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class="vector-toc-link" href="#Champs_d'application"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Champs d'application</span> </div> </a> <button aria-controls="toc-Champs_d'application-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Champs d'application</span> </button> <ul id="toc-Champs_d'application-sublist" class="vector-toc-list"> <li id="toc-En_acoustique" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#En_acoustique"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>En acoustique</span> </div> </a> <ul id="toc-En_acoustique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-En_électromagnétisme" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#En_électromagnétisme"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>En électromagnétisme</span> </div> </a> <ul id="toc-En_électromagnétisme-sublist" class="vector-toc-list"> <li id="toc-Généralités" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Généralités"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Généralités</span> </div> </a> <ul id="toc-Généralités-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemple" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Exemple"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Exemple</span> </div> </a> <ul id="toc-Exemple-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-En_électrostatique" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#En_électrostatique"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>En électrostatique</span> </div> </a> <ul id="toc-En_électrostatique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-En_mécanique_(gravitation)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#En_mécanique_(gravitation)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>En mécanique (gravitation)</span> </div> </a> <ul id="toc-En_mécanique_(gravitation)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interprétation_en_théorie_des_champs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interprétation_en_théorie_des_champs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Interprétation en théorie des champs</span> </div> </a> <ul id="toc-Interprétation_en_théorie_des_champs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notes et références</span> </div> </a> <button aria-controls="toc-Notes_et_références-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Notes et références</span> </button> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Loi en carré inverse</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 31 langues." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-31" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">31 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%A7%D9%84%D8%AA%D8%B1%D8%A8%D9%8A%D8%B9_%D8%A7%D9%84%D8%B9%D9%83%D8%B3%D9%8A" title="قانون التربيع العكسي – arabe" lang="ar" hreflang="ar" data-title="قانون التربيع العكسي" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B0%D0%B4%D0%B2%D0%B0%D1%80%D0%BE%D1%82%D0%BD%D1%8B%D1%85_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B0%D1%9E" title="Закон адваротных квадратаў – biélorusse" lang="be" hreflang="be" data-title="Закон адваротных квадратаў" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_de_l%27invers_del_quadrat" title="Llei de l'invers del quadrat – catalan" lang="ca" hreflang="ca" data-title="Llei de l'invers del quadrat" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Z%C3%A1kon_p%C5%99evr%C3%A1cen%C3%BDch_%C4%8Dtverc%C5%AF" title="Zákon převrácených čtverců – tchèque" lang="cs" hreflang="cs" data-title="Zákon převrácených čtverců" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Afstandskvadratloven" title="Afstandskvadratloven – danois" lang="da" hreflang="da" data-title="Afstandskvadratloven" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abstandsgesetz" title="Abstandsgesetz – allemand" lang="de" hreflang="de" data-title="Abstandsgesetz" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Inverse-square_law" title="Inverse-square law – anglais" lang="en" hreflang="en" data-title="Inverse-square law" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Inversa-kvadrata_le%C4%9Do" title="Inversa-kvadrata leĝo – espéranto" lang="eo" hreflang="eo" data-title="Inversa-kvadrata leĝo" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ley_de_la_inversa_del_cuadrado" title="Ley de la inversa del cuadrado – espagnol" lang="es" hreflang="es" data-title="Ley de la inversa del cuadrado" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D9%85%D8%B1%D8%A8%D8%B9_%D9%85%D8%B9%DA%A9%D9%88%D8%B3" title="قانون مربع معکوس – persan" lang="fa" hreflang="fa" data-title="قانون مربع معکوس" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/K%C3%A4%C3%A4nteisen_neli%C3%B6n_laki" title="Käänteisen neliön laki – finnois" lang="fi" hreflang="fi" data-title="Käänteisen neliön laki" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%AD_an_chearnfhaid_inbh%C3%A9artaigh" title="Dlí an chearnfhaid inbhéartaigh – irlandais" lang="ga" hreflang="ga" data-title="Dlí an chearnfhaid inbhéartaigh" data-language-autonym="Gaeilge" data-language-local-name="irlandais" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE_%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="व्युत्क्रम वर्ग नियम – hindi" lang="hi" hreflang="hi" data-title="व्युत्क्रम वर्ग नियम" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Inverz_n%C3%A9gyzetes_t%C3%B6rv%C3%A9ny" title="Inverz négyzetes törvény – hongrois" lang="hu" hreflang="hu" data-title="Inverz négyzetes törvény" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_kuadrat_terbalik" title="Hukum kuadrat terbalik – indonésien" lang="id" hreflang="id" data-title="Hukum kuadrat terbalik" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Legge_dell%27inverso_del_quadrato" title="Legge dell'inverso del quadrato – italien" lang="it" hreflang="it" data-title="Legge dell'inverso del quadrato" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%862%E4%B9%97%E3%81%AE%E6%B3%95%E5%89%87" title="逆2乗の法則 – japonais" lang="ja" hreflang="ja" data-title="逆2乗の法則" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%AD%EC%A0%9C%EA%B3%B1_%EB%B2%95%EC%B9%99" title="역제곱 법칙 – coréen" lang="ko" hreflang="ko" data-title="역제곱 법칙" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Lex_quadratica_inversa" title="Lex quadratica inversa – latin" lang="la" hreflang="la" data-title="Lex quadratica inversa" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hukum_kuasa_dua_songsang" title="Hukum kuasa dua songsang – malais" lang="ms" hreflang="ms" data-title="Hukum kuasa dua songsang" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Omgekeerde_kwadratenwet" title="Omgekeerde kwadratenwet – néerlandais" lang="nl" hreflang="nl" data-title="Omgekeerde kwadratenwet" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Invers-kvadratisk_lov" title="Invers-kvadratisk lov – norvégien bokmål" lang="nb" hreflang="nb" data-title="Invers-kvadratisk lov" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Lei_do_Inverso_do_Quadrado" title="Lei do Inverso do Quadrado – portugais" lang="pt" hreflang="pt" data-title="Lei do Inverso do Quadrado" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D1%8B%D1%85_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BE%D0%B2" title="Закон обратных квадратов – russe" lang="ru" hreflang="ru" data-title="Закон обратных квадратов" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Inverse-square_law" title="Inverse-square law – Simple English" lang="en-simple" hreflang="en-simple" data-title="Inverse-square law" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Inversa_kvadratlagen" title="Inversa kvadratlagen – suédois" lang="sv" hreflang="sv" data-title="Inversa kvadratlagen" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AF%8D_%E0%AE%87%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AE%9F%E0%AE%BF_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="எதிர் இருமடி விதி – tamoul" lang="ta" hreflang="ta" data-title="எதிர் இருமடி விதி" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ters_kare_yasas%C4%B1" title="Ters kare yasası – turc" lang="tr" hreflang="tr" data-title="Ters kare yasası" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%BE%D0%B1%D0%B5%D1%80%D0%BD%D0%B5%D0%BD%D0%B8%D1%85_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%96%D0%B2" title="Закон обернених квадратів – ukrainien" lang="uk" hreflang="uk" data-title="Закон обернених квадратів" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E5%8F%8D%E6%AF%94%E5%AE%9A%E5%BE%8B" title="平方反比定律 – chinois" lang="zh" hreflang="zh" data-title="平方反比定律" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E5%8F%8D%E6%AF%94%E5%AE%9A%E5%BE%8B" title="平方反比定律 – chinois littéraire" lang="lzh" hreflang="lzh" data-title="平方反比定律" data-language-autonym="文言" data-language-local-name="chinois littéraire" class="interlanguage-link-target"><span>文言</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q333094#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></span></div> </div> </div> </div> </header> 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fichier:Inverse_square_law.