{"id":72,"date":"2014-11-08T02:41:55","date_gmt":"2014-11-08T02:41:55","guid":{"rendered":"http:\/\/underthehood.blogwyrm.com\/?p=72"},"modified":"2022-07-28T06:25:38","modified_gmt":"2022-07-28T10:25:38","slug":"why-dont-we-teach-the-helmholtz-theorem","status":"publish","type":"post","link":"https:\/\/underthehood.blogwyrm.com\/?p=72","title":{"rendered":"Why Don’t We Teach the Helmholtz Theorem?"},"content":{"rendered":"

In an earlier post<\/a>, I outlined a derivation of the Helmholtz Theorem starting from the identity<\/p>\n

\\[ \\nabla^2 \\left( \\frac{1}{|\\vec r – \\vec r \\, {}’|} \\right) = -4 \\pi \\delta( \\vec r – \\vec r \\, {}’ ) .\\]<\/p>\n

It seems hard for me to believe but it was many years after I had studied E&M in graduate school that I\u00a0came across this theorem and to appreciate its power. \u00a0The reason I say it is hard for me to believe is that almost none of the traditional texts talk about it or even have an index entry for it.<\/p>\n

The traditional way we teach electricity and magnetism is to trace\u00a0through the historic development by examining a host of 18th<\/sup> and 19th<\/sup>\u00a0century experiments. The usual course is to introduce integral forms\u00a0of the laws and then show how these can lead to Maxwell’s equation (e.g., from Coulomb\u2019s law to Gauss\u2019s law). \u00a0The road here is long and tortuous, starting with static fields, then layering on the\u00a0time dependence, and then finally sneaking the\u00a0displacement current into Ampere’s law. \u00a0This approach obscures the unity of Maxwell’s equations. \u00a0It also leaves the student bored, confused, and overwhelmed, and incapable of appreciating what follows.<\/p>\n

To such a student is lost the wonder of realizing that the fields can take on a life of their own, independent of the things that created them. \u00a0Lost is the realization that these fields can radiate outwards; that they can reflect and refract (i.e., optics as an inherently electromagnetic phenomenon); and that they can be generated and controlled at will to form the communication network we all use on a daily basis.<\/p>\n

A better approach\u00a0starts with Maxwell\u2019s equations in their full form and then uses the Helmholtz theorem to ‘interrogate’ them to derive the time-honored static field results of Coulomb and Biot-Savart. I believe this approach, which is nearly impossible to find in the usual textbooks, offers a clearer\u00a0view into the unity of electricity and magnetism, at the cost of some slightly\u00a0more mature vector calculus. \u00a0The\u00a0fields are introduced early, and the equations they satisfy are complete. There is no unlearning facts\u00a0later on. \u00a0For example, the traditional textbook results for Coulomb’s law emphasize that the electric field is conservative and that its curl is identically zero. \u00a0Weeks or months go by with that concept firmly emphasized and entrenched and then, and only then, is the student informed that the result doesn’t hold in general.<\/p>\n

There is also a fundamental flaw in the pedagogy of deriving Maxwell’s equations from the integral forms. \u00a0Nowhere along the line is there any explanation as to why knowing the divergence and curl of a vector field is all that is needed to uniquely specify the field.\u00a0After all, why can’t a uniform field be added as a constant of integration?<\/p>\n

At the heart of the traditional approach is the idea of a force field as a physically real object and not\u00a0just a useful mathematical construct. The prototype example is the electric field, which comes from\u00a0the experimental expression for Coulomb’s law, stating that the force on two charges $$q_2$$ due to $$q_1$$ is given by\u00a0(note all equations are expressed in SI units):<\/p>\n

\\[ \\vec F_{21}(\\vec r) = \\frac{1}{4 \\pi \\epsilon_0} \\frac{ q_1 q_2 \\left( \\vec r_2 – \\vec r_1 \\right)}{|\\vec r_2 – \\vec r_1|^3} \\, .\\]<\/p>\n

The usual practice is then to assume one of the charges is a test charge and that the other is smeared into an\u00a0arbitrary charge distribution within a volume $$V$$ and that the resulting electric field is<\/p>\n

\\[ \\vec E(\\vec r) = \\frac{1}{4 \\pi \\epsilon_0} \\int_V d^3 r \\frac{ \\rho(\\vec r \\;’) \\left( \\vec r – \\vec r \\; ‘ \\right)}{|\\vec r – \\vec r \\; ‘|^3} \\, . \\]<\/p>\n

The last step in the the traditional approach involves introducing vector field divergence and curl, the associated theorems they obey, and applying the whole lot to\u00a0electric flux\u00a0to get the\u00a0first of the Maxwell equations<\/p>\n

\\[ \\nabla \\cdot \\vec E (\\vec r) = \\rho(\\vec r) \/ \\epsilon_0 \\, .\\]<\/p>\n

As the traditional program proceeds, magnetostatics follows with the introduction of the Biot-Savart law<\/p>\n

