B

Step 1: Analyze the square ABCD where each side has a resistance of 1 Ω.
The resistance of the wire connecting points will influence the potential at points B and E.
Step 2: Since AB and AD have uniform resistance, the potential difference across AC will also be uniform across those segments.
Step 3: To find the location of point E on CD such that B and E are equipotential, we set up a voltage divider.
Step 4: Define point E such that CE = x and ED = 1 - x. The wire AE connects across to E and must balance the potential difference to ensure there's no potential difference across BE.
Step 5: From symmetry and applying Kirchhoff's laws, we find that the resistance from A to B (2 Ω in series) and A to E (1 + x/1) must satisfy the condition that voltage at B equals voltage at E.
This leads us to derive the equations for the two points to maintain equipotential.
The simplified calculation gives the ratio leading to E being located at 1/3 the length of CD.
Therefore, the location of E that ensures points B and E remain equipotential is at the position of option B, which satisfies the derived conditions for equal potential.