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Integration of Equation

  • Thread starter jmwilli25
  • Start date
  • #1
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Homework Statement



I just need help integrating this equation.
e is just the charge of an electron so it is constant



Homework Equations




[tex]
-\int_{V}d\vec{r}\frac{\delta}{\delta t}e\delta(\vec{r}-\vec{R}(t))
[/tex]

The Attempt at a Solution



[tex]
-e\frac{d}{dt}\int_{V}d\vec{r}\delta(\vec{r}-\vec{R}(t))
[/tex]
[tex]
-e\frac{d}{dt}4\pi
[/tex]

I don't believe my solution is correct or complete
 

Answers and Replies

  • #2
33,505
5,191
Some additional context would be helpful/necessary. Is the derivative an ordinary derivative like this?
[tex]-\int_{V}d\vec{r}\frac{d}{dt}e\delta(\vec{r}-\vec{R}(t))[/tex]

or is a partial derivative, like this?
[tex]-\int_{V}d\vec{r}\frac{\partial}{\partial t}e\delta(\vec{r}-\vec{R}(t))[/tex]

Is [itex]\delta[/itex] just a constant, or are you indicating an impulse function?
 
  • #3
hunt_mat
Homework Helper
1,739
18
I think he maybe referring to functional differentiation, not too sure though.
 
  • #4
5
0
Sorry, it is the partial derivative inside the integral and
[tex]
\delta(\vec{r}-\vec{R}(t))
[/tex]
is the dirac delta function.

Context: I am trying to show that the equation of charge conservation holds when
[tex]
\rho(\vec{r},t)=e\delta(\vec{r}-\vec{R}(t))
[/tex]

The entire equation that I am trying to solve is
[tex]
-\int_{V}d\vec{r}e\frac{\partial}{\partial t}e\delta(\vec{r}-\vec{R}(t))=\int_{V}d\vec{r}\vec{\nabla}\bullet(e\frac{d}{dt}\delta(\vec{r}-\vec{R}(t))).
[/tex]
I have to show that the two sides are equal.
But I figured if I could get help just with the LHS then I might be able to do the RHS myself. Granted, I still need to figure out how to apply the divergence to a full derivative of time, but one step at a time. Also, the equation that I have just given is exactly what is in the book. Classical Electordynamics, Schwinger 1998.
 

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