- #1
vorcil
- 398
- 0
Homework Statement
Suppose the electric field in some region is found to be [tex] \bf{E} = kr^3 \bf{\hat{r}} [/tex], in spherical coordinates, (k is some constant).
(a) find the charge density [tex] \rho [/tex]
(b) find the total charged contained in a sphere of radius R, centered at the origin. (do it two different ways)
Homework Equations
All of them,
[tex] \bf{E(r)} = \frac{1}{4\pi \epsilon_o} \int_V \frac{\rho (\bf{r'})}{\varsigma^2} \hat{\varsigma} d\tau' . [/tex]
[tex] \nabla . \bf{E} = \frac{1}{\epsilon_o} \rho [/tex] <- Gauss's law
noting that [tex] \varsigma = \bf{r - r'} [/tex]
The Attempt at a Solution
re aranging the equation above for rho
[tex] \epsilon_o \nabla . \bf{E} = \rho [/tex]
I know i can use the partial derivative of the vector r for the divergence instead of the traditional partial derivative of x,y,z
[tex] \epsilon_o \frac{\partial}{\partial r} . \bf{E} = \rho [/tex]
substituting in the given E, kr^3 into E
[tex] \epsilon_o \frac{\partial}{\partial r} . (k r^3) = \rho [/tex]
computing the derivative d/dr of r I get,
[tex] \epsilon_0 3kr^2 [/tex]
-
NOW THIS IS WHERE I NEED HELP FOR THE FIRST QUESTION
I've been told that the answer is [tex] 5\epsilon_0 kr^2 [/tex]
can someone please tell me how they got to that?
i'm missing a factor of 2?
-
(b):
using gauss's law,
[tex] \oint \bf{E} . d\bf{a} = \frac{1}{\epsilon_o} Qenc [/tex]
where Qenc is the enclosed charge within the surface/shape/sphere
solving for Qenc,
[tex] \epsilon_o \oint \bf{E} . d\bf{a} = Qenc [/tex]
=
[tex] \epsilon_o \oint (kr^3) . da = Qenc [/tex]
because the E field is a constant,
taking it outside of the integral, leaves me with having to integrate the integral over a closed surface da,
and because it's a sphere, the area of the sphere is just 4pir^2 (if i remember correctly)
making the equation end up as
[tex] \epsilon_o (kr^3) . (4\pi r^2) = Qenc [/tex]
leaving the final charge contained in the sphere to be
[tex] 4\pi \epsilon_o k r^5 [/tex]
- The question asks me to find this equation in two different ways,
I've found it using the only way I know how, using gauss's law
can someone help me think of a different way to find this equation?