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Sang-Hyeon Han
- 9
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If we arrange the polarization spherically, Does it make a uniformly charged sphere radius of R?? If right, How can I find out \vec{P}(\vec{r}) which results in the constant charge density \rho?
You can solve the divergence equation in spherical coordinates to produce a ## \rho_p ## that is constant everywhere, where ## -\nabla \cdot P=\rho_p ##, but polarization still creates an electrically neutral system, so that the surface polarization charge density of ## \sigma_p=P \cdot \hat{n} ## will be such that it neutralizes any positive ## \rho_p ## in the interior. Solving the divergence equation, you get ## P_r(r)=-(\rho_p/3)r ##. ## \\ ## ## \int \rho_p dv=+(4/3) \pi R^3 \rho_p. ## ## \\ ## Now ## P_r=-(\rho_p/3)R ## at ## r=R ##, so that ## \sigma_p=-(\rho_p/3)R ##. ## \\ ## ## \int \sigma_p dA=-(\rho_p/3)R(4 \pi R^2) ## precisely neutralizing ## \int \rho_p dv ##.Sang-Hyeon Han said:If we arrange the polarization spherically, Does it make a uniformly charged sphere radius of R?? If right, How can I find out \vec{P}(\vec{r}) which results in the constant charge density \rho?
very thanks!Charles Link said:You can solve the divergence equation in spherical coordinates to produce a ## \rho_p ## that is constant everywhere, where ## -\nabla \cdot P=\rho_p ##, but polarization still creates an electrically neutral system, so that the surface polarization charge density of ## \sigma_p=P \cdot \hat{n} ## will be such that it neutralizes any positive ## \rho_p ## in the interior. Solving the divergence equation, you get ## P_r(r)=-(\rho_p/3)r ##. ## \\ ## ## \int \rho_p dv=+(4/3) \pi R^3 \rho_p. ## ## \\ ## Now ## P_r=-(\rho_p/3)R ## at ## r=R ##, so that ## \sigma_p=-(\rho_p/3)R ##. ## \\ ## ## \int \sigma_p dA=-(\rho_p/3)R(4 \pi R^2) ## precisely neutralizing ## \int \rho_p dv ##.
Spherically arranged polarization is a phenomenon in which the electric field vectors of a polarized electromagnetic wave are arranged in a spherical pattern around the direction of propagation, rather than being aligned in a specific direction.
In linear polarization, the electric field vectors are aligned in a single direction, while in circular polarization, they rotate around the direction of propagation. Spherically arranged polarization is a combination of both linear and circular polarization, with the vectors arranged in a spherical pattern.
Spherically arranged polarization can be caused by the superposition of two or more polarized waves with different directions of polarization. It can also occur in natural light, such as from the Sun, due to the random orientation of polarized light waves.
Spherically arranged polarization has applications in imaging and microscopy, where it can be used to enhance the contrast and resolution of images. It is also used in telecommunications, as it allows for a larger bandwidth and longer transmission distances compared to linear or circular polarization.
Spherically arranged polarization can be measured using a polarimeter, which measures the intensity and direction of the electric field vectors of a polarized wave. Another method is to use a wave plate, which changes the polarization of light passing through it, to convert spherically arranged polarization into linear or circular polarization for easier measurement.