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Inverse_square_law.svg/500px-Inverse_square_law.svg.png" decoding="async" width="500" height="333" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Inverse_square_law.svg/750px-Inverse_square_law.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Inverse_square_law.svg/1000px-Inverse_square_law.svg.png 2x" data-file-width="480" data-file-height="320" /></a><figcaption>Ce diagramme montre le fonctionnement de la loi en carré inverse. Les lignes représentent le <a href="/wiki/Flux_(math%C3%A9matiques)" title="Flux (mathématiques)">flux</a> émanant de la source. Le nombre total de lignes de flux dépend de l'intensité de la source et est constant avec l'accroissement de la distance. Une densité plus importante de lignes de flux (lignes par unité de surface) est la traduction d'un champ plus intense. La densité de flux est inversement proportionnelle au carré de la distance à la source car l'aire d'un secteur de disque s'accroit avec le carré de son rayon. L'intensité du champ est donc inversement proportionnelle au carré de la distance à la source.</figcaption></figure> <p>En <a href="/wiki/Physique" title="Physique">physique</a>, une <b>loi en carré inverse</b> est une <a href="/wiki/Loi_physique" title="Loi physique">loi physique</a> postulant qu'une <a href="/wiki/Quantit%C3%A9" title="Quantité">quantité</a> physique (énergie, force, ou autre) est inversement proportionnelle au <a href="/wiki/Carr%C3%A9_(alg%C3%A8bre)" title="Carré (algèbre)">carré</a> de la <a href="/wiki/Distance_(math%C3%A9matiques)" title="Distance (mathématiques)">distance</a> de l'origine de cette quantité physique. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Histoire">Histoire</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=1" title="Modifier la section : Histoire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=1" title="Modifier le code source de la section : Histoire"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cette loi fut d'abord suggérée en 1645 par l'astronome français <a href="/wiki/Isma%C3%ABl_Boulliau" title="Ismaël Boulliau">Ismaël Boulliau</a> dans son livre <i>Astronomica Philolaica</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>Note 1<span class="cite_crochet">]</span></a></sup>, puis mise en forme par <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> en 1687 après que <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a> lui eut proposé l'idée dans une lettre datée du <time class="nowrap" datetime="1679-12-09" data-sort-value="1679-12-09">9 décembre 1679</time><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>Note 2<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. Hooke accusa plus tard Newton de <a href="/wiki/Plagiat" title="Plagiat">plagiat</a> lors de la publication en 1687 de ses <i><a href="/wiki/Philosophiae_naturalis_principia_mathematica" class="mw-redirect" title="Philosophiae naturalis principia mathematica">Principia Mathematica</a></i> dans lesquels il introduisait et établissait le même concept. </p> <div class="mw-heading mw-heading2"><h2 id="Champs_d'application"><span id="Champs_d.27application"></span>Champs d'application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=2" title="Modifier la section : Champs d'application" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=2" title="Modifier le code source de la section : Champs d'application"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="En_acoustique">En acoustique</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=3" title="Modifier la section : En acoustique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=3" title="Modifier le code source de la section : En acoustique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Acoustique" title="Acoustique">Acoustique</a>.