The foundations of dimensional analysis were laid down by

Show that $\frac{\text{Joule}}{\text{meter}^2} = \frac{\text{Newton}}{\text{meter}}$.

$A$ bomb explodes at time $t=0$ in a uniform,isotropic medium of density $\rho$ and releases energy $E$,generating a spherical blast wave. The radius $R$ of this blast wave varies with time $t$ as

In terms of potential difference $V$,electric current $I$,permittivity $\varepsilon_0$,permeability $\mu_0$,and speed of light $c$,the dimensionally correct equation$(s)$ is(are):
$(A)$ $\mu_0 I^2 = \varepsilon_0 V^2$
$(B)$ $\varepsilon_0 I = \mu_0 V$
$(C)$ $I = \varepsilon_0 cV$
$(D)$ $\mu_0 cI = V$

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities,namely,the gravitational constant $G$,Planck's constant $h$,and the speed of light $c$,as $Y = c^\alpha h^\beta G^\gamma$. Which of the following is the correct option?

The efficiency of an engine is given by $\eta = \frac{\alpha \beta}{\sin \theta} \cdot \log_{e} \frac{\beta x}{kT}$,where $\alpha$ and $\beta$ are constants. If $T$ is the absolute temperature,$k$ is the Boltzmann constant,$\theta$ is angular displacement,and $x$ is distance,then the incorrect statement is: