The velocity $v$ of a particle at time $t$ is given by $v = at + \frac{b}{t + c}$,where $a, b,$ and $c$ are constants. What are the dimensions of $a, b,$ and $c$ respectively?

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:

Consider the equation $H = \frac{x^p \epsilon^q E^r}{t^s}$, where $H = \text{magnetic field intensity}$, $E = \text{electric field}$, $\epsilon = \text{permittivity}$, $x = \text{distance}$, and $t = \text{time}$. The values of $p, q, r$, and $s$ respectively are:

Which of the following operations can be performed on two physical quantities $A$ and $B$ if they possess different dimensions?

If ${\mu _0}$ and ${\varepsilon _0}$ denote the permeability and permittivity of free space,the dimensions of ${\mu _0}{\varepsilon _0}$ are:

The number of particles crossing a unit area perpendicular to the $X$-axis in unit time is given by $n = -D \frac{n_2 - n_1}{x_2 - x_1}$,where $n_1$ and $n_2$ are the number of particles per unit volume at positions $x_1$ and $x_2$ respectively. Find the dimensions of $D$,which is known as the diffusion constant.