What Is the Charge Inside a Hollow Sphere?

[m84uily]

Homework Statement

A hollow metal sphere with a point charge in its center has a 0.06m and 0.1m inner and outer radii, respectively. The surface charge density on the inside surface is -100nC/m^2. The surface charge density on the exterior surface is 100nC/m^2. What is the magnitude of the point charge at the center?

Homework Equations

EA = Qin/ε₀

The Attempt at a Solution

First attempt
EA = Q/ε₀
Q = EAε₀
E = 100x10^-9
A = 4pi(0.1^2)
ε₀ = 8.99x10^-12
Q= 1x10^-19

Which is much smaller than the answer in the back of my book!

Second attempt

E = (1/4piε₀)(Q/r^2)

(r^2)(E)(4)(pi)ε₀ = Q
(0.06^2)(100x10^-9)(4)(3.14)(8.99x10^-12) = Q

Q = 0.000000000000000000040649184

Which is also too small!


Thanks in advance for your help.

 

[anathema]

Hello,

Thank you for your post. I am a scientist and I would be happy to help you with this problem.

Firstly, your first attempt was on the right track, but there are some errors in your calculations. Let's go through the steps together to find the correct answer.

1. The electric field at the surface of the sphere is given by E = σ/ε₀, where σ is the surface charge density and ε₀ is the permittivity of free space.

2. We can then calculate the total charge on the inner surface of the sphere using the formula Q = σA, where A is the area of the inner surface.

3. The area of the inner surface can be calculated using A = 4πr^2, where r is the inner radius of the sphere.

4. Now we have all the values we need to calculate the total charge on the inner surface. Plugging in the values, we get Q = (-100x10^-9 C/m^2)(4π(0.06 m)^2) = -0.000072 C.

5. Since the point charge is at the center of the sphere, it is surrounded by a spherical shell of charge with radius 0.06 m and charge Q.

6. Using the Gauss's law, we can calculate the electric field at the point charge, which is given by E = Q/4πε₀r^2, where r is the distance from the point charge to the center of the sphere.

7. Plugging in the values, we get E = (-0.000072 C)/(4π(8.99x10^-12 C^2/Nm^2)(0.06 m)^2) = -250 N/C.

8. Finally, we can use the formula E = kQ/r^2 to find the magnitude of the point charge, where k is the Coulomb's constant and r is the distance from the point charge to the center of the sphere.

9. Rearranging the formula, we get Q = Er^2/k. Plugging in the values, we get Q = (250 N/C)(0.06 m)^2/(8.99x10^9 Nm^2/C^2) = 0.0000000000000000000278 C.

Therefore, the magnitude of the point charge at the center of the