</div></div> <p>La loi du carré inverse se trouve en propagation du <a href="/wiki/Son_(physique)" title="Son (physique)">son</a> lorsqu'il s'agit de donner son <a href="/wiki/Intensit%C3%A9_acoustique" title="Intensité acoustique">intensité</a> à une distance donnée d'une source sonore<sup id="cite_ref-Inverse_square_sound_intensity_4-0" class="reference"><a href="#cite_note-Inverse_square_sound_intensity-4"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. </p><p>La <a href="/wiki/Son_(physique)" title="Son (physique)">pression sonore</a> d'un <a href="/wiki/Front_d%27onde" title="Front d'onde">front d'onde</a> <a href="/wiki/Sph%C3%A8re" title="Sphère">sphérique</a> irradiant d'un point source décroit d'un facteur ½ lorsque la distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> est doublée, ou, si elle est exprimée en <a href="/wiki/D%C3%A9cibel" title="Décibel">dB</a>, perdra 6,02 dB. Le comportement de cette pression n'est donc pas en carré inverse, mais simplement inverse : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\propto {\frac {1}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\propto {\frac {1}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec9054dc12b1e74ca14e8d2041cd4faea9f7fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:6.356ex; height:5.176ex;" alt="{\displaystyle p\propto {\frac {1}{r}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p_{1}}{p_{2}}}={\frac {r_{2}}{r_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{1}}{p_{2}}}={\frac {r_{2}}{r_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433fa2c23d6b49d5d131d0decdafd8245cfd048a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.097ex; height:5.343ex;" alt="{\displaystyle {\frac {p_{1}}{p_{2}}}={\frac {r_{2}}{r_{1}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}=p_{2}\cdot r_{2}\cdot {\frac {1}{r_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}=p_{2}\cdot r_{2}\cdot {\frac {1}{r_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ca66215798fd225d8ac0d23cc37f1e2af26b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.089ex; width:16.035ex; height:5.509ex;" alt="{\displaystyle p_{1}=p_{2}\cdot r_{2}\cdot {\frac {1}{r_{1}}}}"></span></dd></dl> <p>Cependant, la remarque est également valable pour la <a href="/wiki/Vitesse_particulaire" title="Vitesse particulaire">vitesse particulaire</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> qui est <a href="/wiki/D%C3%A9phasage" title="Déphasage">en phase</a> avec la pression sonore instantanée <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\propto {\frac {1}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\propto {\frac {1}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015c8de8c907047c4e0d2841cfb5bc0df7afb3de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.225ex; height:5.176ex;" alt="{\displaystyle v\propto {\frac {1}{r}}}"></span></dd></dl> <p>La <a href="/wiki/Forme_quadratique" title="Forme quadratique">composante quadratique</a> de la vitesse particulaire est dans un déphasage de 90° avec la pression sonore seulement en <a href="/wiki/Champ_proche" class="mw-redirect" title="Champ proche">champ proche</a>, ce qui ne contribue pas à l'énergie moyennée temporellement (i.e. intensité sonore). Cette composante se trouve être en carré inverse. L'<a href="/wiki/Intensit%C3%A9_acoustique" title="Intensité acoustique">intensité acoustique</a> est le produit de la <a href="/wiki/Valeur_efficace" title="Valeur efficace">valeur efficace</a> de la pression sonore et de celle de la vitesse particulaire (composante en phase), les deux étant proportionnelles inverses. L'intensité suit donc une loi en carré inverse comme indiqué ci-dessous : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=p\cdot v\propto {\frac {1}{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mi>p</mi> <mo>⋅</mo> <mi>v</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=p\cdot v\propto {\frac {1}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8906c3a6408290b5ebbd1ec61700a2649f74802b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.284ex; height:5.509ex;" alt="{\displaystyle I=p\cdot v\propto {\frac {1}{r^{2}}}}"></span>.</dd></dl> <p>Cette loi en carré inverse a trait à l'intensité sonore. Cependant, les pressions sonores étant plus accessibles à la mesure, cette loi est parfois qualifiée de « loi en inverse de la distance ». </p> <div class="mw-heading mw-heading3"><h3 id="En_électromagnétisme"><span id="En_.C3.A9lectromagn.C3.A9tisme"></span>En électromagnétisme</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=4" title="Modifier la section : En électromagnétisme" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=4" title="Modifier le code source de la section : En électromagnétisme"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/%C3%89lectromagn%C3%A9tisme" title="Électromagnétisme">électromagnétisme</a>.