A physical quantity of the dimensions of length that can be formed out of $c, G$ and $\frac{e^2}{4\pi \varepsilon _0}$ is $[c$ is velocity of light, $G$ is the universal constant of gravitation and $e$ is charge $] $
$\frac{1}{{{c^2}}}$$\sqrt {\frac{{{e^2}}}{{G4\pi \varepsilon_0}}} $
$\frac{1}{{{c^{}}}}\frac{{G{e^2}}}{{4\pi \varepsilon_0}}$
$\frac{1}{{{c^2}}}$$\sqrt {\frac{{G{e^2}}}{{4\pi \varepsilon_0}}} $
${c^2}\;\sqrt {\frac{{G{e^2}}}{{4\pi \varepsilon_0}}} $
A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E} \times \vec{B}) / \mu_0$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $\mu_0$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity (ies)?
$(A)$ $\frac{\text { Energy }}{\text { charge } \times \text { current }}$
$(B)$ $\frac{\text { Force }}{\text { Length } \times \text { Time }}$
$(C)$ $\frac{\text { Energy }}{\text { Volume }}$
$(D)$ $\frac{\text { Power }}{\text { Area }}$
Match List $I$ with List $II$
List $I$ | List $II$ |
$A$ Spring constant | $I$ $(T ^{-1})$ |
$B$ Angular speed | $II$ $(MT ^{-2})$ |
$C$ Angular momentum | $III$ $(ML ^2)$ |
$D$ Moment of Inertia | $IV$ $(ML ^2 T ^{-1})$ |
Choose the correct answer from the options given below
The equation of state of some gases can be expressed as $\left( {P + \frac{a}{{{V^2}}}} \right)\,(V - b) = RT$. Here $P$ is the pressure, $V$ is the volume, $T$ is the absolute temperature and $a,\,b,\,R$ are constants. The dimensions of $'a'$ are
A small steel ball of radius $r$ is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta $. After some time the velocity of the ball attains a constant value known as terminal velocity ${v_T}$. The terminal velocity depends on $(i)$ the mass of the ball $m$, $(ii)$ $\eta $, $(iii)$ $r$ and $(iv)$ acceleration due to gravity $g$. Which of the following relations is dimensionally correct
The equation of stationary wave is
$\mathrm{y}=2 \mathrm{a} \sin \left(\frac{2 \pi \mathrm{nt}}{\lambda}\right) \cos \left(\frac{2 \pi \mathrm{x}}{\lambda}\right)$
Which of the following is NOT correct