Question #43353

A semiconductor wire of length $L$ is placed along the $x$-axis from $x=0$ to $x=L$. The wire is maintained at a constant temperature $T$. Due to a specific doping profile, the free electron concentration in the wire varies as $n(x) = n_0 e^{-\alpha x^2}$, where $n_0$ and $\alpha$ are positive constants. No external voltage source is connected across the wire.

A student uses this wire to find the null point for an ideal cell of electromotive force $\varepsilon$. The positive terminal of the cell is connected to the end $x=0$, and the negative terminal is connected to a jockey that can slide along the wire. Assuming the Einstein relation $D = \frac{kT}{e}\mu$ holds (where $D$ is the diffusion coefficient, $\mu$ is the mobility, $k$ is the Boltzmann constant, and $e$ is the elementary charge), at what distance $l$ from the end $x=0$ will a galvanometer connected in series with the cell show zero deflection?