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Magnetic Vector Potential and conductor

  1. Dec 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that, inside a straight current-carrying conductor of radius R, the vector potential is:
    $$ \vec{A} = \frac{\mu_{0}I}{4\pi}(1-\frac{s^2}{R^2}) $$

    so that ##\vec{A}## is set equal to zero at s = R

    2. Relevant equations

    ## \vec{A} = \frac{\mu_{0}}{4\pi}\int\frac{\vec{I}}{|r'-r|} dl' ##


    3. The attempt at a solution

    I'm really having a hard time even setting it up.
     
  2. jcsd
  3. Dec 13, 2013 #2

    Simon Bridge

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    Start by drawing a picture of a current carrying wire with a significant radius.
    How does the current vary with radius?
     
  4. Dec 13, 2013 #3

    Simon Bridge

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    Hint: cylindrical coordinates.
    Some people like to turn the line integral into an area integral too ... there are lots of approaches.
    For marking - it is usually best to use the method covered in class.
     
  5. Dec 14, 2013 #4

    vanhees71

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    Another hint: The Biot-Savart Law is not very efficient in many problems. It's easier to use the local form of the (magnetostatic) Maxwell Equations:
    [tex]\vec{\nabla} \times \vec{B}=\mu_0 \vec{j}, \quad \vec{\nabla} \cdot \vec{B}=0.[/tex]
    The second equation ("no monopoles") is already solved by the introduction of the vector potential, cf.
    [tex]\vec{B}=\vec{\nabla} \times \vec{A}.[/tex]
    So, just take the appropriate derivatives (in cylindrical coordinates), and check that you get the right current density.
     
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