How Does a Cavity Inside a Charged Sphere Affect Electric Field Calculations?

[sonrie]

An insulating sphere of radius 5.00 cm , centered at the origin, has a uniform volume charge density 3.05 uC/m^3 . There is a spherical cavity cut out of its center of radius 2.00 cm.

Well i know to use Gauss' Law:
Φ = ∫E∙dA = Q/ε
where Φ is the electric flux, E is the electric field, dA is a differential area on the closed surface S with an outward facing surface normal defining its direction, Q is the charge enclosed by the surface, and ε is the electric constant.

1.) What is the electric field at 1.43 cm ?
I found that to be zero.

2.) What is the electric field inside the spherical shell at 2.86 cm ?

Q = (4π/3)*(0.0286^3-0.0200^3)*3.05x10^-6C
The surface area integrated over is: S = 4π*0.0286^2. Hence:
E = Q/εS = (0.0286^3-0.0200^3)*3.05x10^-6/(3ε*0.028...

for surface area i get 1.03*10^-2, for Q i get 4.188*1.53*10^-5*3.05*10^-6= 1.95*10^-10
Then i divide Q/S which is 1.90*10^-12. but i keep getting it wrong. what am i doing wrong?

3.) What is the electric field outside the spherical shell at 6.75 cm ?
Q = (4π/3)*(0.0500^3-0.0200^3)*3.05x10^-6C
The surface area integrated over is: S = 4π*0.0675^2. Hence:
E = Q/εS = (0.0500^3-0.0200^3)*3.05x10^-6/(3ε*0.067...

I did the same as part two equations and i get 3.89*10^-9 which is wrong also. Help!





3. The Attempt at a Solution [/

 

[Nick89]

1) is right, since there is no charge in the cavity.

For the others, I'm getting a headache reading all the digits :p It might be easier to spot a mistake if you write it out in symbols. I'll try that now.

 

[Nick89]

Right,

2) After using gauss' law you should find:
E = \frac{Q}{\epsilon A}
Q is in this case dependend on the charge density rho and the volume V:
Q = \rho V = \rho \frac{4 \pi}{3}(r^3 - {r_0}^3) where r is 2.86 cm and r_0 is the radius of the cavity (2 cm)

E = \frac{\rho \frac{4 \pi}{3}(r^3 - {r_0}^3)}{\epsilon 4 \pi r^2} = \frac{\rho (r^3 - {r_0}^3)}{3 \epsilon r^2}

Entering values:
r = 0.0286, r_0 = 0.02, \epsilon = 8.85 \times 10^{-12}, \rho = 3.05 \times 10^{-6} yields:
E = 2161.9 (provided I didn't make any errors while entering the values in my calculator :p )

What should the answer be according to you?

 

[sonrie]

Thanks A Bunch!