In the expression $A=B+\frac{C}{D+E}$,the dimensions of physical quantities $B$ and $C$ are $[L^{1} M^{0} T^{-1}]$ and $[L^{1} M^{0} T^{0}]$ respectively. The dimensions of quantities $A, D$ and $E$ are

If velocity $(V)$,force $(F)$ and time $(T)$ are chosen as fundamental quantities,then the dimensions of energy are:

$A$ length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material,the Boltzmann constant $(k_B)$,the absolute temperature $(T)$,the number density $(n)$ of certain charged particles,and the charge $(q)$ carried by each of the particles. Which of the following expression$(s)$ for $l$ is(are) dimensionally correct?
$(A)$ $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
$(B)$ $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
$(C)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
$(D)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$

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

An expression of energy density is given by $u = \frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)$,where $\alpha, \beta$ are constants,$x$ is displacement,$k$ is Boltzmann constant,and $t$ is the temperature. The dimensions of $\beta$ will be.

The dimensions of physical quantity $X$ in the equation $\text{Force} = \frac{X}{\text{Density}}$ are given by: