Deriving analytic expression for magnetic field & flow lines of bar magnet.












4












$begingroup$


How can we analytically derive the flow-lines of a normal permanent bar-magnet?




Physics context & own approach:



In classical electromagnetics we have the legendary Maxwell's Equations:



$$begin{align}nabla cdot {bf E} &= frac{rho}{epsilon_0}\nabla cdot {bf B} &=0\nabla times {bf E} &= -frac{partial bf B}{partial t}\nablatimes {bf B} &= mu_0 left( {bf J}+epsilon_0frac{partial {bf E}}{partial t} right)end{align}$$



The situation in a normal permanent magnet far away from any electric field, we should have:



$$begin{align}{bf E} &= 0\bf J &= 0\{bf B} &= cases{c{bf hat y} hspace{1.0cm}text{(inside magnet)} \{bf B}(x,y) text{ (outside)}}end{align}$$
(c is constant : homogeneous magnet with constant microscopic dipole moment)



Can we use this to derive analytic expression for the flow-lines (or vector field $bf B$) around a permanent magnet?





Pictures show color mapped representation of B field around bar and horseshoe magnets acquired from a numerical optimization.






pde vector-analysis physics electromagnetism







share|cite|improve this question











$endgroup$

















    4












    $begingroup$


    How can we analytically derive the flow-lines of a normal permanent bar-magnet?




    Physics context & own approach:



    In classical electromagnetics we have the legendary Maxwell's Equations:



    $$begin{align}nabla cdot {bf E} &= frac{rho}{epsilon_0}\nabla cdot {bf B} &=0\nabla times {bf E} &= -frac{partial bf B}{partial t}\nablatimes {bf B} &= mu_0 left( {bf J}+epsilon_0frac{partial {bf E}}{partial t} right)end{align}$$



    The situation in a normal permanent magnet far away from any electric field, we should have:



    $$begin{align}{bf E} &= 0\bf J &= 0\{bf B} &= cases{c{bf hat y} hspace{1.0cm}text{(inside magnet)} \{bf B}(x,y) text{ (outside)}}end{align}$$
    (c is constant : homogeneous magnet with constant microscopic dipole moment)



    Can we use this to derive analytic expression for the flow-lines (or vector field $bf B$) around a permanent magnet?





    Pictures show color mapped representation of B field around bar and horseshoe magnets acquired from a numerical optimization.






    pde vector-analysis physics electromagnetism







    share|cite|improve this question











    $endgroup$















      4












      4








      4


      5



      $begingroup$


      How can we analytically derive the flow-lines of a normal permanent bar-magnet?




      Physics context & own approach:



      In classical electromagnetics we have the legendary Maxwell's Equations:



      $$begin{align}nabla cdot {bf E} &= frac{rho}{epsilon_0}\nabla cdot {bf B} &=0\nabla times {bf E} &= -frac{partial bf B}{partial t}\nablatimes {bf B} &= mu_0 left( {bf J}+epsilon_0frac{partial {bf E}}{partial t} right)end{align}$$



      The situation in a normal permanent magnet far away from any electric field, we should have:



      $$begin{align}{bf E} &= 0\bf J &= 0\{bf B} &= cases{c{bf hat y} hspace{1.0cm}text{(inside magnet)} \{bf B}(x,y) text{ (outside)}}end{align}$$
      (c is constant : homogeneous magnet with constant microscopic dipole moment)



      Can we use this to derive analytic expression for the flow-lines (or vector field $bf B$) around a permanent magnet?





      Pictures show color mapped representation of B field around bar and horseshoe magnets acquired from a numerical optimization.






      pde vector-analysis physics electromagnetism







      share|cite|improve this question











      $endgroup$




      How can we analytically derive the flow-lines of a normal permanent bar-magnet?




