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Twelve indentical resistance each of ...


Twelve indentical resistance each of resistance R are joined to form a cube as shown in figure. Find resistance of the combination across AB.

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This circuit cannot be solved by considering series and parallel connections.
All three paths from A to B are identical . Let 3l current entered at end A and left at B. Current is distributed in all branches using symmetry.
If `V_(A) - V_(B) = V`, then `R _(AB) = (V)/(3l)` ....(i)
We have
`V_(A) - V_(B) = ( V_(A)- V_(P))+ (V_(P)-V_(Q)) + (V_(Q) -V_(B))`
`implies V = IR + ( l)/(2) R + IR `
`implies V = ( 5 IR)/( 2) `
`implies (V )/( l) = ( 5R)/(2)`
`implies R _(AB) = ( V)/(3l) = ( 5R)/(6)`
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