Determine the furrent drawn from a 12 V suly with internal resistacne `0.5Omega ` by the infinite network shown in Each rsistor has `1Omega` resistance.

Explanation

Here no. of identical units on the left side of line CD and line PQ are same and so equivalent resistance of given infinite ladder between points P and Q as well as between points C and D must be same . Let it be x .
thus , `R_(PQ) = R_(CD) ` = x ... (1)
Now, let us remove part on the left side of line CD and let us replace it by equivalent resistance x.

Now, equivalent resistance between points P and Q in above figure is,
`R_(PQ) = 1 + ((x xx 1)/(x + 1) ) + 1 `
`therefore x =2 + (x)/(x +1) ` [ from equation (1)]
`therefore x (x + 1) = 2 x (x + 1) + x `
`therefore x^(2) + x = 2x + 2 + x `
` therefore x^(2) - 2x - 2 = 0 " "` .... (2)
Comparing above equation with `ax^(2) + bx + c = 0 ` we get a = 1, b = - 2, c = -2
Now,`Delta = b^(2) - 4ac `
= `(-2)^(2) - 4 (1) (-2)`
= 4 + 8
`Delta = 12 = 4 xx 3 " " rArr sqrt(Delta ) = 2 sqrt(3)`
Solution of equation (2) will be,
` x = (- b pm sqrt(Delta ))/( 2 a ) = - ((-2) pm 2 sqrt(3))/(2(1))`
` therefore x = (2 + 2 sqrt(3))/(2 ) " " (because x gt 0 )`
`therefore x = (2(1 + sqrt(3)))/(2 ) = 1 " " sqrt(3) = 1 + 1.732 `
`therefore ` x = 2.732 `Omega " "` .... (3)
Now, as per the statement above resistance is connected across a battery (12 V, 0.5 `Omega` ) and so ,
For adjoining loop,
`I = (epsilon)/(x +r)`
`therefore I = (12)/(2.732 + 0.5) `
` therefore I = (12)/( 3.232)`
`therefore I = 3.713 ` A