A uniform magnetic field exists in the region given by $\overrightarrow{\mathrm{B}}=3\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}+5\widehat{\mathrm{k}}$ .  A rod of length 5 m placed along Y-axis is moved along X-axis with constant speed $1 {\mathrm{ms}}^{-1}$ . Then induced e.m.f in the rod is

$\mathrm{l}=5\mathrm{j}$

$\mathrm{V}=1\mathrm{i}$

$\mathrm{e}=\mathrm{B}\left(\mathrm{l}\times \mathrm{V}\right)$

$=\mathrm{B}\left|\begin{array}{ccc}\widehat{\mathrm{i}}& \widehat{\mathrm{j}}& \widehat{\mathrm{k}}\\ 0& 5& 0\\ 1& 0& 0\end{array}\right|$

$=\mathrm{B}\left(0-5\widehat{\mathrm{K}}\right)$

$\left(3\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}+5\widehat{\mathrm{k}}\right).\left(-5\widehat{\mathrm{k}}\right)$

$=25\mathrm{V}$

Conceptual approach:-

As the rod is placed along Y axis and moving in x direction, the magnetic field has to be perpendicular to both $\overrightarrow{\mathrm{l}}\mathrm{and}\overrightarrow{\mathrm{v}}$ in order to develop motional emf. Hence the z component of the magnetic field should be taken.