- #1

FOIWATER

Gold Member

- 434

- 12

However I am second guessing what I did,

This is the problem:

let rho v be 10/r^2 mC/m^3 between 1<r<4

let rho v be zero elsewhere

a)find the net flux crossing the surface at r=2 m, r= 6m

b)determine D at r=1, r=5

a) I have no issues with, I can simply say that the Flux crossing the surface equals the charge enclosed and solve for the charge enclosed by integrating the volume charge density over the volume. I used triple integration in spherical coordinates with r from 1 to 2, theta from 0 to pi, and phi from 0 to 2pi. I got 40pi mC

Same goes for r=6m, I just integrated the same except r went from 1 to 4 (since there is no more charge after r = 4, or rather, the charge density was given as zero) (but the flux will remain the same at 6m as it was at 4m) 120pi

My issue is with what I did in b), applying gauss' law.

I know gauss' law holds, and for r=1, since there is no charge inside a spherical gaussian surface, there is no flux density on the surface so D=0 but for r=5m, I assumed a gaussian surface (a sphere) of radius 5. I know the flux at 5m is the same as in part b, 120pi. so:

flux = Qenc = surface integral D dot ds, D and ds are both vectors.

How could I, and the book, assume D is constant and take it outside the integral sign?

The charge density inside the sphere was given as 10/r^2, which isn't uniform over the volume?

How could a sphere be perpindicular to the flux lines contain within when the charge density is not uniform?