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Biot-Savart for current density in a volume

  1. Jan 10, 2015 #1
    1. The problem statement, all variables and given/known data
    The Biot-Savart law for a current density j in a volume V is: [itex] d\vec{B} = \frac{\mu_0\vec{j}\times\vec{r}}{4\pi r^3} dV[/itex]

    Derive the formula for the magnetic field created by a single point-like particle carrying charge [itex]q[/itex] moving with velocity [itex]\vec{v}[/itex]. Explain and justify all important steps.
    2. Relevant equations

    [itex] d\vec{B} = \frac{\mu_0\vec{j}\times\vec{r}}{4\pi r^3} dV[/itex]

    3. The attempt at a solution

    If I integrate over the whole volume, that will give me B, but it wont be correct for a moving point charge?

    [itex] \vec{B} = \frac{\mu_0}{4\pi} \int \frac{\vec{j}\times\vec{r}}{ r^3} dV[/itex]
     
  2. jcsd
  3. Jan 10, 2015 #2

    mfb

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    Why not?
    You have to think about how to get a current from a point particle with an "infinite" charge density.
     
  4. Jan 11, 2015 #3
    Sorry, can you elaborate ?
     
  5. Jan 11, 2015 #4

    mfb

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    What is your distribution of j? You'll need it to evaluate your integral.
     
  6. Jan 11, 2015 #5
    I'm not sure, but;

    [itex] \vec{B} = \frac{\mu_0}{4\pi} \int \frac{\vec{j}\times\vec{r}}{ r^3} dV[/itex] then,
    [itex] \vec{B} = \frac{\mu_0}{4\pi} \int \frac{\vec{v}\times\hat{r}}{ r^2} dq[/itex]

    Am I on the right lines?
     
  7. Jan 11, 2015 #6

    mfb

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    j and v are not the same. They have completely different units, for example, and v does not depend on the position. Also, your equation is now independent of q, which clearly shows something is wrong.
     
  8. Jan 11, 2015 #7
    These are all my lecture notes on the Biot-Savart Law.


    BSlaw1.jpg


    BsLaw.jpg

    That's literally all I have to go on, my lecturer barely went over it at all. I have some textbooks as well, but they just provide me with what I wrote earlier, which you said was wrong?
     
  9. Jan 11, 2015 #8

    mfb

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    I don't see any j=v there.

    Let's start with an easier example: you have a ball of total charge Q with radius R and a uniform charge density. Its center is currently at the origin of the coordinate system, and it is moving in x-direction with a speed v. What is the current density for every point in space?
     
  10. Jan 11, 2015 #9

    I/A ?
     
    Last edited: Jan 11, 2015
  11. Jan 11, 2015 #10

    mfb

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    No, whatever I/A is supposed to mean.
     
  12. Jan 11, 2015 #11
    Sigh, then I don't know. I really don't. I know you can't tell me either.

    Those pictures I have included are everything I have been taught (taught is an exceptionally strong word to use) regarding Biot-Savart law, our lecturer did not even derive them as he said you can't 'technically' derive them.

    I have said this in the many other threads I've had to create, I'm not a physics student, I just take a module in Electromagnetism. I am not (necessarily..) asking to be spoon fed, but it's safe assume I know next to nothing regarding the Biot-Savart Law.

    This forum is just another resource I've had to find because my textbooks assume the reader already has a sound base in Physics/maths - which I do not. Alas, I cannot keep using this an excuse.
     
  13. Jan 11, 2015 #12

    mfb

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    The current subproblem has nothing to do with Biot-Savart.

    What do you know about current densities?
    Can you find something like "current density is ... per ... and ..."?
     
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