Fringe Radius Relation for Small Times t

[Ciato]

1. Show that for t<<R, the radius r of a fringe is related to r= (2TR)^1/2

I'm not sure what the relevant equations are, that would be the problem. A gentle shove in the right direction would be appreciated, I've been looking for exactly how to prove it and I can't. ^_^ Thanks for any help!

 

[Doc Al]

This relationship is true in general, not just for fringes. Examine the geometry of the Newton's rings set up: A spherical convex lens placed upon a flat piece of glass. Draw a diagram with the quantities r, t, and R indicated.

 

[baba89]

Firstly, let's define the terms in the equation:
r - radius of a fringe
T - time
R - fringe radius

To prove the relation r= (2TR)^1/2, we can start by looking at the definition of a fringe radius. A fringe is an area of interference between two waves, where the amplitude of the waves is either reinforced or cancelled out. The fringe radius is the distance from the center of the interference pattern to the point where the amplitude of the waves is at its maximum or minimum.

Now, let's consider the case of small times t. This means that the time T is much smaller than the fringe radius R. In other words, the time it takes for the waves to interfere is much smaller than the distance between the interference pattern and the center.

In this case, we can assume that the waves are essentially traveling in a straight line towards the center of the interference pattern. This is known as the "near field" approximation. In this approximation, the distance between the waves is much smaller than the distance between the waves and the center of the interference pattern.

Using the near field approximation, we can apply the Pythagorean theorem to the distance between the waves and the center of the interference pattern. This distance is equal to the fringe radius R. We can write this as:

R= (r^2 + (2TR)^2)^1/2

Where r is the distance between the waves.

Now, since we are considering small times t, we can assume that the distance between the waves r is much smaller than the fringe radius R. This means that we can neglect the r^2 term in the above equation.

R= (2TR)^1/2

This is the same relation as r= (2TR)^1/2. This means that for small times t, the radius of a fringe is related to r= (2TR)^1/2.

In conclusion, we have shown that for t<<R, the radius r of a fringe is related to r= (2TR)^1/2. This relation holds true for the near field approximation, where the time T is much smaller than the fringe radius R.