Universal Expressions for the Polarization and the Depolarization Factor in Homogeneous Dielectric and Magnetic Spheres Subjected to an External Field of <i>Any</i> Form
Spherical structures of dielectric and magnetic materials are studied intensively in basic research and employed widely in applications. The polarization, (<b>P</b> for dielectric and <b>M</b> for magnetic materials), is the parent physical vector of all relevant entities (e....
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-03-01
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| Series: | Crystals |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-4352/15/4/331 |
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| Summary: | Spherical structures of dielectric and magnetic materials are studied intensively in basic research and employed widely in applications. The polarization, (<b>P</b> for dielectric and <b>M</b> for magnetic materials), is the parent physical vector of all relevant entities (e.g., moment, , and force, <b>F</b>), which determine the signals recorded by an experimental setup or diagnostic equipment and configure the motion in real space. Here, we use classical electromagnetism to study the polarization, , of spherical structures of linear and isotropic—however, not necessarily homogeneous—materials subjected to an external vector field, (<b>E</b><sub>ext</sub> for dielectric and <b>H</b><sub>ext</sub> for magnetic materials), dc (static), or even ac of low frequency (quasistatic limit). We tackle an integro-differential equation on the polarization, , able to provide closed-form solutions, determined solely from , on the basis of spherical harmonics, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi mathvariant="normal">Y</mi></mrow><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msubsup></mrow></semantics></math></inline-formula>. These generic equations can be used to calculate analytically the polarization, , directly from an external field, , of any form. The proof of concept is studied in homogeneous dielectric and magnetic spheres. Indeed, the polarization, , can be obtained by universal expressions, directly applicable for <i>any</i> form of the external field, . Notably, we obtain the relation between the extrinsic, , and intrinsic, , susceptibilities (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mo>χ</mo></mrow><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mrow></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mo>χ</mo></mrow><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msubsup></mrow></semantics></math></inline-formula> for dielectric and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mo>χ</mo></mrow><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mrow></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mo>χ</mo></mrow><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msubsup></mrow></semantics></math></inline-formula> for magnetic materials) and clarify the nature of the depolarization factor, , which depends on the degree <i>l</i>—however, not on the order <i>m</i> of the mode <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the applied . Our universal approach can be useful to understand the physics and to facilitate applications of such spherical structures. |
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| ISSN: | 2073-4352 |