Find the capacitance of the infinite ladder shown in figure.

Explanation

As the ladder is infinitely long, the capacitance of the ladder to the right of the points P, Q is the same as that of the ladder to the right of the points A, B the equivalent capacitance of the ladder is `
`C_1`
` the given ladder may be replaced by the connections shown in figure. The equivalent capacitance between A and B is easily found to be `
`C+(CC_1)/(C+ C_1)`
`. But being equivalent to the original ladder, the equivalent capacitance is also `
`C_1`
` Thus, `
`C_1 = C+(CC_1)/(C+ C_1)`
` or, `
`C_1C+ C_1^2 = C^2 + 2 CC_1`
` or `
`C_1^2 - CC_1 /C^2 = 0`
` giving `
`C_1 = (C+sqrtC^2 + 4 C^2)/(2) = (1 + sqrt5)/(2) C`
`. Negative value of C_1 isrejected.