GATE 2020 IN Q.10: solution and answer

Official answer: BSource: Official answer key, GATE 2020 IN (IIT Delhi)

1. A Hall element carries current $I$ along its length. A field $B$ perpendicular to the current pushes the moving carriers sideways with the Lorentz force $F=qvB$.

2. Charge collects on the side faces until the Hall electric field balances the magnetic force: $q{E}_H=qvB$, so $E_H=vB$.

3. The drift velocity follows from the current: $v=\frac{I}{nqA}=\frac{I}{nqwt}$, where $w$ is the width, $t$ is the thickness and $n$ is the carrier density.

4. The Hall voltage across the width is $V_H=E_{H}w=vBw=\frac{IB}{nqt}$.

5. This is $V_H=\frac{R_{H}IB}{t}$ with Hall coefficient $R_H=\frac{1}{nq}$.

6. For a given sensor $n$, $q$ and $t$ are fixed, so $V_H\propto IB$. The Hall voltage is directly proportional to both $I$ and $B$, which is option (B).