- #1
qnach
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Could anyone explain how did Jackson obtain the Taylor distribution of charge distribution at the end of section 1.7 (version 3)?
I think that's correct though I find the conclusion by Jackson a bit too quick, but after some detailed analysis, I think it's correct too.Charles Link said:The print isn't real clear, but I worked the next step. (The exponent in the denominator is ## \frac{5}{2} ##. It looks like ## \frac{3}{2} ##, but correctly it should be ## \frac{5}{2} ##).
## \int\limits_{0}^{+\infty} \frac{3 a^2 r^2}{(r^2+a^2)^{5/2}} \, dr ## can be readily solved by letting ## r=a \tan{\theta} ##.
To resolve the handwaving in post 2, ## \nabla^2 \rho =(\frac{\partial^2{\rho}}{\partial{x^2}}) + ## y and z second partial terms at ## \vec{x} ##, so that ## \nabla^2 \rho =3 (\frac{\partial^2{\rho}}{\partial{x^2}}) ## at ## \vec{x} ##. This ## \nabla^2 ## term is a constant when integrating over dr. It also is useful to look at the integrals of ## \int x^2 \, d^3 r ## vs. ## \int r^2 \, d^3 r ##, etc. Upon working through all the details, I agree with his ## \frac{1}{6} ##.
That's the Landauer big-O symbol. It tells you how in some limit of a variable a function behaves. In this case it's the limit ##a \rightarrow 0##, andqnach said:Could anyone please explain the meaning of the last term of the second last equation in this page.
What does O(a^2,a^2 log a) mean? The O notation contains only one term inside the bracket.
But this guy has two inside.
The expansion of charge on page 35 of Jackson Classical Electrodynamics refers to the mathematical representation of a charge distribution using a series of point charges. This is known as the multipole expansion and is used to simplify the calculation of electric and magnetic fields.
The expansion of charge is important because it allows us to approximate complex charge distributions with simpler point charge distributions. This makes it easier to calculate the electric and magnetic fields, which are crucial in understanding the behavior of electromagnetic waves and particles.
The expansion of charge is closely related to the concept of multipole moments. Each term in the expansion represents a different multipole moment, which is a measure of the distribution of charge in an object. The higher the order of the multipole moment, the more complex the charge distribution is.
Yes, the expansion of charge can be used for any type of charge distribution, as long as it is well-behaved and can be represented as a continuous function. However, for highly irregular charge distributions, the accuracy of the expansion may be limited and higher order terms may be needed.
The expansion of charge is both a theoretical concept and a practical tool. It is an important part of classical electrodynamics theory and is used to understand the behavior of electromagnetic fields. It is also used in practical applications such as antenna design, where the multipole expansion is used to model the radiation pattern of an antenna.