Simple question about polarized materials

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In summary, the electric field in all three regions of a thick spherical shell made of dielectric material with a "frozen-in" polarization is given by $\mathbf{E}=-\nabla \phi-k\hat{\mathbf{r}}$ for $r<a$, and $\mathbf{E}=0$ for $r>b$, where $\phi$ is the scalar potential and $k$ is a constant representing the polarization. This is because in the region $r>b$, the charge enclosed is zero according to Gauss's law.
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Homework Statement


A thick spherical shell of inner radius a and outer radius b is made of dielectric material with a "frozen-in" polarization

P = [tex]\frac{k}{r}[/tex] [tex]\hat{r}[/tex]

where k is a constant and r is the distance from the center. Find the electric field in all three regions.

Homework Equations


Gauss's Law: E*A = [tex]Q_{enc}/\epsilon_{0}[/tex]

The Attempt at a Solution


Basically my question is simple. I've done the whole thing; I have a solution. The problem is that I don't believe the solution. Shouldn't the [tex]Q_{enc}[/tex] for the region where r > b be 0? Since there are no free charges, isn't the material made completely of dipoles because it is a dielectric, and if that is the case shouldn't each positive pole cancel each negative pole causing the enclosed charge to be zero?
 
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  • #2
I'm really confused, and any help would be greatly appreciated.A:The electric field inside a dielectric material is given by$$\mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{P}}{\partial t}$$where $\mathbf{P}$ is the polarization of the material.In this case, the polarization is given by $\mathbf{P}=kr\hat{\mathbf{r}}$, so the electric field is$$\mathbf{E}=-\nabla \phi-k\hat{\mathbf{r}}$$As you can see, there is no dependence on time here, so $\partial \mathbf{P}/\partial t=0$. Therefore, in the region $r>b$, the electric field is just given by the gradient of the scalar potential, i.e., $\mathbf{E}=-\nabla \phi$. Since the charge enclosed must be zero, Gauss's law tells you that $\mathbf{E}=0$.
 
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Yes, you are correct. In a dielectric material, the polarization is caused by the alignment of dipoles, not the presence of free charges. Therefore, in the region where r > b, the enclosed charge should be zero and the electric field should also be zero according to Gauss's Law. This would mean that the solution you have obtained may be incorrect and you should double-check your calculations. Additionally, the polarization should also be taken into account when calculating the electric field in the other two regions, as it will contribute to the total electric field.
 

What are polarized materials?

Polarized materials are materials that have the ability to block certain types of light waves, allowing only light waves that vibrate in one direction to pass through.

What causes materials to become polarized?

Materials become polarized when light waves interact with the atoms or molecules within the material, causing the waves to align in one direction.

What are some examples of polarized materials?

Some common examples of polarized materials are polarized sunglasses, LCD screens, and certain types of plastic films used in photography.

How do polarized materials affect light?

Polarized materials can either block or allow certain types of light waves to pass through, which can affect the intensity and appearance of light. They can also reduce glare and improve visibility.

Can polarized materials be used for scientific research?

Yes, polarized materials are commonly used in scientific research, particularly in fields such as optics, materials science, and chemistry. They can also be used in various techniques such as polarized microscopy and polarized spectroscopy.

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