Sphere with non-uniform charge density

In summary, the conversation discusses finding the value of k for a sphere with non-homogeneous charge distribution, and using Gauss's Law to determine the electric field inside the sphere. The solution involves integrating to find k and using both the differential and integral forms of Gauss's Law, leading to the final result of E = k/(2*epsilon_0).
  • #1
Bonewheel
2
0

Homework Statement


A sphere of radius R carries charge Q. The distribution of the charge inside the sphere, however, is not homogeneous, but decreasing with the distance r from the center, so that ρ(r) = k/r.

1. Find k for given R and Q.

2. Using Gauss’s Law (differential or integral form), find the electric field E inside the sphere, i.e., for r < R.

Homework Equations


[tex]\int_V \rho \, dV = Q[/tex]
[tex]\oint \vec E \cdot d \vec A = \frac {Q_{enclosed}} {\epsilon_0}[/tex]
[tex]\vec {\nabla} \cdot \vec E = \frac {\rho} {{\epsilon}_0}[/tex]

The Attempt at a Solution


1. [tex]\int_V \rho \, dV = \int_{0}^{2 \pi} \int_{0}^{\pi} \int_{0}^R \frac k r r^2 \sin \theta \, dr \, d \theta \, d \phi = 2 k \pi R^2 = Q[/tex]
The units check out here.

2.
Here's where I ran into a problem. I tried using both the differential and integral forms of Gauss's Law, and in both cases the r canceled out, leaving me with an expression for the electric field I know is wrong. Oddly, the units work out here as well.

[tex]\oint \vec E \cdot d \vec A = 4 \pi r^2 E = \frac {Q_{enclosed}} {\epsilon_0} = \frac {2 k \pi r^2} {{\epsilon}_0} [/tex]
[tex]\vec {\nabla} \cdot \vec E = \frac 1 {r^2} \frac {\partial} {\partial r}(r^2 E_r)= \frac {\rho} {{\epsilon}_0} = \frac k {r {{\epsilon}_0}}[/tex]
[tex]E = \frac k {2{{\epsilon}_0}}[/tex]

Thank you so much for any help! Please let me know if you need any further information or edits for clarification.
 
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  • #2
I just realized that I made a very stupid mistake and never found k... Marking as solved.
 

Related to Sphere with non-uniform charge density

1. What is a sphere with non-uniform charge density?

A sphere with non-uniform charge density is a three-dimensional object with varying charge distribution throughout its volume. This means that the charge is not evenly spread out and can be different at different points within the sphere.

2. How is the charge density of a sphere with non-uniform charge density calculated?

The charge density of a sphere with non-uniform charge density is calculated by dividing the total charge by the volume of the sphere. This gives the average charge per unit volume, which can vary at different points within the sphere.

3. What causes a sphere to have non-uniform charge density?

A sphere may have non-uniform charge density due to the presence of different materials or charges within its volume. It can also be a result of external electric fields or forces acting on the sphere.

4. How does non-uniform charge density affect the electric field around a sphere?

The electric field around a sphere with non-uniform charge density is not constant and varies at different points. This means that the strength and direction of the electric field will also vary, depending on the location within the sphere.

5. What are some real-life applications of spheres with non-uniform charge density?

Spheres with non-uniform charge density are commonly used in electrostatics and in the study of electric fields. They can also be found in various technologies such as capacitors, sensors, and particle accelerators.

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