Jackson Electrodynamics problem 6.5b

In summary, for the given conversation, the experts discussed the transformation of the momentum of electromagnetic fields and showed that it can be written as {\bf P_{field}}=\frac{1}{c^2}\int \phi {\bf J} d^3x. They also discussed the expansion of the electrostatic potential in a Taylor series and showed that it can be written as {\bf P_{field}}=\frac{1}{c^2}{\bf E(0)\times m}, where E(0) is the electric field at the current distribution and m is the magnetic moment caused by the current. They also mentioned that this can be proven by writing out the components of the integrals.
  • #1
andrew1982
8
0

Homework Statement


A localized electric charge distribution produces an electrostatic field,
[tex]
{\bf E}=-\nabla \phi
[/tex]
Into this field is placed a small localized time-independent current density J(x) which generates a magnetic field H.
a) show that the momentum of these electromagnetic fields, (6.117), can be transformed to
[tex]
{\bf P_{field}}=\frac{1}{c^2}\int \phi {\bf J} d^3x
[/tex]

b) Assuming that the current distribution is localized to a region small compared to the scale of variation of the electric field, expand the electrostatic potential in a Taylor series and show that
[tex]
{\bf P_{field}}=\frac{1}{c^2}{\bf E(0)\times m}
[/tex]
where E(0) is the electric field at the current distribution and m is the magnetic moment (5.54), caused by the current.

Homework Equations


(6.117):
[tex]
{\bf P_{field}}=\mu_0 \epsilon_0 \int {\bf E \times H} d^3x
[/tex]
(5.54):
[tex]
{\bf m}=\frac{1}{2} \int {\bf x' \times J(x')} d^3x'
[/tex]

The Attempt at a Solution


Part a) was straight forward: subsituting E=- grad phi and integrating by parts gives the answer plus a surface integral that goes to 0 if phi*H goes to 0 faster than 1/r^2.

Part b): This is where I get stuck. I tried to put
[tex]
\phi=\phi(0)+\nabla \phi(0)\cdot{\bf x}
[/tex]
which replaced in the integral for P_field from a) gives
[tex]
{\bf P_{field}}=-\frac{1}{c^2} \int {\bf (E(0)\cdot x) J)} d^3x
[/tex]
if I choose the potential to zero at the origin. Further, using
[tex]
{\bf a\times (b\times c)=(a\cdot c) b-(a\cdot b)c}
[/tex]
on the integrand I get
[tex]
{\bf P_{field}}=\frac{1}{c^2} (\int {\bf E(0)\times (x\times J) }d^3x-\int{\bf (E(0)\cdot J)x}\,d^3x)
[/tex]
The first integral is as far I can see
[tex]
\frac{2}{c^2} {\bf E(0)\times m}
[/tex]
that is, twice the answer. The second integral gets me stuck. I guess I should show that it is equal to minus half of the answer (if I did everything correctly so far), but I don't see how to do this.

I would appreciate if anyone could give me a hint on how to continue or if I'm on the right track at all. Thanks in advance!
 
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  • #2
Somewhere else in Jackson (and other texts) it is shown the second integral
equals EXm. It's in Sec. 5.6 of teh 2nd Edition, if you can follow it.
 
  • #3
If you believe that the second integral is indeed equal to Exm, then why not simply write them both out in components to prove it? Granted it's not really a proper derivation, but you were only asked to show that the formula is true.
 
  • #4
Thanks for your replies, it was very helpful! Using section 5.6 of Jackson as you said I saw that I could show directly that the first integral
[tex]
{\bf P_{field}}=-\frac{1}{c^2} \int {\bf (E(0)\cdot x) J)} d^3x
[/tex]
is equal to the sought answer without using the abc vector rule. Looking below eq. 5.52 (in the 3rd edition) and substituting x by E(0) the whole derivation is there.
 
  • #5
How about the part C
 

1. What is Jackson Electrodynamics problem 6.5b?

Jackson Electrodynamics problem 6.5b refers to a specific problem in the textbook "Classical Electrodynamics" by John David Jackson. It is a physics problem that involves solving for the electric field produced by a moving point charge in the presence of a conducting plane.

2. What is the difficulty level of this problem?

The difficulty level of Jackson Electrodynamics problem 6.5b is considered to be intermediate to advanced. It requires a solid understanding of electromagnetic theory and mathematical skills to solve.

3. How is this problem relevant to real-world applications?

This problem is relevant to real-world applications because it deals with the behavior of electric fields in the presence of conductors. This concept is important in various fields such as electronics, telecommunications, and power systems.

4. What are some tips for solving this problem?

Some tips for solving Jackson Electrodynamics problem 6.5b include carefully reading the problem statement, drawing a diagram to visualize the setup, and breaking down the problem into smaller, manageable steps. It is also important to have a strong understanding of the principles and equations involved.

5. Are there any resources available for help with this problem?

Yes, there are various online resources available for help with Jackson Electrodynamics problem 6.5b. Some options include physics forums, online tutoring services, and study guides specifically for this textbook. It is also helpful to consult with a professor or classmate for clarification and guidance.

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