\\[ \\vec B(\\vec r) = \\frac{\\mu_0}{4 \\pi} \\int_V d^3 r’ \\frac{ \\vec J(\\vec r \\; ‘) \\times (\\vec r – \\vec r \\;’)}{|\\vec r – \\vec r \\; ‘|^3} \\, , \\]<\/p>\n

as the experimental observation for the generation of a magnetic field for a given current density within a volume $$V$$.\u00a0This time the vanishing of the divergence is used to find the vector potential and the curl is\u00a0related to the current density via Ampere’s law<\/p>\n

\\[ \\nabla \\times \\vec B(\\vec r) = \\mu_0 \\vec J(\\vec r) \\,.\\]<\/p>\n

The traditional approach finally gets to time-varying fields when taking up Faraday’s law, requiring the student to unlearn $$\\nabla \\times \\vec E = 0$$ and then finally re-learn Ampere’s equation with the introduction of the displacement current.\u00a0By this time the full Maxwell equations are on display, but the linkage between the\u00a0different facets of each field is highly obscured, and the basic underpinning of the theory — that the divergence and\u00a0curl tells all there is to know about a field — is not to be found. The pedagogy seems to suffer from too many unconnected\u00a0facts with no common framework by which to relate them.<\/p>\n

Using the Helmholtz Theorem in conjunction with an upfront statement of the Maxwell equations offers several advantages in\u00a0teaching electromagnetism. I will content myself with just the derivation of the Coulomb’s and Biot-Savart’s law. \u00a0Additional information can be found in the paper<\/a> and presentation<\/a> I recently wrote for the Fall meeting of the Chesapeake Section of AAPT.<\/p>\n

Start by\u00a0considering the Maxwell equations, presented here in vacuum, as<\/p>\n

\\[ \\nabla \\cdot \\vec E(\\vec r,t) = \\rho(\\vec r,t) \/ \\epsilon_0 \\, ,\\]<\/p>\n

\\[ \\nabla \\cdot \\vec B(\\vec r,t) = 0 \\, , \\]<\/p>\n

\\[ \\nabla \\times \\vec E(\\vec r,t) = -\\frac{\\partial \\vec B (\\vec r,t)}{\\partial t} \\, , \\]<\/p>\n

and<\/p>\n

\\[ \\nabla \\times \\vec B(\\vec r,t) = \\mu_0 \\vec J (\\vec r,t) + \\epsilon_0 \\mu_0 \\frac{\\partial \\vec E(\\vec r,t)}{\\partial t} \\, .\\]<\/p>\n

Since the Coulomb and Biot-Savart laws are in the domain of the electro- and magnetostatics, all terms\u00a0in Maxwell’s equations\u00a0involving time derivatives are set equal to zero and all $$t$$’s are eliminated to yield
\n\\[ \\nabla \\cdot \\vec E(\\vec r) = \\rho(\\vec r) \/ \\epsilon_0 \\, ,\\]
\n\\[ \\nabla \\cdot \\vec B(\\vec r) = 0 \\, , \\]
\n\\[ \\nabla \\times \\vec E(\\vec r) = 0 \\, , \\]<\/p>\n

and
\n\\[ \\nabla \\times \\vec B(\\vec r) = \\mu_0 \\vec J (\\vec r) \\, .\\]<\/p>\n

Now substituting the electric and magnetic field divergences and curls into the $$U(\\vec r)$$ and $$\\vec W(\\vec r)$$ expressions in Helmholtz’s theorem yields the usual scalar potential<\/p>\n

\\[U(\\vec r) = \\frac{1}{4 \\pi \\epsilon_0}\\int_V d^3 r’ \\frac{\\rho(\\vec r \\;’)}{|\\vec r – \\vec r \\;’|} \\]<\/p>\n

for the electric field and the usual vector potential<\/p>\n

\\[\\vec W(\\vec r) = \\frac{\\mu_0}{4\\pi} \\int_{V} d^3r’ \\frac{ \\vec J(\\vec r) }{|\\vec r – \\vec r\\;’|}\\]<\/p>\n

for the magnetic field.<\/p>\n

On the whole, I think this approach enhances the physical understanding of Maxwell’s equations in ways the traditional approach can’t. \u00a0It’s not without its downside, but none of the problems present much difficulty. \u00a0Further details can be found in my paper<\/a>.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

In an earlier post, I outlined a derivation of the Helmholtz Theorem starting from the identity \\[ \\nabla^2 \\left( \\frac{1}{|\\vec r – \\vec r \\, {}’|} \\right) = -4 \\pi… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[4,2,3],"class_list":["post-72","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-delta-function","tag-em","tag-helmholtz"],"_links":{"self":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/72","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=72"}],"version-history":[{"count":30,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/72\/revisions"}],"predecessor-version":[{"id":106,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/72\/revisions\/106"}],"wp:attachment":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=72"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=72"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=72"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}