Apparent paradox with Ampère's law (bringing about questions with other laws)












2














Let’s say we have a current wire with a current $I$ flowing. We know there is a field of $B=frac{mu_0I}{2pi r}$ by using Ampère's law, and a simple integration path which goes circularly around the wire. Now if we take the path of integration as so the surface spans doesn’t intercept the wire we trivially get a $B=0$ which is obviously incorrect.



I see that I have essentially treated it as if there is no current even present. But a similar argument is used in other situations without fault.



Take for example a conducting cylinder with a hollow, cylindrical shaped space inside. By the same argument there is no field inside.



To further illustrate my point, the derivation of the B field inside of a solenoid requires you to intercept the currents. You can’t simply do the loop inside of the air gap.



This, at least to me, seems like the same thing, and I can’t justify why one is incorrect and the other is incorrect. Please point out why I am stupid.






electromagnetism







share|cite|improve this question





























    2














    Let’s say we have a current wire with a current $I$ flowing. We know there is a field of $B=frac{mu_0I}{2pi r}$ by using Ampère's law, and a simple integration path which goes circularly around the wire. Now if we take the path of integration as so the surface spans doesn’t intercept the wire we trivially get a $B=0$ which is obviously incorrect.



    I see that I have essentially treated it as if there is no current even present. But a similar argument is used in other situations without fault.



    Take for example a conducting cylinder with a hollow, cylindrical shaped space inside. By the same argument there is no field inside.



    To further illustrate my point, the derivation of the B field inside of a solenoid requires you to intercept the currents. You can’t simply do the loop inside of the air gap.



    This, at least to me, seems like the same thing, and I can’t justify why one is incorrect and the other is incorrect. Please point out why I am stupid.






    electromagnetism







    share|cite|improve this question



























      2












      2








      2







      Let’s say we have a current wire with a current $I$ flowing. We know there is a field of $B=frac{mu_0I}{2pi r}$ by using Ampère's law, and a simple integration path which goes circularly around the wire. Now if we take the path of integration as so the surface spans doesn’t intercept the wire we trivially get a $B=0$ which is obviously incorrect.



      I see that I have essentially treated it as if there is no current even present. But a similar argument is used in other situations without fault.



      Take for example a conducting cylinder with a hollow, cylindrical shaped space inside. By the same argument there is no field inside.



      To further illustrate my point, the derivation of the B field inside of a solenoid requires you to intercept the currents. You can’t simply do the loop inside of the air gap.



      This, at least to me, seems like the same thing, and I can’t justify why one is incorrect and the other is incorrect. Please point out why I am stupid.






      electromagnetism







      share|cite|improve this question















      Let’s say we have a current wire with a current $I$ flowing. We know there is a field of $B=frac{mu_0I}{2pi r}$ by using Ampère's law, and a simple integration path which goes circularly around the wire. Now if we take the path of integration as so the surface spans doesn’t intercept the wire we trivially get a $B=0$ which is obviously incorrect.



      I see that I have essentially treated it as if there is no current even present. But a similar argument is used in other situations without fault.



      Take for example a conducting cylinder with a hollow, cylindrical shaped space inside. By the same argument there is no field inside.



      To further illustrate my point, the derivation of the B field inside of a solenoid requires you to intercept the currents. You can’t simply do the loop inside of the air gap.



      This, at least to me, seems like the same thing, and I can’t justify why one is incorrect and the other is incorrect. Please point out why I am stupid.






      electromagnetism




      electromagnetism






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 34 mins ago









      Peter Mortensen

      1,92811323




      1,92811323










      asked 8 hours ago









      Jake RoseJake Rose

      7218




      7218






















          2 Answers
          2






          active

          oldest

          votes


















          5














          You aren't stupid, you are just learning things the same way we all did: the school of hard knocks. Ampere's law says that the integral around that closed path is zero, not that the field is zero at every point. What the law tells us is that the field is sometimes "positive" and sometimes "negative" on that path, and when we add up contributions from everywhere on the path, we get zero.






          share|cite|improve this answer





















          • What about if you take a circularly symmetric path in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0..
            – Jake Rose
            8 hours ago










