In the Van der Waals equation $\left[ P + \frac{a}{V^2} \right] [V - b] = RT$,where $P$ is pressure,$V$ is volume,$R$ is the universal gas constant,and $T$ is temperature,the ratio of constants $\frac{a}{b}$ is dimensionally equal to:

The potential energy of a particle varies with distance $x$ from a fixed origin as $U = \frac{A\sqrt{x}}{x^2 + B}$,where $A$ and $B$ are dimensional constants. Find the dimensional formula for $A/B$.

If energy $E = G^p h^q c^r$,where $G$ is the universal gravitational constant,$h$ is Planck's constant,and $c$ is the speed of light,find the values of $p, q$,and $r$ respectively.

The equation of motion of a damped oscillator is given by $m \frac{d^2 x}{d t^2}+b \frac{d x}{d t}+k x=0$. The dimensional formula of $\frac{b}{\sqrt{k m}}$ is

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

The dimensional formulas for acceleration,velocity,and length are $\alpha \beta^{-2}$,$\alpha \beta^{-1}$,and $\alpha \gamma$ respectively. What is the dimensional formula for the coefficient of friction?