In our Lecture-I we discussed the phenomena of aberrations that arise because of a discrepancy of a first order theory and the 3rd order theory as depicted by the Maclaurin series; where we saw that first order theory represents the so called paraxial optical systems. Please have a look of the linked article to get a basic view of the ground on which we are discussing this topic. At-least going half-way through the lecture and stopping short of the derivation will do well.
We discussed that there are two kind of aberrations. Monochromatic and Chromatic. As the name suggests the monochromatic aberrations are a result of the discrepancy when we considered our incoming ray to be having a single wavelength of light. The chromatic rays on the other hand can have multiple colors or wavelength of light. The monochromatic aberrations are also called as Seidel or Primary aberrations and we will shed more light on them today. The chromatic aberrations were dealt in greater detail — eg the derivations pertained to the chromatic aberrations. We did so because the chromatic aberrations are simple to understand. More…
All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.
First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.
fixed: which changes only if estimated wrong.
That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics. More…
Optical systems are studied under two assumptions
object point does not lie far away from the axis of the optical system.
rays taking part in image formation make a small angle with the axis of the optical system.
The domain of optics where above two assumptions are valid is called as Paraxial optics. Paraxial systems are highly idealized and in reality do not perfectly represent the situation. The consequential errors in image reconstruction are known as aberrations.
The paraxial assumption can be represented by truncating at the first term of the polynomial expansion of the sin function by the Maclaurin series. More…
Finally I am successful in calculating pi value — less than 0.3% error, by using random number generation. Although my computer needs some fixation on its compiler or path definition etc, there are very good online compilers which helps in testing and running c++ codes: try the given link.
Computing the value of pi using std::rand()
Enter number of trials: 10000
Enter number of random (x,y) points per trial: 10
pi = 3.14376 +- 0.00519107
average – exact = 0.00216735
CPU time = 0.004027 secs
Here is the code I found by searching a good deal on the web. Yes I did tinker around but only because my own compiler (Turbo C++ on windows 10, 64 bits) was throwing some exceptions on the included headers.
using namespace std;
double pi_estimate(const unsigned long points) More…
A long and technical discourse on Quantum Wave Function.
A 64 slide presentation styled discourse on the Quantum Wave Function. It consists of detailed solution of 5 important and interesting problems, apart from a threadbare discussion of the concepts.
We need to understand first that Force can be categorized into two types. One is called tangential or collinear force. This component of the force is always along the direction of motion and changes speed of an object. It can change direction once the velocity of the object has become zero. Its NOT centripetal force. It can never make an object go in a plane or 3 D trajectory, as the motion is limited to only one dimension. The object can only go back and forth.
Now look at the other component. Its called a radial force. Its always perpendicular to the direction of motion. This force is called centripetal force, always. Note that its different from what we call central forces.
In consequence, both tangential and radial forces can be central. More…