          • @JakeRose In that case $vec{B}$ is perpendicular to $dvec{l}$, so the integral is 0 (without the field being 0)
            – Poon Levi
            8 hours ago



















          4














          Your argument is incorrect. $oint vec Bcdot dvec ell$ is $0$ when no current is enclosed but this does not imply $B=0$: you cannot use $oint vec Bcdot dvec ell= Btimes 2pi r =mu_0 I_{encl}$ since $vec B$ is not constant on the loop defined by the contour: in other words, $ointvec Bcdot dvec ell$ is not $Btimes 2pi r$ unless the contour is one where $vec Bcdot dvec ell$ is constant.






          share|cite|improve this answer























          • What if you take a circularly symmetric loop in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0
            – Jake Rose
            8 hours ago










          • @JakeRose Yes... Ampere's law is always true, but it's now always useful in recovering $vec B$: only in some specialized symmetric situations can one do this (see physics.stackexchange.com/q/318183/36194).
            – ZeroTheHero
            8 hours ago











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "151"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: false,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: null,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f452776%2fapparent-paradox-with-amp%25c3%25a8res-law-bringing-about-questions-with-other-laws%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          5














          You aren't stupid, you are just learning things the same way we all did: the school of hard knocks. Ampere's law says that the integral around that closed path is zero, not that the field is zero at every point. What the law tells us is that the field is sometimes "positive" and sometimes "negative" on that path, and when we add up contributions from everywhere on the path, we get zero.






          share|cite|improve this answer





















          • What about if you take a circularly symmetric path in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0..
            – Jake Rose
            8 hours ago










          • @JakeRose In that case $vec{B}$ is perpendicular to $dvec{l}$, so the integral is 0 (without the field being 0)
            – Poon Levi
            8 hours ago
















          5














          You aren't stupid, you are just learning things the same way we all did: the school of hard knocks. Ampere's law says that the integral around that closed path is zero, not that the field is zero at every point. What the law tells us is that the field is sometimes "positive" and sometimes "negative" on that path, and when we add up contributions from everywhere on the path, we get zero.






          share|cite|improve this answer





















          • What about if you take a circularly symmetric path in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0..
            – Jake Rose
            8 hours ago










          • @JakeRose In that case $vec{B}$ is perpendicular to $dvec{l}$, so the integral is 0 (without the field being 0)
            – Poon Levi
            8 hours ago














          5












          5








          5






          You aren't stupid, you are just learning things the same way we all did: the school of hard knocks. Ampere's law says that the integral around that closed path is zero, not that the field is zero at every point. What the law tells us is that the field is sometimes "positive" and sometimes "negative" on that path, and when we add up contributions from everywhere on the path, we get zero.






          share|cite|improve this answer












          You aren't stupid, you are just learning things the same way we all did: the school of hard knocks. Ampere's law says that the integral around that closed path is zero, not that the field is zero at every point. What the law tells us is that the field is sometimes "positive" and sometimes "negative" on that path, and when we add up contributions from everywhere on the path, we get zero.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 8 hours ago









          garypgaryp

          16.6k12962




          16.6k12962












          • What about if you take a circularly symmetric path in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0..
            – Jake Rose
            8 hours ago










          • @JakeRose In that case $vec{B}$ is perpendicular to $dvec{l}$, so the integral is 0 (without the field being 0)
            – Poon Levi
            8 hours ago


















          • What about if you take a circularly symmetric path in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0..
            – Jake Rose
            8 hours ago










          • @JakeRose In that case $vec{B}$ is perpendicular to $dvec{l}$, so the integral is 0 (without the field being 0)
            – Poon Levi
            8 hours ago
















          What about if you take a circularly symmetric path in the solenoid?
          – Jake Rose
          8 hours ago




          What about if you take a circularly symmetric path in the solenoid?
          – Jake Rose
          8 hours ago












          Ahhh, the field is perpendicular, so B (in that direction) really does = 0..
          – Jake Rose
          8 hours ago




          Ahhh, the field is perpendicular, so B (in that direction) really does = 0..
          – Jake Rose
          8 hours ago












