ABCD is a square where each side is a uniform wire of resistance `1 Omega`. Find a point E on CD such that if a uniform wire of resistance `1 Omega` is connected across AE and a constant potential is applied across A and C, the points B and E will be equipotential.

Explanation

Let resistance `DE = x Omega` and that between `CE = ( 1 - x)Omega`.The points B and E will be equipotential, when effective resistance acros AE = effective, resistance across CE
i.e., `((1 + x) xx 1)/((1 + x)+1) = (1 - x) or (1 + x) = (1 - x) xx (2 + x)`
or `1 +x = 2 + x - 2x 2x - x^(2) or x^(2) + 2x - 1 =0`
On solving it, we get, `x = sqrt(2)-1`
Now, `1 - x = 2 - (sqrt(2)-1) = 2 - sqrt(2) =(sqrt(2)-1)`
`:. (CE)/(ED) = (1-x)/(x) = (sqrt2(sqrt2-1))/((sqrt2 - 1)) = sqrt(2)`