thin rod

logistic_guy

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A thin rod of length 2L\displaystyle 2L is centered on the x\displaystyle x axis. The rod carries a uniformly distributed charge Q\displaystyle Q. Determine the potential V\displaystyle V as a function of y\displaystyle y for points along the y\displaystyle y axis. Let V=0\displaystyle V = 0 at infinity.
 
A thin rod of length 2L\displaystyle 2L is centered on the x\displaystyle x axis. The rod carries a uniformly distributed charge Q\displaystyle Q. Determine the potential V\displaystyle V as a function of y\displaystyle y for points along the y\displaystyle y axis. Let V=0\displaystyle V = 0 at infinity.
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From accumulated experience when the charge Q\displaystyle Q is uniformly distributed then the density per unit length is:

λ=dQdx\displaystyle \lambda = \frac{dQ}{dx}

I am assuming that I placed the thin rod on a coordinate system! Then the integral becomes:

V=k1r dQ=λk1r dx\displaystyle V = \int k\frac{1}{r} \ dQ = \lambda k \int \frac{1}{r} \ dx

It is still tough but I made a progress.

🤩🙌
 
With a sketch, I figured out how to write the variable r\displaystyle r in terms of the variable x\displaystyle x. I am just lazy to show it here!

V=λk1r dx=λk1x2+y2 dx\displaystyle V = \lambda k \int \frac{1}{r} \ dx = \lambda k \int \frac{1}{\sqrt{x^2 + y^2}} \ dx

😴
 
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