          @JakeRose In that case $vec{B}$ is perpendicular to $dvec{l}$, so the integral is 0 (without the field being 0)
          – Poon Levi
          8 hours ago




          @JakeRose In that case $vec{B}$ is perpendicular to $dvec{l}$, so the integral is 0 (without the field being 0)
          – Poon Levi
          8 hours ago











          4














          Your argument is incorrect. $oint vec Bcdot dvec ell$ is $0$ when no current is enclosed but this does not imply $B=0$: you cannot use $oint vec Bcdot dvec ell= Btimes 2pi r =mu_0 I_{encl}$ since $vec B$ is not constant on the loop defined by the contour: in other words, $ointvec Bcdot dvec ell$ is not $Btimes 2pi r$ unless the contour is one where $vec Bcdot dvec ell$ is constant.






          share|cite|improve this answer























          • What if you take a circularly symmetric loop in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0
            – Jake Rose
            8 hours ago










          • @JakeRose Yes... Ampere's law is always true, but it's now always useful in recovering $vec B$: only in some specialized symmetric situations can one do this (see physics.stackexchange.com/q/318183/36194).
            – ZeroTheHero
            8 hours ago
















          4














          Your argument is incorrect. $oint vec Bcdot dvec ell$ is $0$ when no current is enclosed but this does not imply $B=0$: you cannot use $oint vec Bcdot dvec ell= Btimes 2pi r =mu_0 I_{encl}$ since $vec B$ is not constant on the loop defined by the contour: in other words, $ointvec Bcdot dvec ell$ is not $Btimes 2pi r$ unless the contour is one where $vec Bcdot dvec ell$ is constant.






          share|cite|improve this answer























          • What if you take a circularly symmetric loop in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0
            – Jake Rose
            8 hours ago










          • @JakeRose Yes... Ampere's law is always true, but it's now always useful in recovering $vec B$: only in some specialized symmetric situations can one do this (see physics.stackexchange.com/q/318183/36194).
            – ZeroTheHero
            8 hours ago














          4












          4








          4






          Your argument is incorrect. $oint vec Bcdot dvec ell$ is $0$ when no current is enclosed but this does not imply $B=0$: you cannot use $oint vec Bcdot dvec ell= Btimes 2pi r =mu_0 I_{encl}$ since $vec B$ is not constant on the loop defined by the contour: in other words, $ointvec Bcdot dvec ell$ is not $Btimes 2pi r$ unless the contour is one where $vec Bcdot dvec ell$ is constant.






          share|cite|improve this answer














          Your argument is incorrect. $oint vec Bcdot dvec ell$ is $0$ when no current is enclosed but this does not imply $B=0$: you cannot use $oint vec Bcdot dvec ell= Btimes 2pi r =mu_0 I_{encl}$ since $vec B$ is not constant on the loop defined by the contour: in other words, $ointvec Bcdot dvec ell$ is not $Btimes 2pi r$ unless the contour is one where $vec Bcdot dvec ell$ is constant.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 1 hour ago

























          answered 8 hours ago









          ZeroTheHeroZeroTheHero

          18.9k52956




          18.9k52956












          • What if you take a circularly symmetric loop in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0
            – Jake Rose
            8 hours ago










          • @JakeRose Yes... Ampere's law is always true, but it's now always useful in recovering $vec B$: only in some specialized symmetric situations can one do this (see physics.stackexchange.com/q/318183/36194).
            – ZeroTheHero
            8 hours ago


















          • What if you take a circularly symmetric loop in the solenoid?
            – Jake Rose
            8 hours ago










          • Ahhh, the field is perpendicular, so B (in that direction) really does = 0
            – Jake Rose
            8 hours ago










          • @JakeRose Yes... Ampere's law is always true, but it's now always useful in recovering $vec B$: only in some specialized symmetric situations can one do this (see physics.stackexchange.com/q/318183/36194).
            – ZeroTheHero
            8 hours ago
















          What if you take a circularly symmetric loop in the solenoid?
          – Jake Rose
          8 hours ago




          What if you take a circularly symmetric loop in the solenoid?
          – Jake Rose
          8 hours ago












