To find the distance $d$ over which a signal can be seen clearly in foggy conditions,a railway engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog,intensity (power/area) $S$ of the light from the signal,and its frequency $f$. The engineer finds that $d$ is proportional to $S^{1/n}$. The value of $n$ is:

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. Then,the dimensional formula for $AB$ is:

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 the acceleration due to gravity is $10\,m/s^2$ and the units of length and time are changed to kilometer and hour respectively,the numerical value of the acceleration is

What could be a correct expression for the speed of ocean waves in terms of its wavelength $\lambda$,the depth $h$ of the ocean,the density $\rho$ of sea water,and the acceleration of free fall $g$?

The expressions below give current $I$ through an electronic component as a function of applied potential $V$. $I_0$ and $V_0$ are constants having dimensions of current and potential respectively. Which of the following are dimensionally incorrect?
$(A)$ $I=I_0\left(e^{\frac{2 V}{V_0}}+1\right)$
$(B)$ $I=I_0\left(e^{\frac{V}{2 V_0}}-1\right)$
$(C)$ $I=I_0 V_0\left(e^{\frac{V}{V_0}}-1\right)$
$(D)$ $I=I_0\left(\frac{V}{V_0}\right)\left(e^{\frac{V}{V_0}}-1\right)$