A uniform resistance wire AB of resistance R₀ is connected across an ideal DC voltage source of V₀ . A resistor of unknown resistance R is connected between end A and the midpoint C of the wire. If the measured potential difference across the resistor R is V₀ 4 , what is the value of R in terms of R₀ ?

Step-by-step solution

The wire AB is divided into two equal halves, AC and CB, each having a resistance of R₀ 2 . The resistor R is in parallel with the segment AC. The equivalent resistance of this parallel combination is: R_ AC = R R₀ 2 R + R₀ 2 = R R₀ 2R + R₀ The total resistance of the circuit is the sum of R_ AC and the resistance of segment CB: R_ eq = R_ AC + R_ CB = R R₀ 2R + R₀ + R₀ 2 = 4R R₀ + R₀^2 2(2R + R₀) The potential difference across the parallel combination (and thus across R ) is given by the voltage divider rule: V_