Question

A metal sample carrying a current along $X-$axis with density $J_x$ is subjected to a magnetic field $B_z$ (along $z-$axis). The electric field $E_y$ developed along $Y-$axis is directly proportional to $J_x$ as well as $B_z$. The constant of proportionality has $SI$ unit.

• A. $\frac{m^2}{A}$
• B. $\frac{m^3}{As}$
• C. $\frac{m^2}{As}$
• D. $\frac{As}{m^3}$

Answer 1

Answer: (B)

$\frac{m^3}{As}$

Given: $E_y\alpha J_x$ and $E_y\alpha B_Z$

$E_y\alpha J_x{B}_Z$

$\Rightarrow E_y=K{J}_x{B}_Z$ (Where K = constant of proportionality)

By Dimensional analysis we have;

$\left[E_y\right]=\left[K{J}_x{B}_Z\right]$

$\Rightarrow \left[K\right]=\frac{\left[E_y\right]}{\left[J_x{B}_Z\right]}=\frac{\left[\right]M^1{L}^1{T}^3{A}^{-1}]}{\left[L^{-2}{A}^A\right]\left[M^1{T}^{-2}{A}^{-1}\right]}$

$\Rightarrow \left[K\right]=\frac{\left[L^3\right]}{\left[A^1{T}^1\right]}\to$ Dimensional formual of K

So, S.I unit of K will be : $\frac{m^3}{As}$