</div></div> <div class="mw-heading mw-heading4"><h4 id="Généralités"><span id="G.C3.A9n.C3.A9ralit.C3.A9s"></span>Généralités</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=5" title="Modifier la section : Généralités" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=5" title="Modifier le code source de la section : Généralités"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'<a href="/wiki/Intensit%C3%A9" class="mw-disambig" title="Intensité">intensité</a> (ou <a href="/wiki/%C3%89clairement_lumineux" title="Éclairement lumineux">éclairement lumineux</a> ou <a href="/wiki/Irradiance" class="mw-redirect" title="Irradiance">irradiance</a>) de la <a href="/wiki/Lumi%C3%A8re" title="Lumière">lumière</a> ou d'autres ondes linéaires se propageant à partir d'une <a href="/w/index.php?title=Source_ponctuelle&action=edit&redlink=1" class="new" title="Source ponctuelle (page inexistante)">source ponctuelle</a> (énergie par unité de surface perpendiculaire à la source) est inversement proportionnelle au carré de la distance à la source, ce qui fait qu'un objet (de même taille) placé deux fois plus loin recevra seulement un quart de l'<a href="/wiki/%C3%89nergie" title="Énergie">énergie</a> émise (pour la même période).<br /> Plus généralement, l'irradiance, <i>i.e.</i> l'intensité (ou <a href="/wiki/Puissance_(physique)" title="Puissance (physique)">puissance</a> par unité de surface dans la direction de <a href="/wiki/Propagation_des_ondes" title="Propagation des ondes">propagation</a>) d'un <a href="/wiki/Front_d%27onde" title="Front d'onde">front d'onde</a> <a href="/wiki/Sph%C3%A9rique" class="mw-redirect" title="Sphérique">sphérique</a> varie en raison inverse de la distance à la source (en postulant qu'il n'y ait pas de pertes dues à l'<a href="/wiki/Absorption_(optique)" title="Absorption (optique)">optique</a> ou à la <a href="/wiki/Diffusion_des_ondes" title="Diffusion des ondes">diffusion</a>).<br /> Ainsi par exemple, l'intensité de la lumière émise par le <a href="/wiki/Soleil" title="Soleil">Soleil</a> est de 9140 <a href="/wiki/Watt" title="Watt">watts</a> par <a href="/wiki/M%C3%A8tre_carr%C3%A9" title="Mètre carré">mètre carré</a> à la distance de <a href="/wiki/Mercure_(plan%C3%A8te)" title="Mercure (planète)">Mercure</a> (0,387 UA), mais seulement de 1 370 <abbr class="abbr" title="watt par mètre carré">W/m<sup>2</sup></abbr> à la distance de la <a href="/wiki/Terre" title="Terre">Terre</a> (1 UA), une multiplication par trois environ de la distance résultant en une division par neuf environ de l'intensité lumineuse.<br /> Les photographes et les éclairagistes utilisent cette loi afin de déterminer la localisation optimale des sources de <a href="/wiki/Lumi%C3%A8re" title="Lumière">lumière</a> afin que le sujet soit convenablement éclairé. Cette loi est aussi particulièrement importante en <a href="/wiki/Radiographie" title="Radiographie">radiographie</a>, dans la planification des traitements en <a href="/wiki/Radioth%C3%A9rapie" title="Radiothérapie">radiothérapie</a> et en <a href="/wiki/Radioprotection" title="Radioprotection">radioprotection</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Exemple">Exemple</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=6" title="Modifier la section : Exemple" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=6" title="Modifier le code source de la section : Exemple"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prenons comme exemple la puissance totale irradiée à partir d'un point source (comme une <a href="/wiki/Antenne_isotrope" title="Antenne isotrope">antenne isotrope</a>), notée <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. Pour des distances à la source importantes (relativement à la taille de la source), cette puissance est distribuée sur des surfaces sphériques de plus en plus importantes lorsque la distance s'accroît. Puisque la superficie d'une sphère de rayon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> est <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca925588619ce34da35b2c2ffb5b267e313d50ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.439ex; height:2.676ex;" alt="{\displaystyle A=4\pi r^{2}}"></span>, l'<a href="/wiki/Intensit%C3%A9" class="mw-disambig" title="Intensité">intensité</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> de