Question

In the following circuit diagram, an infinite series of resistances is shown. Equivalent resistance between points A and B is ______.

Options

• Infinite

• Zero

• 2Ω

• 30Ω

Answer 1

In the following circuit diagram, an infinite series of resistances is shown. Equivalent resistance between points A and B is 2Ω.

Explanation:

Denote the equivalent resistance between points A and B as R_eq.

Since the circuit extends infinitely to the right and is symmetric, the equivalent resistance of the resistances after the first branch is also R_eq.

Therefore the simplified circuit can be represented as the circuit below.

Since the resistance of 2Ω and R_eq are in parallel the equivalent resistance can be written as `(2R_(eq))/(2 + R_(eq))`

Since the resistances of 1Ω and 2R_eq/2 + R_eq are in series, the equivalent resistance between A and B can be written as follows:

`R_(eq) = 1 + (2R_(eq))/(2 + R_e)`

`R_(eq) * (2  + R_(eq)) = 2 + R_(eq) + 2R_(eq)`

`R_(eq)^2 - R_(eq) - 2` = 0

or

`R_(eq)` = 2 or `R_(eq)` = – 1

Since the value of resistance cannot be negative, the value of R_eq is 2Ω.