Question

How to solve the following circuit of resistors?

Twelve resistors each of resistance 1 Ω are connected in the circuit shown in the adjoining figure. Net resistance between points A and H would be :-

Answer 1

`3/4 Omega`

Explanation:

By symmetry, the currents in the resistors must be distributed as shown below

Applying Kirchoff's Voltage law to the loop ACEBA gives :

`y+z-x/2-x = 0 implies y+z=3/2 x`

while the loop CHGEC gives

`y-z-(x+2z)-(x/2+z)-z = 0 implies y-4z = 3/2x`

The two equations together give `z=0` and so

`y = 3/2x implies I = 2y+x = (2+2/3)y = 8/3y implies y = 3/8I`

The net voltage drop across thee circuit = the voltage drop along the path ACH =

`2y times 1\ Omega = 3/4I times 1 Omega`

Thus, the effective resistance between A and H is `3/4 Omega`