How Does Charge Distribution Affect Electric Fields in Nested Spheres?

[roam]

Homework Statement

A conducting sphere of radius, R = 5.5 cm with an excess charge of Q = -35.5 nC is surrounded by a concentric, conducting, spherical shell of inner radius, R_in = 9.5 cm and outer radius, R_out = 11.5 cm that carries an excess charge of q = -13.0 nC.

[PLAIN]http://img571.imageshack.us/img571/7821/imagex.gif

Determine the electric field at the following radii for the aforementioned arrangement:

(a) r = 42.5 cm.

(b) r = 8.5 cm.

The Attempt at a Solution

(a) In the indicated region r>R_out. Therefore I model the charge distribution as a sphere with charge -Q and the expression for the field in this region would be

$$E=-k_e \frac{Q}{r^2}$$

$$-(9 \times 10^9) \frac{35.5}{42.5^2}=176885813.1$$

even if I convert r to meters I still get the wrong answer (correct answer: 2420)

(b) Again, I can apply Gauss's law to find the electric field. Since R<r<R_in I think I should use:

$$k_e \frac{Q}{r^2}=(9 \times 10^9) \frac{35.5}{8.5^2} = 4422145329$$

The correct answer is 44200 N/C, did I forget to convert something?
Thanks.

 

[kuruman]

What is the charge enclosed by a Gaussian surface at 42.5 cm? Don't forget that both the sphere and the shell have charge on them. Also, how many coulombs to a nano-coulomb are there?

 

[roam]

What is the charge enclosed by a Gaussian surface at 42.5 cm? Don't forget that both the sphere and the shell have charge on them. Also, how many coulombs to a nano-coulomb are there?

Yes, I got all the unit conversions correct but I'm still getting the wrong answer:

$$-(9 \times 10^{9})\frac{35.5 \times 10^{-9}}{(0.425)^2}=-1768.8$$

Why is that?

 

[kuruman]

Yes, I got all the unit conversions correct but I'm still getting the wrong answer:

$$-(9 \times 10^{9})\frac{35.5 \times 10^{-9}}{(0.425)^2}=-1768.8$$

Why is that?

It is because the charge enclosed by a spherical Gaussian surface at 42.5 cm is not -35.5 nC. What is it?

 

[rwisz]

I would like to commend you on your attempt at solving this problem. However, it seems that you may have made a few errors in your calculations. Let's go through each part separately to see where the mistakes may have occurred.

(a) In this part, you correctly identified that the charge distribution can be modeled as a single point charge with charge -Q. However, you seem to have made an error in the value of k_e, which is the Coulomb's constant. It should be 9 x 10^9 Nm^2/C^2, not 9 x 10^9. So, your final answer should be -2420 N/C, which is the correct answer.

(b) For this part, you correctly identified that the charge distribution can be modeled as a single point charge with charge Q. However, you seem to have made an error in the value of k_e again. It should be 9 x 10^9 Nm^2/C^2. Also, remember that for a point charge, the electric field at a distance r is given by E = k_eQ/r^2. So, your final answer should be 44200 N/C, which is the correct answer.

In summary, it seems that the main errors were in the values of k_e, which is a common mistake. Always double check your units and values when working with Coulomb's law. Keep up the good work and continue to practice your problem-solving skills.