(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm trying to prove that Electric field away from a line segment of charge, is equivalent to the field away from a point charge, provided I observe from far enough.

2. Relevant equations

Ignoring all the constants:

potential_line = log( (sqrt(r^2 + a^2) + a) / (sqrt(r^2 + b^2) - b) )

potential_charge = 1/r

Here a+b is the length of the line segment, such that a and b are the parts of the line segment 'above' and 'below' the line of sight of the observer, assuming the line is vertical one. 'r' is the distance to the line along the line of sight.

Trying to show that the two equations become equivalent (close) when r is much bigger than a+b.

3. The attempt at a solution

I graphed with WorlframAlpha both formulas and the graphs look the same. Here are the two links. I took a=2, b=1 for an experiment.

http://www.wolframalpha.com/input/?i=y=log((sqrt(x^2+4)+2)/(sqrt(x^2+1)-1))++from+1+to+100

http://www.wolframalpha.com/input/?i=3/x+from+1+to+100

The graph looks very close, which is good. But how do I show this algebraically ? I tried to simplify that sqrt(r^2 + a^2) = sqrt(r^2 + b^2) = r, when r >> a+b. The graphs continue to look similar. I tried to simplify the numerator, and got this:

log( 1 + (a+b)/r + ab/r^2 )

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# Trouble proving equivalence at limit

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