1. Identify the Effective Circuit

Based on the mathematical form of the answers, the problem requires calculating the voltage V needed to drive current I into an infinite ladder of resistors. The ladder has:

• Series Resistance (R_s): 4r (from text “Each resistor has resistance 4r”)
• Shunt Resistance (R_p): r (from “internal resistance r”)

2. Calculate Equivalent Resistance (R_eq)

For an infinite ladder:

R_eq = R_s + (R_p || R_eq)

Substituting values:

R_eq = 4r + (r * R_eq) / (r + R_eq)

R_eq² + rR_eq = 4r² + 4rR_eq + rR_eq

R_eq² – 4rR_eq – 4r² = 0

3. Solve the Quadratic Equation

Using the quadratic formula:

R_eq = [4r ± √(16r² – 4(1)(-4r²))] / 2

R_eq = [4r ± √(32r²)] / 2

R_eq = [4r + 4r√2] / 2 = 2r(1 + √2)

4. Calculate Voltage V

Using Ohm’s Law V = I * R_eq:

V = I * [2r(1 + √2)]

V = (2 + 2√2)Ir