      Physics context & own approach:



      In classical electromagnetics we have the legendary Maxwell's Equations:



      $$begin{align}nabla cdot {bf E} &= frac{rho}{epsilon_0}\nabla cdot {bf B} &=0\nabla times {bf E} &= -frac{partial bf B}{partial t}\nablatimes {bf B} &= mu_0 left( {bf J}+epsilon_0frac{partial {bf E}}{partial t} right)end{align}$$



      The situation in a normal permanent magnet far away from any electric field, we should have:



      $$begin{align}{bf E} &= 0\bf J &= 0\{bf B} &= cases{c{bf hat y} hspace{1.0cm}text{(inside magnet)} \{bf B}(x,y) text{ (outside)}}end{align}$$
      (c is constant : homogeneous magnet with constant microscopic dipole moment)



      Can we use this to derive analytic expression for the flow-lines (or vector field $bf B$) around a permanent magnet?





      Pictures show color mapped representation of B field around bar and horseshoe magnets acquired from a numerical optimization.






      pde vector-analysis physics electromagnetism




      pde vector-analysis physics electromagnetism






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 30 at 18:09







      mathreadler

















      asked Jan 26 at 14:29









      mathreadlermathreadler

      15.2k72263




      15.2k72263






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          Here is what I suspect is a standard approach to such problems:



          We have



          $nabla times mathbf B = 0; tag 1$



          then



          $nabla times (nabla times mathbf B) = 0; tag 2$



          there is a standard identity from vector calculus which asserts that



          $nabla times (nabla times mathbf B) = nabla(nabla cdot mathbf B) - nabla^2 mathbf B; tag3$



          using this in concert with the Maxwell equation



          $nabla cdot mathbf B = 0 tag 4$



          yields



          $nabla times (nabla times mathbf B) = - nabla^2 mathbf B; tag5$



          (2) thus becomes



          $nabla^2 mathbf B = 0, tag 6$



          where it is understood that $nabla^2$ is applied to $mathbf B$ component-wise; thus we have



          $nabla^2 mathbf B_x = nabla^2 mathbf B_y = nabla^2 mathbf B_z = 0; tag 7$



          we may impose boundary conditions on the components of by by assuming



          $mathbf B = c hat{mathbf y} tag 8$



          in the iron bar, as is given in the text of the problem. We also assume that



          $mathbf B to 0 ; text{at infinity}; tag 9$



          since the components of $mathbf B$ satisfy Laplace's equation, this should be sufficient information to assure the existence of a unique solution for $mathbf B$; but I strongly suspect it will not be very nice, analytically speaking.



          To conclude, there is probably no "nice" formula for the field lines, but they can of course be plotted via numerical methods once we have found the vector field $mathbf B$.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            This is a very nice start. Thank you Robert.
            $endgroup$
            – mathreadler
            Jan 30 at 18:13






          • 1




            $begingroup$
            @mathreadler: Finding the solution will be tough, I'll warrant. Probably need a Green's function on the exterior of the bar. Probably get messy. You might check out J.D. Jackson's book, it has a lot of stuff on problems like this. Do you know it? Cheers!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:16






          • 1




            $begingroup$
            Yep I suppose it would be hard, that is why I asked it. I am an applied maths engineer and have gotten to a level I can write code and find my own numerical solutions by various methods, but my level of solving PDE:s analytically never got nearly as high. Thank you for the book hint! Will see if I can check it out!
            $endgroup$
            – mathreadler
            Jan 30 at 18:40






          • 1




            $begingroup$
            @mathreadler: it is a classic standard graduate level text--quite difficult but very thorough. You should have no trouble locating a copy!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:57











          Your Answer





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          2












          $begingroup$

          Here is what I suspect is a standard approach to such problems:



