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(C) According to the principle of homogeneity,a physical equation is dimensionally correct only if the dimensions of all terms on both sides of the eq

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$C$

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(C) According to the principle of homogeneity,a physical equation is dimensionally correct only if the dimensions of all terms on both sides of the equation are identical. The argument of an exponential function must be dimensionless. In all given options,the term $\frac{V}{V_0}$ is dimensionless,so the exponential terms are valid.Now,we check the dimensions of the coefficients:For $(A)$: $[I] = [I_0]$,which is correct.For $(B)$: $[I] = [I_0]$,which is correct.For $(C)$: $[I] = [I_0 V_0]$. Since $[V_0]$ is potential,$[I] 
eq [I_0 V_0]$. Thus,$(C)$ is dimensionally incorrect.For $(D)$: $[I] = [I_0] 	imes [\frac{V}{V_0}]$. Since $[\frac{V}{V_0}]$ is dimensionless,$[I] = [I_0]$,which is correct.Therefore,option $(C)$ is the dimensionally incorrect expression.

List-$I$	List-$II$
$A$. Meter $(L)$	$I$. $\sqrt{\frac{hc}{G}}$
$B$. Second $(S)$	$II$. $\sqrt{\frac{Gh}{c^5}}$
$C$. Kilogram $(M)$	$III$. $\sqrt{\frac{L^2c^3}{Gh}}$
$D$. Kelvin $(K)$	$IV$. $\sqrt{\frac{Gh}{c^3}}$

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