1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to Convert Maxwell's Equations into Integral Form

  1. Jan 12, 2017 #1
    1. The problem statement, all variables and given/known data
    I'd like to know how to convert Maxwell's Equations from Differencial form to Integral form.
    2. Relevant equations
    upload_2017-1-12_19-57-50.png Gauss' Law
    upload_2017-1-12_19-58-28.png Gauss' Law for Magnetism
    upload_2017-1-12_19-59-6.png Faraday's Law
    upload_2017-1-12_20-0-5.png The Ampere-Maxwell Law

    3. The attempt at a solution
    Convert using properties of vector analysis (as Divergence and Curl). May I convert them, using the opposite process that I would use to convert the integral form to differential form?
     
    Last edited: Jan 12, 2017
  2. jcsd
  3. Jan 13, 2017 #2

    Charles Link

    User Avatar
    Homework Helper

    There are a couple of ways to get integral forms of Maxwell's equations. Gauss' law can be applied to the equation ## \nabla \cdot E=\rho/\epsilon_o ## to give ## \int \nabla \cdot E \, d^3x= \int E \cdot \, dA=Q/\epsilon_o ## where ## Q ## is the total charge enclosed, and the flux of the electric field is over the surface of the enclosed volume. An alternative integral form can be written for this equation: ## E(x)=\int \frac{\rho(x')(x-x')}{4 \pi \epsilon_o |x-x'|^3} \, d^3x' ## . This second form is essentially Coulomb's law (inverse square law) for the electric charge distribution ## \rho(x) ##. ## \\ ## Gauss' law can also be employed on the equation ## \nabla \cdot B=0 ##, but usually this one is just used in a qualitative form where the lines of flux for the magnetic field ## B ## are said to be continuous. In integral form, it reads ## \int B \cdot \, dA=0 ##. ## \\ ## ## \\ ## For the curl equations, Stokes theorem (## \int \nabla \times E \cdot \, dA=\oint E \cdot \, ds ##), is usually employed, e.g. Faradays law becomes ##\varepsilon= \oint E \cdot \, ds=-\int (\frac{\partial{B}}{\partial{t}}) \cdot \, dA ##. ## \\ ## For the curl B, in the steady state, again Stokes law is often employed to give ## \oint B \cdot ds=\mu_o I ##. An alternative integral form does exist for this one also in the steady state which is the Biot-Savart Law: ## B(x)=\int \frac{\mu_o J(x') \times (x-x')}{4 \pi |x-x'|^3} \, d^3x' ##. For the non-steady state, I believe the solutions of Maxwell's equations are found by the Lienard-Wiechert method, but that is likely to be beyond the scope of what you are presently doing. ## \\ ## It is worth noting that the integral forms of these, which is sometimes treated in courses in vector calculus, are a little more complicated than simply going from differentiation to integration.
     
    Last edited: Jan 13, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: How to Convert Maxwell's Equations into Integral Form
Loading...