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How to solve this without infinitesimals

  1. Sep 14, 2013 #1
    I'm trying to find a way to use calculus without infinitesimals and I'm stuck on this physics problem.

    It's a uniform charge distribution question. Basically a half circle with radius [itex]r[/itex] and you have to find the electric field at a point that is along its x-axis. The [itex]E_y[/itex] component will be 0 because of symmetry. So all you need is is [itex]E_x[/itex]. The equation ends up being:
    [tex]E_x=∫dEcosθ[/tex]
    The only way I know how to solve this is using infinitesimals and finding one in terms of the other, like this:
    [tex]dE=k\frac{dQ}{r^2}[/tex]
    [tex]dQ=\frac{Q}{πr}dy[/tex]
    [tex]dy=rdθ[/tex]
    Then substituting those in all the way up so I have my integral in terms of θ.

    So how would I go about solving this without using infinitesimals? Thanks
     
  2. jcsd
  3. Sep 14, 2013 #2

    verty

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    Homework Helper

    This is always difficult to explain.

    ##E_x = \int cos(\theta(E)) dE##.

    On the one hand, ##E_x## is the area under ##y = cos(\theta(E))##, a Riemann sum of rectangles having width ##dE##. This seems to imply that ##dE## is a width, an infinitesimal width.

    On the other hand, we have a rate of change ##dE## and a related rate of change ##cos(\theta(E)) dE## which when integrated, gives ##E_x##. The rate of change of ##E_x## is ##cos(\theta(E)) dE##, it is related to ##dE##, the rate of change of ##E##.

    Which is it? Is ##dE## a width or is it a rate? This is a little like asking what is a number. One person will say, the thing you count with, another will say, an equivalence class of sets by cardinality, another will say, a Church numeral, etc, etc. There is no answer to that question. In the same way, it doesn't matter what ##dE## is, what matters is how differentials are used.

    Of course, there is also the view that ##dE## is a finite width and ##cos(\theta(E))## is the gradient of the tangent to ##E_x## at ##E##, so ##cos(\theta(E)) dE## is the ##E_x## offset of the tangent at distance ##dE## from ##E##.
     
  4. Sep 14, 2013 #3
    I think the general answer to this question (as it pertains to setting up integrals) is "Set up an approximating Riemann Sum and take a limit." Most of the time it's just a matter of changing all of the [itex]d[/itex]s to [itex]\Delta[/itex]s. In my experience, the physicists aren't using infinitesimals so much as they are being a bit lazy with limits and abusing notation. It gets the job done, though. So more power to 'em.
     
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