The expressions below give current $I$ through an electronic component as a function of applied potential $V…
The expressions below give current $I$ through an electronic component as a function of applied potential $V . I_0$ and $V_0$ are constants having dimensions of current and potential respectively. Which of the following are dimensionally incorrect?
(A) $I=I_0\left(e^{\frac{2 V}{V_0}}+1\right)$
(B) $I=I_0\left(e^{\frac{v}{2 V_0}}-1\right)$
(C) $I=I_0 V_0\left(e^{\frac{V}{V_0}}-1\right)$
(D) $I=I_0\left(\frac{V}{V_0}\right)\left(e^{\frac{V}{V_0}}-1\right)$
$\mathrm{A}$
$\mathrm{B}$
$\mathrm{C}$
$\mathrm{D}$
Solution
According to the principle of homogeneity, a physical equation will be dimensionally correct only if the dimensions of all the terms occuring on both side of the equation are the same.
Hence, in option (c)
$I=I_0 V_0\left(e^{\frac{v}{V_0}}-1\right)$
Dimension of $I \neq$ dimensions of $I_0 V_0$ Therefore, equation shown in option (c) is not dimensionally correct.