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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| author | Dimosthenis Stamopoulos |
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| description | 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. |
| format | Article |
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| spelling | doaj-art-d8371f6409f94fbf894486e3678fb2a32025-08-20T03:13:54ZengMDPI AGCrystals2073-43522025-03-0115433110.3390/cryst15040331Universal Expressions for the Polarization and the Depolarization Factor in Homogeneous Dielectric and Magnetic Spheres Subjected to an External Field of <i>Any</i> FormDimosthenis Stamopoulos0Department of Physics, School of Science, National and Kapodistrian University of Athens, Zografou Panepistimioupolis, 15784 Athens, GreeceSpherical 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.https://www.mdpi.com/2073-4352/15/4/331electric polarizationmagnetic polarizationdepolarization factordielectric spheresmagnetic spheresdepolarizing factor |
| spellingShingle | Dimosthenis Stamopoulos 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 Crystals electric polarization magnetic polarization depolarization factor dielectric spheres magnetic spheres depolarizing factor |
| title | 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 |
| title_full | 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 |
| title_fullStr | 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 |
| title_full_unstemmed | 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 |
| title_short | 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 |
| title_sort | 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 |
| topic | electric polarization magnetic polarization depolarization factor dielectric spheres magnetic spheres depolarizing factor |
| url | https://www.mdpi.com/2073-4352/15/4/331 |
| work_keys_str_mv | AT dimosthenisstamopoulos universalexpressionsforthepolarizationandthedepolarizationfactorinhomogeneousdielectricandmagneticspheressubjectedtoanexternalfieldofianyiform |