- #1
yungman
- 5,759
- 292
I read in the book regarding a point charge at the origin where [itex]Q(t)= \rho_{(t)}Δv'\;[/itex]. The wave eq is.
[tex]\nabla^2V-\mu\epsilon\frac{\partial^2 V}{\partial t^2}= -\frac {\rho_v}{\epsilon}[/tex]
For point charge at origin, spherical coordinates are used where:
[tex] \nabla^2V=\frac 1 {R^2}\frac {\partial}{\partial R}\left( R^2 \frac {\partial V}{\partial R}\right)[/tex]
This is because point charge at origin, [itex]\frac {\partial}{\partial \theta} \hbox{ and }\; \frac {\partial}{\partial \phi}[/itex] are all zero.
My question is this:
The book then said EXCEPT AT THE ORIGIN, V satisfies the following homogeneous equation:
[tex]\frac 1 {R^2}\frac {\partial}{\partial R}\left( R^2 \frac {\partial V}{\partial R}\right)-\mu\epsilon \frac {\partial^2 V}{\partial t^2}=0[/tex]
The only reason I can think of why this equation has to exclude origin is because R=0 and origin and this won't work. Am I correct or there's another reason?
Thanks
Alan
[tex]\nabla^2V-\mu\epsilon\frac{\partial^2 V}{\partial t^2}= -\frac {\rho_v}{\epsilon}[/tex]
For point charge at origin, spherical coordinates are used where:
[tex] \nabla^2V=\frac 1 {R^2}\frac {\partial}{\partial R}\left( R^2 \frac {\partial V}{\partial R}\right)[/tex]
This is because point charge at origin, [itex]\frac {\partial}{\partial \theta} \hbox{ and }\; \frac {\partial}{\partial \phi}[/itex] are all zero.
My question is this:
The book then said EXCEPT AT THE ORIGIN, V satisfies the following homogeneous equation:
[tex]\frac 1 {R^2}\frac {\partial}{\partial R}\left( R^2 \frac {\partial V}{\partial R}\right)-\mu\epsilon \frac {\partial^2 V}{\partial t^2}=0[/tex]
The only reason I can think of why this equation has to exclude origin is because R=0 and origin and this won't work. Am I correct or there's another reason?
Thanks
Alan