- #1
albega
- 75
- 0
So part of the idea presented in my book is that:
div(r/r3)=0 everywhere, but looking at this vector field it should not be expected. We would expect some divergence at the origin and zero divergence everywhere else.
However I don't understand why we would expect it to be zero everywhere but the centre, because if you draw it, the arrows get smaller as we move out radially. If you consider placing a little cube somewhere in the field not at the centre, the arrows entering that cube would be larger than those leaving. Surely that would give a negative divergence at these points. I obviously understand why it should be large at the centre.
This leads me to my other point. Consider the r/r2 field - it is similar to the above field, just falling off less rapidly. This field has 1/r2 divergence however. My first issue is, why does it not give negative divergences by my above argument of arrows into a little cube, and because of the similarities for the above field, why does it not give zero divergence apart from at the origin? Secondly, why does the maths behave so much more nicely for such a similar field?
Thanks for any help :)
div(r/r3)=0 everywhere, but looking at this vector field it should not be expected. We would expect some divergence at the origin and zero divergence everywhere else.
However I don't understand why we would expect it to be zero everywhere but the centre, because if you draw it, the arrows get smaller as we move out radially. If you consider placing a little cube somewhere in the field not at the centre, the arrows entering that cube would be larger than those leaving. Surely that would give a negative divergence at these points. I obviously understand why it should be large at the centre.
This leads me to my other point. Consider the r/r2 field - it is similar to the above field, just falling off less rapidly. This field has 1/r2 divergence however. My first issue is, why does it not give negative divergences by my above argument of arrows into a little cube, and because of the similarities for the above field, why does it not give zero divergence apart from at the origin? Secondly, why does the maths behave so much more nicely for such a similar field?
Thanks for any help :)