The dimensional formula of polarization $P$ is . . . . . . .

$A$ function $f(\theta)$ is defined as $f(\theta) = 1 - \theta + \frac{\theta^2}{2!} - \frac{\theta^3}{3!} + \frac{\theta^4}{4!} - \dots$. Why is it necessary for $f(\theta)$ to be a dimensionless quantity?

The dimensional formula of a physical quantity represented by $\frac{e^2}{4 \pi \varepsilon_0 \hbar}$ is (where $e$ is the charge of an electron,$\varepsilon_0$ is the permittivity of free space,and $\hbar$ is the reduced Planck's constant). Note: The expression $\frac{e^2}{4 \pi \varepsilon_0 \hbar}$ is equivalent to the fine-structure constant $\alpha$ multiplied by the speed of light $c$.

An athletic coach told his team that muscle times speed equals power. What dimensions does he view for muscle?

The wave equation is given by $Y = A \sin \omega \left( \frac{x}{v} - k \right)$,where $\omega$ is the angular velocity and $v$ is the linear velocity. What is the dimension of $k$?

What is the dimensional formula of an electron volt $(eV)$, and what is its value in Joules $(J)$?