Jackson Classical Electrodynamics: Deriving W_{int} from (1.57) to (1.58)

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In summary, the conversation is about the derivation of the interaction energy W_{int} between two point charges in the example provided on page 42 of the 3rd edition of Classical Electrodynamics by Jackson. The speaker is asking for clarification on how to get from equation (1.57) to (1.58) and mentions that they do not have their copy of Jackson with them. Another person suggests that it is a straightforward substitution and offers to help if the speaker gets stuck. Some people in the conversation have earlier editions of the book with different equation numbering.
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americanforest
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Can someone explain to me how Jackson, on page 42 of the 3rd ed. of Classical Electrodynamics, when he is deriving the interaction energy [tex]W_{int}[/tex] in his example involving two point charges, gets from equation (1.57) to (1.58). I thought about typing up the TeX but I'm sure most of you have this book. I know he makes that substitution he mentions but I'm not sure how to go about doing that. Help!
 
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  • #2
I keep my copy of Jackson at the office and not at home. I do not need him to pervade my life any more than is possible.
 
  • #3
Some of us are old and have earlier editions of Jackson. Presumably the equation numbering is different.
 
  • #4
It's just a straight substitution: Solve for [itex]\mathbf{x}[/itex] in terms of [itex]\mathbf{\rho}[/itex], and substitute. If you get stuck, post your work and we'll be able to help you better.
 

FAQ: Jackson Classical Electrodynamics: Deriving W_{int} from (1.57) to (1.58)

1. What is the significance of deriving W_{int} from (1.57) to (1.58) in Jackson Classical Electrodynamics?

Deriving W_{int} from (1.57) to (1.58) in Jackson Classical Electrodynamics helps to understand the relationship between the electric and magnetic fields in an electromagnetic field and how they contribute to the energy density of the system. It also allows for the calculation of the total energy stored in the field.

2. How is the energy density of an electromagnetic field related to W_{int}?

The energy density of an electromagnetic field is directly related to W_{int}, as it is a measure of the amount of energy stored in the field per unit volume. W_{int} is the integration of the energy density over the entire volume of the field.

3. What is the mathematical expression for W_{int} in terms of the electric and magnetic fields?

The mathematical expression for W_{int} is given by the integral of 1/2(E^2 + B^2) over the volume of the electromagnetic field. This represents the sum of the electric and magnetic energy densities.

4. How does W_{int} relate to the Poynting vector and energy flow in an electromagnetic field?

W_{int} is related to the Poynting vector, as it is the energy flow per unit area of the field. The Poynting vector is defined as the cross product of the electric and magnetic fields, and it represents the direction and magnitude of energy flow in the field. W_{int} is the integration of the Poynting vector over the entire volume of the field.

5. What are some real-world applications of understanding W_{int} in Jackson Classical Electrodynamics?

Understanding W_{int} can have various real-world applications, such as in the design and optimization of electromagnetic devices, calculating the energy efficiency of electrical systems, and studying the behavior of electromagnetic fields in different materials. It can also aid in the development of new technologies, such as wireless power transfer and electromagnetic shielding.

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