la radiation à la distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> est : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4bc12fb1cc12752d0a57aaa45596b57a23bd45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.031ex; height:5.509ex;" alt="{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\propto {\frac {1}{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\propto {\frac {1}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d966f7c79f2a116000c0dd6f85838a06e982bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.209ex; height:5.509ex;" alt="{\displaystyle I\propto {\frac {1}{r^{2}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {I_{1}}{I_{2}}}={\frac {{r_{2}}^{2}}{{r_{1}}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {I_{1}}{I_{2}}}={\frac {{r_{2}}^{2}}{{r_{1}}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3fe4ced59b8d9153c2c347872799266a3411759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.005ex; height:6.176ex;" alt="{\displaystyle {\frac {I_{1}}{I_{2}}}={\frac {{r_{2}}^{2}}{{r_{1}}^{2}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}=I_{2}\cdot {r_{2}^{2}}\cdot {\frac {1}{{r_{1}}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}=I_{2}\cdot {r_{2}^{2}}\cdot {\frac {1}{{r_{1}}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84527b47209f711ce64f5255c782dd2d1956311f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.707ex; height:5.676ex;" alt="{\displaystyle I_{1}=I_{2}\cdot {r_{2}^{2}}\cdot {\frac {1}{{r_{1}}^{2}}}}"></span></dd></dl> <p>L'énergie ou l'intensité décroît d'un facteur ¼ lorsque la distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> est doublée. C'est la raison fondamentale pour laquelle l'intensité d'une <a href="/wiki/Rayonnement" title="Rayonnement">radiation</a>, qu'elle soit <a href="/wiki/%C3%89lectromagn%C3%A9tique" class="mw-redirect" title="Électromagnétique">électromagnétique</a> ou <a href="/wiki/Acoustique" title="Acoustique">acoustique</a>, suit une loi en carré inverse, et ceci au moins dans un contexte idéalement tridimensionnel (une propagation en deux dimensions suivrait une loi inverse de la distance et en une dimension, l'<a href="/wiki/Onde_plane" title="Onde plane">onde plane</a> resterait constante en amplitude alors même que la distance à la source change). </p> <div class="mw-heading mw-heading3"><h3 id="En_électrostatique"><span id="En_.C3.A9lectrostatique"></span>En électrostatique</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=7" title="Modifier la section : En électrostatique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=7" title="Modifier le code source de la section : En électrostatique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Articles détaillés : <a href="/wiki/%C3%89lectrostatique" title="Électrostatique">électrostatique</a> et <a href="/wiki/Loi_de_Coulomb_(%C3%A9lectrostatique)" title="Loi de Coulomb (électrostatique)">Loi de Coulomb (électrostatique)</a>.</div></div> <p>En électrostatique, la force agissant sur un objet chargé par un autre objet chargé est une <a href="/wiki/Force_centrale" title="Force centrale">force centrale</a> dont l'expression est la suivante : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{1\rightarrow 2}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}{\|{\vec {r}}_{12}\|}^{2}}}.{\frac {{\vec {r}}_{12}}{\|{\vec {r}}_{12}\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo stretchy="false">→</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{1\rightarrow 2}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}{\|{\vec {r}}_{12}\|}^{2}}}.{\frac {{\vec {r}}_{12}}{\|{\vec {r}}_{12}\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5457fe8433825bfdc3378e0ffbe01892d929499c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.63ex; height:6.676ex;" alt="{\displaystyle {\vec {F}}_{1\rightarrow 2}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}{\|{\vec {r}}_{12}\|}^{2}}}.