Physics

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.


ANSWER

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


SOLUTION
Given: $$E_{y} \alpha J_{x}$$  and $$E_{y} \alpha B_{Z}$$
$$E_{y} \alpha J_{x}B_{Z}$$
$$\Rightarrow E_{y}=KJ_{x}B_{Z}$$ (Where K = constant of proportionality)
By Dimensional analysis we have; 
$$\left [ E_{y} \right ] = [KJ_{x}B_{Z}]$$
$$\Rightarrow \left [ K \right ] = \dfrac{\left [ E_{y} \right ]}{\left [ J_{x}B_{Z} \right ]}=\dfrac{[]M^{1}L^{1}T^{3}A^{-1}]}{[L^{-2}A^{A}][M^{1}T^{-2}A^{-1}]}$$
$$\Rightarrow [K]=\dfrac{[L^{3}]}{[A^{1}T^{1}]}\rightarrow $$ Dimensional formual of K 
So, S.I unit of K will be : $$ \dfrac{m^{3}}{As}$$
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