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Homework Help: Potential due to a rod

  1. Oct 29, 2005 #1
    A rod of length L lies along the x-axis with the left end at the origin. It has non-uniform charge density given by [tex]\lambda = \alpha x[/tex]. where [tex]\alpha[/tex] is a positive constant. Calculate the potential at a position A, which is a distance d to the left of the origin.

    Here is what i did:

    we have [tex] dV = \frac{k_e dq}{r}[/tex]

    then i made use of the [tex]\lambda = \alpha x[/tex]

    and we have [tex]V=\int\frac{k_e\lambda dx}{x} = \int\frac{k_e\alpha x dx}{x} = k_e\alpha (d+L-d)[/tex] where my limits of integration were L to d+L in the second integral

    however the answer provided says that [tex]V=k_e\alpha (L-d ln(1 + \frac{L}{d})[/tex].

    where did i go wrong?

    thanks
     
    Last edited: Oct 29, 2005
  2. jcsd
  3. Oct 29, 2005 #2

    Astronuc

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    [tex]V=\int\frac{k_e\lambda dx}{x} = \int\frac{k_e\alpha x dr}{r}[/tex]

    then determine the relationship between r and x, i.e. r = r(x), which will also determine the relationship between dr and dx.

    x is the distance along the rod, r is the separation between the incremental charge dq and the point at which the potential is being measured.
     
  4. Oct 29, 2005 #3
    so i should put r = d + x???
     
  5. Oct 29, 2005 #4

    Astronuc

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    Given: A rod of length L lies along the x-axis with the left end at the origin.

    Calculate the potential at a position A, which is a distance d to the left of the origin.

    Perhaps one can draw a diagram.

    x [0,L]
    d to left of origin
    r [d,L+d] = d+x

    so what is dr if 'd' is constant?
     
  6. Oct 30, 2005 #5
    if d is constant dr is just d+x, this is correct right? and i have a diagram that came with the question but i still dont see what to put into the integral or how...
     
  7. Oct 30, 2005 #6
    ok i got it it was really simple actually.
     
  8. Oct 30, 2005 #7

    Tide

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    Incidentally, it's not the "potential due to a rod." It's the electrical potential due to electrical charge that happens to reside on a rod.
     
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