Question #42017

A Wheatstone bridge consists of four arms forming a quadrilateral $ABCD$. A battery of EMF $V = 20\ \text{V}$ and negligible internal resistance is connected across $AC$, with $A$ connected to the positive terminal. Arms $AB$, $BC$, and $CD$ are made of standard resistors, each of resistance $R_0 = 100\ \Omega$. Arm $DA$ contains a semiconductor resistor with a temperature-dependent resistance $R_s = R_0 [1 - \alpha (T - T_0)]$, where $\alpha = 1.2 \times 10^{-4}\ \text{K}^{-1}$ and $T_0$ is the ambient temperature.

A galvanometer of resistance $R_g = 50\ \Omega$ is connected across $BD$ and reads positive for current flowing from $D$ to $B$. The semiconductor dissipates heat to the surroundings at $T_0$ with a thermal conductance $K = 10\ \text{mW/K}$. Due to a mechanical defect, the galvanometer has a constant zero error $I_z$. In the steady state, the galvanometer reads $I_{\text{read}} = 550\ \mu\text{A}$.

Assuming the change in resistance is small ($\Delta R \ll R_0$) so that first-order perturbation is valid, what is the zero error $I_z$?