In the relation $P = \frac{\alpha}{\beta} e^{-\frac{\alpha Z}{k\theta}}$,$P$ is pressure,$Z$ is distance,$k$ is Boltzmann constant,and $\theta$ is temperature. The dimensional formula of $\beta$ is:

Consider a simple pendulum,having a bob attached to a string,that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length $(l)$,mass of the bob $(m)$,and acceleration due to gravity $(g)$. Derive the expression for its time period using the method of dimensions.

Which of the following relations can be derived using dimensional analysis?

Consider the equation $H = \frac{x^p \epsilon^q E^r}{t^s}$,where $H = \text{magnetic field}$,$E = \text{electric field}$,$\epsilon = \text{permittivity}$,$x = \text{distance}$,and $t = \text{time}$. The values of $p, q, r$,and $s$ respectively are:

What is dimensional analysis? Write the limitations of dimensional analysis.

If momentum $(P)$,area $(A)$ and time $(T)$ are taken to be the fundamental quantities,then the dimensional formula for energy is: