12 identical resistance R each form a cube . Resistance across its face diagonal corners is?

Please refer to the following image

Here, 12 resistance are arranged such that they form a a cube. Now each of the 12 wires represents a resistor of value 'R'.

We need to calculate the resistance across face diagonal corners.

Now, let assume assume that we have attached a voltage source across (or emf V) the ends of points A and B and thus current starts to flow in the network. We have also assumed that, to our convenience, a current of '6x' flows through point A and thus it splits up equally into three parts of current '2x' each (as resistance is same in each arm) and each '2x' current further splits into two equal parts of current 'x' each.

We need to calculate the equivalent resistance (say r) across the terminal AG which is the corner diagonal of the cube.

Applying Kirchoff's voltage law in loop ABCGA, we get

V = 2xR + xR + 2xrR

so,

V = 5xR  ..................(1)

now, we also know that from Ohm's Law

V = r.6x .................(2)

here, 'V' is the potential difference applied, '6x' is the total current supplied and 'r' is the net resistance of the circuit.

so,

by putting value of V from (1) in (2), we get

5xR = r.6x

thus,

r = (5/6)R

 

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