Geometric Optics

Geometric Optics for EM wave

EM wave propagation An EM wave goes through the athmosphere of Earth (considered inhomogeneous) having an index of refraction of $n(z)$. This index of refraction depends on altitude $z$ as: $n(z) =1+a*exp(-\frac {z}{H})$ where $a =3*10^{-4}$ and $H = 9 km$ is the altitude. Please find a system the  equations for the position and wave

Hermitian

Hermitian or not?

Hermitian definition Please find if $(\overrightarrow p*\overrightarrow r)\overrightarrow L$ is hermitian or not. Please state the definition of a hermitian matrix? Which are the matrixes or $\hat r$, $\hat p$ and $\hat L$? We write $\hat p=-i\hbar\nabla =-i\hbar (\frac{\partial }{\partial x}, \frac{\partial }{\partial y}.\frac{\partial }{\partial x})$ $\hat r= r= (x,y,z)$ and thus $\hat r*\hat p=-i\hbar

Flight Centrifuge

Flight Centrifuge

The flight centrifuge A flight centrifuge was designed  to simulate the  flight in a space vehicle. The arm which is $d=40 ft$, rotates about axis A-A. The cockpit rotates itself about the axis C-C. Inside the cockpit the seat rotates about the axis B-B.  Motors control these rotations and different situations of flight are simulated. 

EM theory

Electron in classical EM theory

Classical electron We know from theory that a charge (an electron) that is accelerating in an atom, gives out energy at a rate of $W=\frac{2}{3}\frac{a^{2}e^{2}}{c^{3}}  (ergs/s)$  where $c$ is the light speed and $a$ is the particle acceleration. Let us say that at the initial moment $t=0$, the radius of the H atom is 1

Falling Cat

Falling Cat

You have a symbolic falling cat. This cat is made up of two cylinders, a heavy one (the body) and a smaller one (the tail) as in the figure. You know the mass and dimensions of the falling cat and its tail. The tail rotates around the cat body to allow the cat to fall

Hydrogen

Electron position in H atom

For the ground state of Hydrogen please find the most probable value of  the electron position $r$. The probability that the electron is found between $r$ and $r+dr$ is $P=|R_{nl} (r) |^2*r^2 dr$ Here above $R_{nl} (r)$ is the radial function for hydrogen in the n, l state and $r^2*dr$ comes from the volume element

Empty Sphere Field

Empty Sphere field and potential

An empty sphere is charged. (the inner and outer radii of the sphere are $R_0$ and $R$). Please find the empty sphere field and potential as functions of $R$. What is the difference between the cases when the sphere is free in space and when the sphere is grounded? When the sphere is not grounded,