Gaussian solid sphere Q

In summary, the problem involves a solid sphere with a non-uniform charge distribution of p=Ar^2 and the task is to find the total charge Q within the volume of the sphere. Gauss's Law can be used for symmetric cases but in this problem, we must integrate dq due to the non-uniform charge distribution. The volume element dV is given by dV = (4πr^2)dr and the amount of charge dQ in dV is dQ = 4∙A∙π∙r^4∙dr where A is a constant and ρ(r) = A∙r^2. Integrating this over the sphere gives the total charge Q, which is 4/
  • #1
lonewolf219
186
2
Question:

A solid sphere of radius R has a non-uniform charge distribution of p=Ar^2, where A is constant. Find total charge Q within the volume of the sphere.


p=roe
p=Q/dV

EdA=qenclosed/Enaught

Can you use Gauss' Law for this problem when sphere is solid? If so, how?
Since p is non-uniform, we must integrate dq, correct? The answer to this question in the book is 4/5pieAr^5.

But how to get the answer? I think dq=Ar^2dV. But to have dV=4pier^2 is incorrect since the charge is not on the surface of the sphere (it is not a conductor), am I wrong?

Some help would be appreciated! Thanks!
 
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  • #2
lonewolf219 said:
Question:

A solid sphere of radius R has a non-uniform charge distribution of p=Ar^2, where A is constant. Find total charge Q within the volume of the sphere.

p=roe
p=Q/dV

EdA=qenclosed/Enaught

Can you use Gauss' Law for this problem when sphere is solid? If so, how?
Since p is non-uniform, we must integrate dq, correct? The answer to this question in the book is 4/5pieAr^5.

But how to get the answer? I think dq=Ar^2dV. But to have dV=4pier^2 is incorrect since the charge is not on the surface of the sphere (it is not a conductor), am I wrong?

Some help would be appreciated! Thanks!
Gauss's Law gives the Electric field flux through a surface, which in very symmetric cases can give the Electric field itself. It won't help with this problem.

As you state, the volume element (differential), dV, is given by dV = (4πr2)dr.

The amount of charge, dQ, in the volume element, dV, is dQ = ρ(r)∙dV = ρ(r)∙4πr2∙dr , where ρ(r) is the volume charge density as a function of r and is given by ρ(r) = A∙r2 for some constant A. Thus dQ is given by:
dQ = 4∙A∙π∙r4∙dr​
Integrate that over the sphere to find the total charge.
 
  • #3
Thanks Sammy. So Guauss' Law is not used to find a non-uniform charge? If the sphere was a uniform charge, could we rearrange the variables of Gauss' Law to solve for q enclosed?
 
  • #4
lonewolf219 said:
Thanks Sammy. So Guauss' Law is not used to find a non-uniform charge? If the sphere was a uniform charge, could we rearrange the variables of Gauss' Law to solve for q enclosed?
Indeed, for this case, Gauss;s Law Could be used to find the magnitude of the electric field if that's what was asked for.

All they're asking you to do is find the total charge of the sphere.
 
  • #5


A. Yes, Gauss' Law can still be used for this problem even though the sphere is solid. We can use the integral form of Gauss' Law, which states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space (ε0). In this case, we can choose a Gaussian surface in the shape of a sphere with radius r, centered at the center of the solid sphere.

The charge enclosed by this Gaussian surface can be calculated by integrating the non-uniform charge density over the volume of the sphere. This is represented by the equation p=Ar^2, where A is the constant charge density and r is the distance from the center of the sphere. Integrating this over the volume of the sphere gives us the total charge Q=4/5πε0Ar^5.

To get the answer, we can use the fact that the electric flux through a closed surface is equal to the charge enclosed divided by ε0. This gives us the equation Q/ε0=4πr^2E, where E is the electric field at the surface of the sphere. We can then use this equation to solve for E, and use the result to find the total charge Q within the volume of the sphere.

In summary, Gauss' Law can be used for this problem by choosing a Gaussian surface in the shape of a sphere and integrating the non-uniform charge density over the volume of the sphere. This will give us the total charge Q within the volume of the sphere, which can be calculated using the equation Q=4/5πε0Ar^5.
 

1. What is a Gaussian solid sphere Q?

A Gaussian solid sphere Q is a mathematical concept used in statistical mechanics to describe the probability distribution of a particle's position in a solid sphere. It is based on the Gaussian distribution curve, also known as the normal distribution curve, which is commonly used to describe the distribution of data in many natural phenomena.

2. How is Gaussian solid sphere Q calculated?

Gaussian solid sphere Q is calculated using the formula Q = (3/4πσ^3)exp[-(r^2/2σ^2)], where Q represents the probability of finding a particle within a certain radius r in the solid sphere, and σ represents the standard deviation of the distribution. This formula is derived from the Gaussian distribution curve and takes into account the three-dimensional nature of the solid sphere.

3. What is the significance of Gaussian solid sphere Q in statistical mechanics?

In statistical mechanics, Gaussian solid sphere Q is used to describe the behavior of a system of particles within a solid sphere. It helps to predict the probability of finding a particle at a specific location within the sphere, which is important in understanding the overall behavior and properties of the system. It is also a fundamental concept in the study of phase transitions and critical phenomena.

4. Can Gaussian solid sphere Q be applied to real-world systems?

Yes, Gaussian solid sphere Q can be applied to real-world systems, such as the behavior of atoms in a solid crystal or the distribution of particles in a spherical nanoparticle. It is a useful tool for understanding the statistical behavior of microscopic particles in various systems.

5. Are there any limitations to using Gaussian solid sphere Q?

One limitation of Gaussian solid sphere Q is that it assumes a perfectly symmetrical and uniform distribution of particles within the solid sphere. In reality, many systems may have uneven distributions or other factors that can affect the accuracy of the calculations. Additionally, this model does not take into account any interactions between particles, which may be important in some systems.

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