{\frac {{\vec {r}}_{12}}{\|{\vec {r}}_{12}\|}}}"></span></dd></dl> <p>dans laquelle <i>ε</i><sub>0</sub> = 8,854 × 10<sup>−12</sup> <abbr class="abbr" title="farad par mètre">F/m</abbr> est une constante universelle appelée <a href="/wiki/Permittivit%C3%A9" title="Permittivité">permittivité électrique</a> du vide, et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{12}={\overrightarrow {M_{1}M_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{12}={\overrightarrow {M_{1}M_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19586437e3449618ee24372d51d81f7463b2f24a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.447ex; width:12.944ex; height:4.176ex;" alt="{\displaystyle {\vec {r}}_{12}={\overrightarrow {M_{1}M_{2}}}}"></span> est le <a href="/wiki/Vecteur_position" title="Vecteur position">vecteur position</a> qui relie le premier corps au deuxième, les lettres <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> désignant les charges des deux objets. </p> <div class="mw-heading mw-heading3"><h3 id="En_mécanique_(gravitation)"><span id="En_m.C3.A9canique_.28gravitation.29"></span>En mécanique (gravitation)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=8" title="Modifier la section : En mécanique (gravitation)" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=8" title="Modifier le code source de la section : En mécanique (gravitation)"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Articles détaillés : <a href="/wiki/Gravitation" title="Gravitation">gravitation</a> et <a href="/wiki/Potentiel_newtonien" title="Potentiel newtonien">potentiel newtonien</a>.</div></div> <p>Dans le cadre de la <a href="/wiki/M%C3%A9canique_newtonienne" title="Mécanique newtonienne">mécanique newtonienne</a>, la force de gravitation s'exprime comme une force centrale dépendante de l'inverse du carré de la distance à l'objet exerçant l'attraction de manière tout à fait analogue à la loi de Coulomb : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{12}=-G{\frac {m_{1}m_{2}}{d^{2}}}{\vec {u}}_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{12}=-G{\frac {m_{1}m_{2}}{d^{2}}}{\vec {u}}_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71abeb4d06bb93b2eb964152db2f4fd8e98ef9b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.612ex; height:5.009ex;" alt="{\displaystyle {\vec {F}}_{12}=-G{\frac {m_{1}m_{2}}{d^{2}}}{\vec {u}}_{12}}"></span></dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02692d91696638b124e69ce91adc174c9eb1b825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.647ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{12}}"></span> étant la <a href="/wiki/Force_(physique)" title="Force (physique)">force</a> gravitationnelle exercée par le corps 1 sur le corps 2 (en <a href="/wiki/Newton_(unit%C3%A9)" title="Newton (unité)">newtons</a> ou <abbr class="abbr" title="mètre kilogramme par seconde carré">m kg s<sup>−2</sup></abbr>) ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, la <a href="/wiki/Constante_gravitationnelle" title="Constante gravitationnelle">constante gravitationnelle</a>, qui vaut 6,674 2 × 10<sup>−11</sup> <abbr class="abbr" title="newton mètre carré par kilogramme carré">N m<sup>2</sup> kg<sup>−2</sup></abbr> (ou <abbr class="abbr" title="mètre cube par kilogramme seconde carré">m<sup>3</sup> kg<sup>−1</sup> s<sup>−2</sup></abbr>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecebe334d5cadc3ffcf245eb02919034d7a2ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}"></span>, les masses des deux corps en présence (en <a href="/wiki/Kilogramme" title="Kilogramme">kilogrammes</a>) ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>, la distance entre les deux corps (en <a href="/wiki/M%C3%A8tre" title="Mètre">mètres</a>) ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeed63f5d047e34525c2195a9e659c15df0cfb8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.676ex;" alt="{\displaystyle {\vec {u}}_{12}}"></span> est un <a href="/wiki/Vecteur" title="Vecteur">vecteur</a> unitaire dirigé du corps 1 vers le corps 2 ;</li> <li>le signe – indique que le corps 2 est attiré par le corps 1.</li></ul> <p>Il convient de noter que ce type de forces (force de gravitation et force électrostatique) génèrent des <a href="/wiki/Trajectoire" title="Trajectoire">trajectoires</a> qui sont ce qu'on appelle en mathématiques des <a href="/wiki/Conique" title="Conique">coniques</a> (<a href="/wiki/Ellipse_(math%C3%A9matiques)" title="Ellipse (mathématiques)">ellipses</a>, <a href="/wiki/Parabole" title="Parabole">paraboles</a> et <a href="/wiki/Hyperbole_(math%C3%A9matiques)" title="Hyperbole (mathématiques)">hyperboles</a>), ce qui se retrouve en particulier dans les mouvements des objets célestes lorsque les effets relativistes sont peu perceptibles. </p> <div class="mw-heading mw-heading2"><h2 id="Interprétation_en_théorie_des_champs"><span id="Interpr.C3.A9tation_en_th.C3.A9orie_des_champs"></span>Interprétation en théorie des champs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=9" title="Modifier la section : Interprétation en théorie des champs" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=9" title="Modifier le code source de la section : Interprétation en théorie des champs"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour un <a href="/wiki/Champ_irrotationnel" class="mw-redirect" title="Champ irrotationnel">champ de vecteurs irrotationnel</a>, une loi en carré inverse correspond à une <a href="/wiki/Divergence_(analyse_vectorielle)" title="Divergence (analyse vectorielle)">divergence</a> nulle hors de la source. