent and is not in a potential.
Let’s generalize this equation a little…
For a photon,
We’re now gonna assume it to be true for particles. It turns out to be a good assumption…
We know that there is gonna be a set of energies so we want to know the Eigen value equation.
This time we’ll find an E by generating an ω
Let’s look at the wave function to do that…
As you might have guessed it by now, we’re gonna take the derivative with respect to time to pull it out front!
o account for the minus sign(-) we will simply use -i.i=1
And that’s same as the equation we had before…
Please note that this equation is only for one spatial dimension! But we need to consider all 3 dimensions. The three dimensions in Cartesian Coordinate System are orthogonal to each other. That means, they will not interfere with each other.
Each of these contributes to the total energy.
So, in three dimensions, the time dependent wave equation becomes…
And the time independent version becomes…
This can more simplified as…
Where ∇ is what we call Nabla.
To include Potential(V) we modify it a bit further like this…
where Ĥ is what we call Hamiltonian.
is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system (this addition is the total energy of the system in most of the cases under analysis ( Wiki )
So, the full Schrödinger’s Wave Equation becomes…
general relativity field equation
First part copied from my page at www.kinberg.net/wordpress/stellan/einsteins-field-equations-of-general-relativity/
As is correctly explained in Wikipedia
- Rμν is the Ricci curvature tensor,
the Ricci tensor is the part of the curvature of spacetime that determines the degree to which matter will tend to converge or diverge in time
- R is the scalar curvature,
the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.
- gμν is the metric tensor,
- Λ is the cosmological constant,
- G is Newton’s gravitational constant,
- c is the speed of light in vacuum, and
- Tμν is the stress–energy tensor.
I found this Youtube that explains the general relativity field equation. I like it because of the good graphic representations.
As with the tensor youtube made by Dan Fleisch, this made by Eugene Khutoryansky show graphically how a tensor change depending on curvature of space. Very well done video.