          We have



          $nabla times mathbf B = 0; tag 1$



          then



          $nabla times (nabla times mathbf B) = 0; tag 2$



          there is a standard identity from vector calculus which asserts that



          $nabla times (nabla times mathbf B) = nabla(nabla cdot mathbf B) - nabla^2 mathbf B; tag3$



          using this in concert with the Maxwell equation



          $nabla cdot mathbf B = 0 tag 4$



          yields



          $nabla times (nabla times mathbf B) = - nabla^2 mathbf B; tag5$



          (2) thus becomes



          $nabla^2 mathbf B = 0, tag 6$



          where it is understood that $nabla^2$ is applied to $mathbf B$ component-wise; thus we have



          $nabla^2 mathbf B_x = nabla^2 mathbf B_y = nabla^2 mathbf B_z = 0; tag 7$



          we may impose boundary conditions on the components of by by assuming



          $mathbf B = c hat{mathbf y} tag 8$



          in the iron bar, as is given in the text of the problem. We also assume that



          $mathbf B to 0 ; text{at infinity}; tag 9$



          since the components of $mathbf B$ satisfy Laplace's equation, this should be sufficient information to assure the existence of a unique solution for $mathbf B$; but I strongly suspect it will not be very nice, analytically speaking.



          To conclude, there is probably no "nice" formula for the field lines, but they can of course be plotted via numerical methods once we have found the vector field $mathbf B$.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            This is a very nice start. Thank you Robert.
            $endgroup$
            – mathreadler
            Jan 30 at 18:13






          • 1




            $begingroup$
            @mathreadler: Finding the solution will be tough, I'll warrant. Probably need a Green's function on the exterior of the bar. Probably get messy. You might check out J.D. Jackson's book, it has a lot of stuff on problems like this. Do you know it? Cheers!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:16






          • 1




            $begingroup$
            Yep I suppose it would be hard, that is why I asked it. I am an applied maths engineer and have gotten to a level I can write code and find my own numerical solutions by various methods, but my level of solving PDE:s analytically never got nearly as high. Thank you for the book hint! Will see if I can check it out!
            $endgroup$
            – mathreadler
            Jan 30 at 18:40






          • 1




            $begingroup$
            @mathreadler: it is a classic standard graduate level text--quite difficult but very thorough. You should have no trouble locating a copy!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:57
















          2












          $begingroup$

          Here is what I suspect is a standard approach to such problems:



          We have



          $nabla times mathbf B = 0; tag 1$



          then



          $nabla times (nabla times mathbf B) = 0; tag 2$



          there is a standard identity from vector calculus which asserts that



          $nabla times (nabla times mathbf B) = nabla(nabla cdot mathbf B) - nabla^2 mathbf B; tag3$



          using this in concert with the Maxwell equation



          $nabla cdot mathbf B = 0 tag 4$



          yields



          $nabla times (nabla times mathbf B) = - nabla^2 mathbf B; tag5$



          (2) thus becomes



          $nabla^2 mathbf B = 0, tag 6$



          where it is understood that $nabla^2$ is applied to $mathbf B$ component-wise; thus we have



          $nabla^2 mathbf B_x = nabla^2 mathbf B_y = nabla^2 mathbf B_z = 0; tag 7$



          we may impose boundary conditions on the components of by by assuming



          $mathbf B = c hat{mathbf y} tag 8$



          in the iron bar, as is given in the text of the problem. We also assume that



          $mathbf B to 0 ; text{at infinity}; tag 9$



          since the components of $mathbf B$ satisfy Laplace's equation, this should be sufficient information to assure the existence of a unique solution for $mathbf B$; but I strongly suspect it will not be very nice, analytically speaking.



