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Suggestions for practice problems in E&M

  1. Jan 5, 2017 #1
    Hey all,
    I am working my way through a couple of emag books (Griffiths, Jackson, and Schwinger) and I was wondering if any of y'all have suggestions for problems that you thought were particularly physically insightful or useful.

    Cheers,
    IR
     
  2. jcsd
  3. Jan 5, 2017 #2

    Charles Link

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    Try these problems/posts that were recently on Physics Forums:
    https://www.physicsforums.com/threads/electric-field-of-a-charged-dielectric-sphere.890319/
    https://www.physicsforums.com/threa...field-of-a-uniformly-polarized-sphere.877891/
    https://www.physicsforums.com/threads/magnetic-field-of-a-ferromagnetic-cylinder.863066/
    https://www.physicsforums.com/threads/potential-of-sphere-given-potential-of-surface.887477/
     
  4. Jan 5, 2017 #3

    Charles Link

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    @IxRxPhysicist I have one more E&M problem that I don't think is included in the above that you might find useful. Begin with the equation ## B=\mu_o H +M ## which is an equation that comes out of the "pole" model of magnetostatics. (J.D. Jackson emphasizes the "pole" model.) (Sometimes you will see this equation as ## B=\mu_o H +\mu_o M' ## where ## M'=M/\mu_o ##.) Upon taking the divergence of both sides ## \nabla \cdot B=\mu_o \nabla \cdot H +\nabla \cdot M ##. But ## \nabla \cdot B=0 ## so that ## \mu_o \nabla \cdot H=-\nabla \cdot M ##. You might recognize the right side as ## -\nabla \cdot M=\rho_m ## where ## \rho_m ## is the magnetic charge density (fictitious). The problem is to solve this for ## H ##. ##\\ ## The result is that ## H ## has an integral solution with the inverse square law ## H(x)=\int \frac{1}{4 \pi \mu_o} \frac{\rho_m(x')(x-x')}{|x-x'|^3} \, d^3x' ##. The question is, where is the current in conductor contribution to ## H ## which is absent from this solution? I will give you a hint: The inhomogeneous differential equation ## \nabla \cdot H =\frac{\rho_m}{\mu_o} ## can also have a solution to the homogeneous equation as the complete solution. (The current in conductor contribution to ## H ## can be found using Biot-Savart's law. The Biot-Savart solution obeys ## \nabla \cdot H=0 ##.) ## \\ ## Note: A similar problem is encountered if you take the curl of both sides of the above equation. ## \nabla \times B=\mu_o J_{total} ## (in the steady-state case) where ## J_{total}=J_{free}+J_m ## and ## \nabla \times M=\mu_o J_m ## so that ## \nabla \times H=J_{free} ##. This has a Biot-Savart type integral for ## H ##, but the question is where did the magnetic "pole" contribution go that we found above with the ## \nabla \cdot H ## equation? And the answer is again similar: This time, the homogeneous ## \nabla \times H =0 ## needs to be considered as having a contribution to the complete solution for ## H ##.
     
    Last edited: Jan 5, 2017
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