How could electric susceptibilbility depend on position?

In summary, Griffiths is discussing the effective surface charge of a homogeneous and isotropic dielectric. He explains that the bound-charge density within the dielectric is given by the surface divergence of the bound polarization vector. This results in bound-surface charges at the surface of the dielectric, which can be calculated using the surface-normal unity vector and the bound polarization vector.
  • #1
Ahmed1029
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In the statement encircled, what does Griffiths actually mean?
Screenshot_2022-06-07-15-17-53-45_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg
 
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  • #2
It's a bit "nebulous". I guess what he considers is the effective (bound) surface charge of a homogeneous and isotropic dielectric.

[edit: corrected in view of #3]
The bound-charge density within the dielectric is
$$\rho=-\vec{\nabla} \cdot \vec{P},$$
which is 0, for ##\vec{P}=\text{const}##, within the dielectric. Trivially it's also 0 outside the dielectric, where is vacuum, i.e., no charges at all.

At the surface you have, however bound-surface charges, which you get by taking the "surface divergence". Let ##\vec{n}## be the surface-normal unity vector pointing out of the material. Then with a Gaussian pillbox with two sides parallel to the boundary of the dielectric, you get
$$\sigma=-\mathrm{Div} \vec{P}=\vec{n} \cdot \vec{P}.$$
 
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  • #3
vanhees71 said:
which is, for P→=0, within the dielectric
You probably mean ##\vec{P}=\text{constant}## inside the dielectric because if it was zero then ##\vec{D}=\epsilon_0\vec{E}+\vec{P}=\epsilon_0\vec{E}## inside the dielectric which doesn't look right...
 
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  • #4
At the boundary the susceptibility has different values on the two sides of the boundary.
 
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Related to How could electric susceptibilbility depend on position?

1. How does the electric susceptibility vary with position?

The electric susceptibility, denoted by χ, is a measure of how easily a material can be polarized in response to an applied electric field. It is a tensor quantity and can vary with position due to variations in the material's composition, structure, and external factors such as temperature and pressure.

2. What factors can cause the electric susceptibility to change with position?

The electric susceptibility of a material can be influenced by various factors, including its chemical composition, crystal structure, defects or impurities, and external conditions such as temperature and pressure. These factors can lead to variations in the material's polarizability and affect its electric susceptibility at different positions.

3. How does the electric susceptibility affect the behavior of a material?

The electric susceptibility is a crucial parameter that determines the response of a material to an applied electric field. A material with a high electric susceptibility will be highly polarizable and exhibit a strong response to an electric field, while a material with low susceptibility will be less affected by the field. The electric susceptibility also plays a role in determining the dielectric constant and refractive index of a material.

4. Can the electric susceptibility be engineered or controlled at different positions?

Yes, the electric susceptibility can be engineered or controlled at different positions by altering the material's composition, structure, or external conditions. This can be achieved through techniques such as doping, strain engineering, or applying an external electric field. By controlling the electric susceptibility, one can tailor the properties of a material for specific applications, such as in electronic devices or sensors.

5. How is the electric susceptibility measured and characterized at different positions?

The electric susceptibility can be measured and characterized using various experimental techniques, such as dielectric spectroscopy, ellipsometry, or polarimetry. These techniques involve applying an electric field to the material and measuring its response, which can then be used to calculate the electric susceptibility. The susceptibility can also be simulated and predicted using theoretical models, such as density functional theory or molecular dynamics simulations.

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