          To conclude, there is probably no "nice" formula for the field lines, but they can of course be plotted via numerical methods once we have found the vector field $mathbf B$.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            This is a very nice start. Thank you Robert.
            $endgroup$
            – mathreadler
            Jan 30 at 18:13






          • 1




            $begingroup$
            @mathreadler: Finding the solution will be tough, I'll warrant. Probably need a Green's function on the exterior of the bar. Probably get messy. You might check out J.D. Jackson's book, it has a lot of stuff on problems like this. Do you know it? Cheers!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:16






          • 1




            $begingroup$
            Yep I suppose it would be hard, that is why I asked it. I am an applied maths engineer and have gotten to a level I can write code and find my own numerical solutions by various methods, but my level of solving PDE:s analytically never got nearly as high. Thank you for the book hint! Will see if I can check it out!
            $endgroup$
            – mathreadler
            Jan 30 at 18:40






          • 1




            $begingroup$
            @mathreadler: it is a classic standard graduate level text--quite difficult but very thorough. You should have no trouble locating a copy!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:57














          2












          2








          2





          $begingroup$

          Here is what I suspect is a standard approach to such problems:



          We have



          $nabla times mathbf B = 0; tag 1$



          then



          $nabla times (nabla times mathbf B) = 0; tag 2$



          there is a standard identity from vector calculus which asserts that



          $nabla times (nabla times mathbf B) = nabla(nabla cdot mathbf B) - nabla^2 mathbf B; tag3$



          using this in concert with the Maxwell equation



          $nabla cdot mathbf B = 0 tag 4$



          yields



          $nabla times (nabla times mathbf B) = - nabla^2 mathbf B; tag5$



          (2) thus becomes



          $nabla^2 mathbf B = 0, tag 6$



          where it is understood that $nabla^2$ is applied to $mathbf B$ component-wise; thus we have



          $nabla^2 mathbf B_x = nabla^2 mathbf B_y = nabla^2 mathbf B_z = 0; tag 7$



          we may impose boundary conditions on the components of by by assuming



          $mathbf B = c hat{mathbf y} tag 8$



          in the iron bar, as is given in the text of the problem. We also assume that



          $mathbf B to 0 ; text{at infinity}; tag 9$



          since the components of $mathbf B$ satisfy Laplace's equation, this should be sufficient information to assure the existence of a unique solution for $mathbf B$; but I strongly suspect it will not be very nice, analytically speaking.



          To conclude, there is probably no "nice" formula for the field lines, but they can of course be plotted via numerical methods once we have found the vector field $mathbf B$.






          share|cite|improve this answer











          $endgroup$



          Here is what I suspect is a standard approach to such problems:



          We have



          $nabla times mathbf B = 0; tag 1$



          then



          $nabla times (nabla times mathbf B) = 0; tag 2$



          there is a standard identity from vector calculus which asserts that



          $nabla times (nabla times mathbf B) = nabla(nabla cdot mathbf B) - nabla^2 mathbf B; tag3$



          using this in concert with the Maxwell equation



          $nabla cdot mathbf B = 0 tag 4$



          yields



          $nabla times (nabla times mathbf B) = - nabla^2 mathbf B; tag5$



          (2) thus becomes



          $nabla^2 mathbf B = 0, tag 6$



          where it is understood that $nabla^2$ is applied to $mathbf B$ component-wise; thus we have



          $nabla^2 mathbf B_x = nabla^2 mathbf B_y = nabla^2 mathbf B_z = 0; tag 7$



          we may impose boundary conditions on the components of by by assuming



          $mathbf B = c hat{mathbf y} tag 8$



          in the iron bar, as is given in the text of the problem. We also assume that



          $mathbf B to 0 ; text{at infinity}; tag 9$



          since the components of $mathbf B$ satisfy Laplace's equation, this should be sufficient information to assure the existence of a unique solution for $mathbf B$; but I strongly suspect it will not be very nice, analytically speaking.



