If $\mu_0$ and $\varepsilon_0$ are the permeability and permittivity of free space,respectively,then the dimension of $\left(\frac{1}{\mu_0 \varepsilon_0}\right)$ is

The energy $E$ of a system is a function of time $t$ and is given by $E(t) = \alpha t - \beta t^3$,where $\alpha$ and $\beta$ are constants. The dimensions of $\alpha$ and $\beta$ are

The equation of state of some gases can be expressed as $\left( {P + \frac{a}{{{V^2}}}} \right)\,(V - b) = RT$. Here $P$ is the pressure,$V$ is the volume,$T$ is the absolute temperature and $a, b, R$ are constants. The dimensions of $a$ are

$A$ great physicist of this century ($P.A.M.$ Dirac) loved playing with numerical values of fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics ($c, e,$ mass of electron, mass of proton) and the gravitational constant $G$, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe ($\approx 15$ billion years). From the table of fundamental constants in this book, try to see if you too can construct this number. If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?

The velocity of water waves $v$ may depend upon their wavelength $\lambda$,the density of water $\rho$,and the acceleration due to gravity $g$. The method of dimensions gives the relation between these quantities as:

The Young's modulus of a wire is given by $Y = \frac{F}{A} \cdot \frac{L}{\Delta L}$,where $L$ is length,$A$ is cross-sectional area,and $\Delta L$ is the change in length. By what factor must we multiply to convert the value from $CGS$ units to $MKS$ units?