Charge on the capacitor is given by Q=I αe^-(t)/(ΔV ε0 β), where αand βare constant, t= time, I=

step 1 In this question we will learn about some basic concept of the units and measurement. The questi

Charge on the capacitor is given by Q=I αe^-(t)/(ΔV ε0 β),...

Charge on the capacitor is given by $Q=I \alpha e^{-\frac{t}{\Delta V \varepsilon_{0} \beta}}$, where $\alpha$ and $\beta$ are constant, $t=$ time, $I=$ current, $\Delta V=$ Potential difference then, dimension of $\frac{\beta}{\alpha}$ is same as dimension of (A) $\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}$ (B) $\mu_{0} \varepsilon_{0}$ (C) $\frac{\mu_{0}}{\varepsilon_{0}}$ (D) $\frac{1}{\mu_{0} \varepsilon_{0}}$

Charge on the capacitor is given by $Q=I \alpha e^{-\frac{t}{\Delta V \varepsilon_{0} \beta}}$, where $\alpha$ and $\beta$ are constant, $t=$ time, $I=$ current, $\Delta V=$ Potential difference then, dimension of $\frac{\beta}{\alpha}$ is same as dimension of
(A) $\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}$
(B) $\mu_{0} \varepsilon_{0}$
(C) $\frac{\mu_{0}}{\varepsilon_{0}}$
(D) $\frac{1}{\mu_{0} \varepsilon_{0}}$

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Key Concepts

Dimensional Analysis

Dimensional analysis is the method of checking the consistency of physical equations by comparing the dimensions on both sides. In problems like this one, it ensures that each term has consistent units (for example, charge being in Coulombs) and that derived quantities, such as the ratio of constants, correctly match the dimensions of known physical quantities.

Dimensionless Exponents

In any formula involving an exponential function, the argument of the exponential must be dimensionless. This requirement forces the parameters in the exponent (in this case, the combination t/(?V ?? ?)) to multiply together so that their resulting units cancel out. This principle helps determine the necessary dimensions of the constants involved, such as ?, by setting up the condition that its product with other factors yields a dimensionless quantity.

Vacuum Permittivity and Permeability

Vacuum permittivity (??) and permeability (??) are fundamental constants in electromagnetism that characterize how electric and magnetic fields propagate in free space. Their dimensions are interconnected by the relation c = 1/?(????), where c is the speed of light (with dimensions of length per time). This relation is often used in dimensional analysis to compare ratios involving these constants with other physical quantities such as speed.

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