- #1
xSilja
- 3
- 0
Homework Statement
a) and b) are no problem.
I need help to solve c) and d)
Homework Equations
c) Delta dirac function
Gauss' law
d) Gauss' law
## \int_V {\rho \, d\tau} = Q_{enclosed} ##
The Attempt at a Solution
By taking laplace on the potential I get:
## \rho(\mathbf{r}) = \frac{q_0}{4 \, \pi \, r} \, e^{-r/\lambda} \, \left( \frac{cos^2(\theta)}{\lambda^2} + \frac{2}{r^2} (1-3 \, cos^2(\theta)) \right)##
c) I got a hint that it was a good idea to use the dirac delta function along with the charge distribution.
But I'm not exactly sure why. As I understand it the dirac delta function "picks out" the value of a function at zero. So I'd get:
## \int {\rho(\mathbf{r}) \, \delta(r) \, dr} = \rho(0) ##
I realize that there must be a dimensional problem here, but I'm not sure how to use a delta function in 3D and spherical coordinates.
Also how will it help me to find the density at the origin? Can I apply Gauss' law here and let the radius go towards zero to get the charge in the origin?
d) I want to solve the integral
## Q = \int_0^\pi \int_0^{2 \, \pi} \int_a^{\infty} \, \rho(\mathbf{r}) \, r^2 \, \sin(\theta) \, dr \, d\theta \, d\phi ##
I tried evaluating this with Maple.
By assuming a>0 I get a complex function multiplied by infinity, which is not of much use.
If I also assume lambda>0 (as it says in the problem) I get rid of the infinity, but get exponential integrals instead.
I'm not sure how to move on from here. I suspect I need to modify my function for charge distribution by assumptions, to make it simpler.