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Using Gauss' Law to Calculate electric field near rod.

  • Thread starter Nathan B
  • Start date

Nathan B

1. Homework Statement
No variables, just a conceptual question.

2. Homework Equations
Flux = EA = Q/ε
3. The Attempt at a Solution
Given a uniformly charged rod of FINITE length, could we use Gauss' law for electric flux to calculate the field at a point p a distance x away from the rod, so long as the whole rod is enclosed and x lies on the surface area of the enclosing gaussian surface? I tried it with the equation E = λ L/(ε A), but it didn't work. I also found multiple different A's could be used, but none of them gave the right answer. Could someone please explain to me where I'm going wrong with this?
 
Gauss's Law states that
$$\oint\mathbf{E}\cdot d\mathbf{a}=\frac{Q_{enc}}{\epsilon}$$
Normally, there is some kind of symmetry argument which can be made that allows us to know the direction of ##\mathbf{E}##. If the rod was infinitely long, then you could use mirror symmetry and translational symmetry to argue that only the radial component of ##\mathbf{E}## is non-zero at all points. In that case, the dot product ##\mathbf{E}\cdot d\mathbf{a}=E\hat{r}\cdot da\hat{r}=Eda## and we can evaluate the integral. In the case of a finite rod, do these symmetry arguments hold? If they don't then can you evaluate ##\mathbf{E}\cdot d\mathbf{a}##?
 

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