QB_PH_SEC7_009: Band Theory of Solids & Semiconductors

A semiconductor ribbon of thickness $t = 0.5\text{ mm}$ carrying an electric current $I = 10\text{ mA}$ is placed in a transverse magnetic field $B = 0.5\text{ T}$. If the measured Hall voltage is $V_H = 2.5\text{ mV}$, what is the magnitude of the Hall coefficient $R_H$ in $10^{-4}\text{ m}^3/\text{C}$?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Hall Voltage Expression**: $V_H = \frac{R_H I B}{t}$ where $t$ is the specimen thickness along the magnetic field direction. 2. **Isolating the Hall Coefficient**: $R_H = \frac{V_H t}{I B}$ 3. **Unit Consistency & Evaluation**: - $V_H = 2.5\text{ mV} = 2.5 \times 10^{-3}\text{ V}$ - $t = 0.5\text{ mm} = 0.5 \times 10^{-3}\text{ m}$ - $I = 10\text{ mA} = 10 \times 10^{-3}\text{ A} = 10^{-2}\text{ A}$ - $B = 0.5\text{ T}$ $R_H = \frac{(2.5 \times 10^{-3})(0.5 \times 10^{-3})}{(10^{-2})(0.5)} = \frac{1.25 \times 10^{-6}}{5.0 \times 10^{-3}} = 0.25 \times 10^{-3} = 2.5 \times 10^{-4}\text{ m}^3/\text{C}$ 4. **Official Integer/Decimal Answer**: 2.5.