As shown in the above figure, a uniform magnetic field B is directed perpendicular to the plane of a 3l long wooden stick which is rotated about an end with constant angular velocity $\omega \text{. }$ Find the potential difference across the whole length of the stick if the upper ${\left(\frac{1}{3}\right)}^{rd}$ of its length is coated with copper:

The primary concept behind potential difference due to motion of a conductor is the drift of free charged particles (like electrons) inside the conductor. Due to this reason, potential difference will be developed only in the copper part and not in the wooden part as wood is an insulator and does not have free charge carriers.
EMF induced in the copper stick of length I is given by

$\mathrm{\epsilon }={\int }_{2\mathrm{l}}^{3\mathrm{l}} (\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})\cdot \mathrm{d}\overrightarrow{\mathrm{l}}$

$\Rightarrow \mathrm{\epsilon }={\int }_{2\mathrm{l}}^{3\mathrm{l}} \mathrm{B\omega ldl}(\because \mathrm{v}=\mathrm{l\omega })$

$\Rightarrow \mathrm{\epsilon }=\mathrm{B\omega }\left[\frac{9{\mathrm{l}}^2}{2}-\frac{4{\mathrm{l}}^2}{2}\right]$

$\Rightarrow \mathrm{\epsilon }=\frac{5}{2}{\mathrm{B\omega l}}^2$