Show that the Hall voltage across wires made of the same material, carrying identical currents, and subjected to the same magnetic field is inversely proportional to their diameters. (Hint: Consider how drift velocity depends on wire diameter.)

The Hall effect is defined as the generation of a voltage differential (the Hall voltage) across an electrical conductor that is transverse to the current and perpendicular to the applied magnetic field.

We'll take into account the Hall voltage, which will be

\({\rm{\varepsilon }} = {\rm{BL}}{{\rm{v}}_{\rm{D}}}{\rm{ }}\)

and

\({{\text{v}}_{\text{D}}} = \frac{{\text{I}}}{{{\text{nqA}}}} = \frac{{\text{I}}}{{{\text{nq\pi }}{{\text{r}}^{\text{2}}}}}{\text{ }}\)

Where,

\({\text{L}} = {\text{2r}}\)

Therefore,

\({\rm{\varepsilon }} = \frac{{{\rm{BL}}}}{{{\rm{nq\pi }}{{\rm{r}}^{\rm{2}}}}}\)

\( = \frac{{{\rm{B}} \times {\rm{2r}}}}{{{\rm{nq\pi }}{{\rm{r}}^{\rm{2}}}}}\)

\( = \frac{{{\rm{4B}}}}{{{\rm{nq\pi }}}} \times \frac{{\rm{1}}}{{{\rm{2r}}}}\)

When

\(k = \frac{{{\text{4B}}}}{{{\text{nq\pi }}}}\)

\({\rm{\varepsilon }} = \frac{k}{{{\text{2}}r}}\) ………………….(1)

Therefore, the hall voltage is inversely proportional to the diameter of the wire.