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=10" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=10" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Cet article utilise du contenu du <i>Federal Standard 1037C</i>, qui, en tant que travail du Gouvernement américain, est dans le domaine public.</li></ul> <ul><li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé <span class="plainlinks">« <a class="external text" href="https://en.wikipedia.org/wiki/Inverse-square_law?oldid=156944800">Inverse-square law</a> » <small>(<a class="external text" href="https://en.wikipedia.org/wiki/Inverse-square_law?action=history">voir la liste des auteurs</a>)</small></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=11" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=11" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Isaac Newton était un lecteur d'Ismaël Boulliau et il citera ses données en 1687 dans ses <i>Principia Mathematica</i></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text">La lettre de Hooke à Newton été vendue aux enchères chez <a href="/wiki/Sotheby%27s" title="Sotheby's">Sotheby's</a> en avril 1918. Publiée en 1952, elle est maintenant la propriété de la bibliothèque de l'<a href="/wiki/Universit%C3%A9_Yale" title="Université Yale">Université de Yale</a> à <a href="/wiki/New_Haven" title="New Haven">New Haven</a> (<a href="/wiki/Connecticut" title="Connecticut">Connecticut</a>)</span> </li> </ol></div> </div> <div class="mw-heading mw-heading3"><h3 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&veaction=edit&section=12" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Loi_en_carr%C3%A9_inverse&action=edit&section=12" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Koyré1952"><span class="ouvrage" id="Alexandre_Koyré1952"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Alexandre Koyré, « <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/227384"><cite style="font-style:normal;" lang="en">An Unpublished Letter of Robert Hooke to Isaac Newton : Une lettre non publiée de Robert Hooke à Isaac Newton</cite></a> », sur <span class="italique"><a href="/wiki/JSTOR" title="JSTOR">jstor.org</a></span>, <time class="nowrap" datetime="1952-12-04" data-sort-value="1952-12-04">4 décembre 1952</time> <small style="line-height:1em;">(consulté le <time class="nowrap" datetime="2020-09-22" data-sort-value="2020-09-22">22 septembre 2020</time>)</small></span></span>.</span> </li> <li id="cite_note-Inverse_square_sound_intensity-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-Inverse_square_sound_intensity_4-0">↑</a> </span><span class="reference-text"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/invsqs.html">Inverse-Square law for sound</a></span> </li> </ol></div> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Physique" title="Portail de la physique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/36px-Circle-icons-physics-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/48px-Circle-icons-physics-logo.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Physique" title="Portail:Physique">Portail de la physique</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:T%C3%A9l%C3%A9communications" title="Portail des télécommunications"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Circle-icons-phone.svg/24px-Circle-icons-phone.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Circle-icons-phone.svg/36px-Circle-icons-phone.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Circle-icons-phone.svg/48px-Circle-icons-phone.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:T%C3%A9l%C3%A9communications" title="Portail:Télécommunications">Portail des télécommunications</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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