A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material. Boltzmann constant $\left(k_B\right)$, the absolute temperature $(T)$, the number per unit volune $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particless. Which of the following expression($s$) for $l$ is(are) dimensionally correct?

($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$

($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$

($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$

($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$

A massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega$. The power $P$ radiated by it as gravitational waves is given by $P=G c^{-5} m^{x} R^{y} \omega^{z}$, where $c$ and $G$ are speed of light in free space and the universal gravitational constant, respectively. Then,

if Energy is given by $U = \frac{{A\sqrt x }}{{{x^2} + B}},\,$, then dimensions of $AB$ is

The $SI$ unit of energy is $J=k g\, m^{2} \,s^{-2} ;$ that of speed $v$ is $m s^{-1}$ and of acceleration $a$ is $m s ^{-2} .$ Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ( $m$ stands for the mass of the body ):

$(a)$ $K=m^{2} v^{3}$

$(b)$ $K=(1 / 2) m v^{2}$

$(c)$ $K=m a$

$(d)$ $K=(3 / 16) m v^{2}$

$(e)$ $K=(1 / 2) m v^{2}+m a$

The equation of a circle is given by $x^2+y^2=a^2$, where $a$ is the radius. If the equation is modified to change the origin other than $(0,0)$, then find out the correct dimensions of $A$ and $B$ in a new equation: $(x-A t)^2+\left(y-\frac{t}{B}\right)^2=a^2$.The dimensions of $t$ is given as $\left[ T ^{-1}\right]$.

If force $({F})$, length $({L})$ and time $({T})$ are taken as the fundamental quantities. Then what will be the dimension of density