          To conclude, there is probably no "nice" formula for the field lines, but they can of course be plotted via numerical methods once we have found the vector field $mathbf B$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Feb 18 at 17:18









          mathreadler

          15.2k72263




          15.2k72263










          answered Jan 28 at 21:49









          Robert LewisRobert Lewis

          48.3k23167




          48.3k23167








          • 1




            $begingroup$
            This is a very nice start. Thank you Robert.
            $endgroup$
            – mathreadler
            Jan 30 at 18:13






          • 1




            $begingroup$
            @mathreadler: Finding the solution will be tough, I'll warrant. Probably need a Green's function on the exterior of the bar. Probably get messy. You might check out J.D. Jackson's book, it has a lot of stuff on problems like this. Do you know it? Cheers!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:16






          • 1




            $begingroup$
            Yep I suppose it would be hard, that is why I asked it. I am an applied maths engineer and have gotten to a level I can write code and find my own numerical solutions by various methods, but my level of solving PDE:s analytically never got nearly as high. Thank you for the book hint! Will see if I can check it out!
            $endgroup$
            – mathreadler
            Jan 30 at 18:40






          • 1




            $begingroup$
            @mathreadler: it is a classic standard graduate level text--quite difficult but very thorough. You should have no trouble locating a copy!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:57














          • 1




            $begingroup$
            This is a very nice start. Thank you Robert.
            $endgroup$
            – mathreadler
            Jan 30 at 18:13






          • 1




            $begingroup$
            @mathreadler: Finding the solution will be tough, I'll warrant. Probably need a Green's function on the exterior of the bar. Probably get messy. You might check out J.D. Jackson's book, it has a lot of stuff on problems like this. Do you know it? Cheers!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:16






          • 1




            $begingroup$
            Yep I suppose it would be hard, that is why I asked it. I am an applied maths engineer and have gotten to a level I can write code and find my own numerical solutions by various methods, but my level of solving PDE:s analytically never got nearly as high. Thank you for the book hint! Will see if I can check it out!
            $endgroup$
            – mathreadler
            Jan 30 at 18:40






          • 1




            $begingroup$
            @mathreadler: it is a classic standard graduate level text--quite difficult but very thorough. You should have no trouble locating a copy!
            $endgroup$
            – Robert Lewis
            Jan 30 at 18:57








          1




          1




          $begingroup$
          This is a very nice start. Thank you Robert.
          $endgroup$
          – mathreadler
          Jan 30 at 18:13




          $begingroup$
          This is a very nice start. Thank you Robert.
          $endgroup$
          – mathreadler
          Jan 30 at 18:13




          1




          1




          $begingroup$
          @mathreadler: Finding the solution will be tough, I'll warrant. Probably need a Green's function on the exterior of the bar. Probably get messy. You might check out J.D. Jackson's book, it has a lot of stuff on problems like this. Do you know it? Cheers!
          $endgroup$
          – Robert Lewis
          Jan 30 at 18:16




          $begingroup$
          @mathreadler: Finding the solution will be tough, I'll warrant. Probably need a Green's function on the exterior of the bar. Probably get messy. You might check out J.D. Jackson's book, it has a lot of stuff on problems like this. Do you know it? Cheers!
          $endgroup$
          – Robert Lewis
          Jan 30 at 18:16




          1




          1




          $begingroup$
          Yep I suppose it would be hard, that is why I asked it. I am an applied maths engineer and have gotten to a level I can write code and find my own numerical solutions by various methods, but my level of solving PDE:s analytically never got nearly as high. Thank you for the book hint! Will see if I can check it out!
          $endgroup$
          – mathreadler
          Jan 30 at 18:40




          $begingroup$
          Yep I suppose it would be hard, that is why I asked it. I am an applied maths engineer and have gotten to a level I can write code and find my own numerical solutions by various methods, but my level of solving PDE:s analytically never got nearly as high. Thank you for the book hint! Will see if I can check it out!
          $endgroup$
          – mathreadler
          Jan 30 at 18:40




          1




          1




          $begingroup$
          @mathreadler: it is a classic standard graduate level text--quite difficult but very thorough. You should have no trouble locating a copy!
          $endgroup$
          – Robert Lewis
          Jan 30 at 18:57




          $begingroup$
          @mathreadler: it is a classic standard graduate level text--quite difficult but very thorough. You should have no trouble locating a copy!
          $endgroup$
          – Robert Lewis
          Jan 30 at 18:57


















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