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Homework Help: Lorrentz force law & vector notation

  1. Aug 22, 2014 #1
    1. The problem statement, all variables and given/known data
    A current element is 3.0mm long, centerd on the origin of cartesian coordinates, and carries a current of 2.5A in the direction ez.
    What field does it produce at the point (3.0, 0, 4.0)

    2. Relevant equations
    This is obviously a Lorentz force law question
    so . . . ∂B = (I ∂I (r-r0)) / I r-r0 I^3

    where B = magnetic field
    I - current
    r - point of origin
    r0 - point being considered

    3. The attempt at a solution
    B = 4∏x10^-7 / 4∏ x 2.5A x 3.0x10^-3 )(ez(3ex + 4ez)) /5^3

    cancelling 4∏'s
    B = 10^-7 x 2.5A x 3.0x10^-3 (ez(3ex + 4ez)) /5^3

    The number bit is easy, but the problem I have is the vector notation.

    Please, how do I do the (ez(3ex + 4ez)) /5^3 bit?

    I have the final answer as
    1.8x10^-11 T ey
  2. jcsd
  3. Aug 22, 2014 #2
    I might be missing something, but shouldn't you be considering the Biot–Savart law instead?

    It seems to me like you're being asked to determine the magnetic field produced by a current and not the force on a point charge in an EM field.

    Edit: I see hints of the Biot–Savart law in your expressions, so I guess it was just a typo.

    If you start out with the differential:
    d\mathbf{B} = \frac{u_0}{4\pi}\frac{I d\mathbf{s}\times \mathbf{\hat{r}}}{r^2}
    where ##d\mathbf{s}## is directed along the current and ##\mathbf{\hat{r}}## is a unit vector towards the point you're considering.

    Then since ##d\mathbf{s}\times \mathbf{\hat{r}}## is always directed along ##\mathbf{e}_y##, you have:
    d\mathbf{B} = \frac{u_0}{4\pi}\frac{I |d\mathbf{s}\times \mathbf{\hat{r}}|}{r^2}\mathbf{e}_y
    The problem is then to find expressions for ##d\mathbf{s}## and ##r## in terms of a common variable you can use to integrate over the whole of the current element.

    Any help?
    Last edited: Aug 22, 2014
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