Finding the Scalar Potential of a Sphere with Non-Uniform Charge Density

And putting that in terms of Q:http://img180.imageshack.us/img180/7451/59work5.gif [Broken][/QUOTE]In summary, the problem asks to find the scalar potential at all points inside and outside the sphere using the formula http://img17.imageshack.us/img17/4628/wangsnesseqn572.gif [Broken] and to express the results in terms of the total charge Q of the sphere. The attempt at a solution involves trying to simplify the problem by visualizing the sphere as a series of infinitely thin shells, but this method does not take into account the varying charge density on each shell. The correct approach is to treat each infinitesimal piece of charge as a point charge and integrate the
  • #1
MarcZero
2
0

Homework Statement



A sphere of radius a has a charge density which varies with distance r from the center according to http://img14.imageshack.us/img14/9577/wangsnessproblem594.gif [Broken] where A is a constant and http://img35.imageshack.us/img35/555/wangsnessproblem593.gif [Broken].[/URL] Find the scalar potential http://img35.imageshack.us/img35/9326/phi2.gif [Broken] at all points inside and outside the sphere by using the following formula:

http://img17.imageshack.us/img17/4628/wangsnesseqn572.gif [Broken]

Express your results in terms of the total charge Q of the sphere.


Homework Equations



Volume of the Sphere: http://img10.imageshack.us/img10/8776/spherevolume.gif [Broken]

Vector Capital R: (From Source to Point) http://img21.imageshack.us/img21/2586/captialr.gif [Broken]


The Attempt at a Solution



I'm not sure if I'm over-simplifying this but I figured since the charge density is directly related and varies according to the volume, that is, that I can think of the sphere as containing a series of spheres layered on each other that are infinitely thin, I could take the integral and relate Q to the charge density and the volume. So:

http://img197.imageshack.us/img197/5717/59work1.gif [Broken]

Then:

http://img197.imageshack.us/img197/3756/59work2.gif [Broken]

Finding that:

http://img197.imageshack.us/img197/5041/59work3.gif [Broken]

Plugging that into the main formula above, I get:

http://img197.imageshack.us/img197/9933/59work4.gif [Broken]

And putting that in terms of Q:

http://img180.imageshack.us/img180/7451/59work5.gif [Broken]


So if I am reading the question correctly, this is the final formula it asks for. I am really suspicious that I missed something and am forgetting a concept that will make this problem more complex than I think it is. I might also be over-thinking things as well. Any insight would be appreciated. Thank you for your time in advance and I apologize about the pictures, as I am still learning MathType.
 
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  • #2
Hi MarcZero, welcome to PF!:smile:

MarcZero said:
I'm not sure if I'm over-simplifying this but I figured since the charge density is directly related and varies according to the volume, that is, that I can think of the sphere as containing a series of spheres layered on each other that are infinitely thin, I could take the integral and relate Q to the charge density and the volume. So:

http://img197.imageshack.us/img197/5717/59work1.gif [Broken]
[/URL]

You are definitely trying to oversimplify things. There is nothing wrong with visualizing the solid sphere as a bunch of spherical shells (which is what I assume you meant) layered on top of each other. But the charge density on each shell will be different, since each shell is a different distance from the center, and the charge density depends on the distance from the center.

In general, [itex]Q=\int_{\mathcal{V}}\rho(r')d\tau'\neq\rho V[/itex]...it is only when the charge density is uniform throughout the volume that you can make this conclusion. In any case, calculating the total charge on the sphere does not really help you to calculate the scalar potential.

Plugging that into the main formula above, I get:

http://img197.imageshack.us/img197/9933/59work4.gif [Broken]
[/URL]

No, the formula

[tex]\Phi(\textbf{r})=\frac{1}{4\pi\epsilon_0}\int_{\mathcal{V}}\frac{\rho(\textbf{r}')d\tau'}{R}[/tex]

tells you to treat each infinitesimal piece of charge [itex]dq'=\rho(\textbf{r}')d\tau'[/itex] as a point charge, located at [itex]\textbf{r'}[/itex], and integrate (or add up) the contribution to the potential of each piece

[tex]d\Phi(\textbf{r})=\frac{1}{4\pi\epsilon_0}\frac{dq'}{R}=\frac{1}{4\pi\epsilon_0}\frac{\rho(\textbf{r}')d\tau'}{R}[/tex]

Each piece of charge will be at a different distance [itex]r'[/itex] from the center, and hence will have a different charge density and will also be a different distance [itex]R=|\textbf{r}-\textbf{r}'|[/itex] from the field point and therefor have a different contribution to the potential.

This means that [tex]\Phi(\textbf{r})\neq\frac{1}{4\pi\epsilon_0}\frac{Q}{r}[/tex]...You will actually need to integrate in order to find out what the potential really is!
 
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1. What is a scalar potential?

A scalar potential is a mathematical concept in physics that describes the energy associated with a system in terms of a scalar field. It is a fundamental concept in electrostatics and is used to calculate the electric potential at any point in space due to a charge distribution.

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

A sphere with non-uniform charge density is a spherical object with a distribution of electric charge that is not constant throughout its volume. This means that the charge density varies at different points on the surface and within the sphere.

3. Why is it important to find the scalar potential of a sphere with non-uniform charge density?

Knowing the scalar potential of a sphere with non-uniform charge density allows us to understand the behavior of electric fields and the resulting forces in this system. It is a crucial step in solving many real-world problems related to electrostatics and can help in the design and analysis of various devices such as capacitors and conductors.

4. How do you find the scalar potential of a sphere with non-uniform charge density?

The scalar potential of a sphere with non-uniform charge density can be found by using the formula V = k∫ρ(r')/|r - r'| dV', where V is the scalar potential, k is the Coulomb's constant, ρ(r') is the charge density at a point r', and dV' is an element of volume. This integral can be solved by using appropriate mathematical techniques such as Gauss's law or the method of images.

5. What are some real-world applications of finding the scalar potential of a sphere with non-uniform charge density?

Finding the scalar potential of a sphere with non-uniform charge density has numerous applications in various fields such as electronics, telecommunications, and medical imaging. For example, the calculation of the electric potential in a human body with different charge densities can help in understanding the flow of currents and designing medical devices like pacemakers. It is also essential in the study and development of electronic devices such as transistors and capacitors.

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