Could someone clear up the paper on casimir effect

In summary, the conversation discusses the calculations and integration involved in a change of coordinates from Cartesian to polar, specifically in the context of a journal article. The mention of substitution leading to an expression of l^2/pie^2 and the presence of xdx at (3.7) is explained as being related to the use of radial variables and the integration over positive values of nx and ny.
  • #1
epislon58
49
1
Hello,

could someone please explain to me what happens between (3.6) and (3.7). In specific, I don't understand how substitution results in l^2/pie^2 and also what do they mean integrating over radial angle. And the x next to the dx at (3.7), where did it come from.

http://aphyr.com/data/journals/113/comps.pdf

Thank you!
 
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  • #2
anyone?
 
  • #3
I didn't look that carefully but xdx is because x is a radial variable, so in the change of coordinates from Cartesian to polar, we usually get rdrdθ, where r is the Jacobian for the change of variables - they use x where I wrote r. The dθ doesn't appear becuase it has been integrated over already. Originally, they integrated over positive values of nx and ny, which are the original Cartesian coordinates. This is the first quadrant, ie. θ=0 to θ=∏/2.
 
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FAQ: Could someone clear up the paper on casimir effect

What is the Casimir Effect?

The Casimir Effect is a physical phenomenon that occurs when two uncharged metal plates are placed in close proximity to each other. This causes a force to be exerted between the plates, which is believed to be caused by the fluctuations of virtual particles in the quantum vacuum.

How was the Casimir Effect discovered?

The Casimir Effect was first predicted in 1948 by Dutch physicist Hendrik Casimir. He proposed that the energy density of the vacuum between two plates would be slightly lower than the energy density of the vacuum outside the plates, resulting in a net force between them.

How is the Casimir Effect measured?

The Casimir Effect is typically measured using a device called a Casimir force balance. This involves suspending two plates in a vacuum and measuring the force between them. Other methods, such as using atomic force microscopy, have also been used to measure the Casimir Effect.

What are the practical applications of the Casimir Effect?

One potential application of the Casimir Effect is in nanotechnology, where it could be used to control the movement of small particles and devices. It has also been suggested that the Casimir Effect could be used to create a new type of energy source, although this is still in the early stages of research.

Can the Casimir Effect be fully explained by current theories?

The Casimir Effect is still an active area of research and there is still much to be understood about its underlying mechanisms. While it can be partially explained by current theories, there are still some discrepancies that need to be resolved. Scientists continue to study the Casimir Effect in order to gain a better understanding of this intriguing phenomenon.

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