          Ahhh, the field is perpendicular, so B (in that direction) really does = 0
          – Jake Rose
          8 hours ago




          Ahhh, the field is perpendicular, so B (in that direction) really does = 0
          – Jake Rose
          8 hours ago












          @JakeRose Yes... Ampere's law is always true, but it's now always useful in recovering $vec B$: only in some specialized symmetric situations can one do this (see physics.stackexchange.com/q/318183/36194).
          – ZeroTheHero
          8 hours ago




          @JakeRose Yes... Ampere's law is always true, but it's now always useful in recovering $vec B$: only in some specialized symmetric situations can one do this (see physics.stackexchange.com/q/318183/36194).
          – ZeroTheHero
          8 hours ago


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Physics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f452776%2fapparent-paradox-with-amp%25c3%25a8res-law-bringing-about-questions-with-other-laws%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          flock() on closed filehandle LOCK_FILE at /usr/bin/apt-mirror

          Image
          0 I recently upgraded an Ubuntu 16.04 virtual machine to 18.04. I regularly use this VM to run apt-mirror at school on their high-speed internet, then rsync these files to my rack-mount server at home for updating multiple computers at home without wasting the tiny amount of bandwidth I have there. Since updating I can't get apt-mirror to run on my VM. It keeps giving the error: flock() on closed filehandle LOCK_FILE at /usr/bin/apt-mirror line 214 immediately followed by: apt-mirror is already running, exiting at /usr/bin/apt-mirror line 217 apt-mirror is NOT running, and does not have a cron job. I've tried rebooting, and even looked for the old-style lockfile to delete, but I guess that method is no longer used. Is there some way to force unlock this so I can run apt-mirror? UPDATE: So... I f
          Read more

          Mangá

          Image
            Nota:  Não confundir com Manga. Mangás e Animes Mangá Mangaká • Tankōbon Film comic La nouvelle manga Mangá original em inglês Anime História Animação influenciada por animes • Filler • OVA • Seiyū Públicos Kodomo • Shōjo • Shōnen Josei • Seinen Gêneros Mahō shōjo • Mecha • Harém Ecchi • Hentai ( Yaoi • Yuri Futanari • Shotacon • Lolicon Toddlercon ) Termos Bishōjo • Chibi • Fan service Gaiden • Kawaii • Kaitō • Light novel Mahō shōnen • Moe (antropomorfismo • Kemonomimi Nekomimi ) • Super deformed Omake • Tsundere • Yonkoma Listas Mangás • Animes • Ecchi • Mangaka • Estúdios • Revistas Fãs ( otaku ) Anime Music Video Convenções • Cosplay Dōjinshi • Fansub • Fandub • Fujoshi OS-tan • Scanlation Portal de Anime e mangá v d e O mangá (português brasileiro) ou manga (português europeu) [ 1 ] (em japonês: 漫画 , manga ? , lit. “história em quadrinhos”) , é a
          Read more

          Eduardo VII do Reino Unido

          Image
          Eduardo VII Rei do Reino Unido da Grã-Bretanha e Irlanda e dos Domínios Britânicos de Além-mar Imperador da Índia Rei do Reino Unido e Imperador da Índia Reinado 22 de janeiro de 1901 a 6 de maio de 1910 Coroação 9 de agosto de 1902 Delhi Durbar 1 de janeiro de 1903 Predecessora Vitória Sucessor Jorge V   Esposa Alexandra da Dinamarca Descendência Alberto Vitor, Duque de Clarence e Avondale Jorge V do Reino Unido Luísa, Princesa Real Vitória de Gales Maud de Gales Alexandre João de Gales Casa Saxe-Coburgo-Gota Nome completo Alberto Eduardo Nascimento 9 de novembro de 1841   Palácio de Buckingham, Londres, Reino Unido Morte 6 de maio de 1910 (68 anos)   Palácio de Buckingham, Londres, Reino Unido Enterro Capela de São Jorge, Windsor, Berkshire, Reino Unido 20 de maio de 1910 Pai Alberto de Saxe-Coburgo-Gota Mãe Vitória do Reino Unido Assinatura
          Read more