Question #42747

A modified metre bridge is constructed with a uniform wire AB of length $L = 100 \text{ cm}$ and total resistance $5 R_0$. Two resistors $R_1 = 4 R_0$ and $R_2 = R_0$ are connected in the left gap (between A and D) and right gap (between D and B) respectively. An ideal battery of emf $E$ is connected across the ends A and B (with the positive terminal at A). A non-ideal galvanometer is connected between the junction D and a jockey J that slides on the wire. The galvanometer has a resistance $R_g = 0.4 R_0$ and a mechanical zero error, such that it shows zero deflection only when a constant current $i_0 = \frac{E}{40 R_0}$ flows through it from D to J. Due to the mechanical design of the track, the jockey exerts a variable pressure on the wire, creating a position-dependent contact resistance $R_c$ in series with the galvanometer. For a jockey position $x$ (measured from A in cm), this contact resistance is given by $R_c(x) = R_0 \left[ 1.05 + 13 \left( \frac{x - 80}{100} \right) \right]$. Find the apparent null point position $x$ where the galvanometer shows zero deflection.