CBSE Class 12-science Answered

Let us assume a potential difference V is applied between A and B as shown in figure.

Let i be the current drawn by the given resistor network .

At point C , input current i is split into i₁ and i₂ . By symmetry at H , i₁ and i₂ combine to give i .

At point D , current i₂ is split into i₃ and i₄ . By symmetry at G , i₃ and i₄ combine to give i₂ .

At point E , current i₁  and i₃ combine to give current i_5. By symmetry at F , current i₅ is split into i₁ and i₃ .

By applying Kirchoff's voltage law to the loop CIJEC , we get

;  Hence we get ,   ............................. (1)

Similarly by applying Kirchoff's voltage law to the loop DEFGMND , we get

..............................(2)

By applying Kirchoff's voltage law to ACDEFGHBOPA , we get

Hence we get from above expression ,

............................(3)

Using eqn.(1) , we get from above equation as

............................(4)

By substituting i₅ in eqn.(2) , we get

  .........................(5)

At junction E , by applying Kirchoff's current law , we get

Using eqn.(4) and (5) , we get current i₁ from above expression as

By substituting i₃ and i₅ in eqn.(3) , we get

Hence total current drawn by given resistor network is

Hence the effective resistance between points A and B is (r/2)