Question

Consider three quantities $x=E/B,y=\sqrt{1/{\mu }_0{\mathrm{\epsilon }}_0}$ and $z=\frac{l}{CR}$. Here, l is the length of a wire, C is a capacitance and R is a resistance. All other symbols have standard meanings.
(a) x, y have the same dimensions.
(b) y, z have the same dimensions.
(c) z, x have the same dimensions.
(d) None of the three pairs have the same dimensions.

Answer 1

(a) x, y have the same dimensions.
(b) y, z have the same dimensions.
(c) z, x have the same dimensions.

Lorentz Force:
$$\begin{gathered} qvB=qE \\ \Rightarrow \mathrm{Dimensions}\mathrm{of}x=\left[v\right]=\left[\frac{E}{B}\right]=\left[{\mathrm{LT}}^{-1}\right] \end{gathered}$$
$$\begin{gathered} y=\frac{1}{\sqrt{{\mathrm{\mu }}_{\mathrm{o}}{\mathrm{\epsilon }}_{\mathrm{o}}}}=\sqrt{\frac{4\mathrm{\pi }}{{\mathrm{\mu }}_{\mathrm{o}}}\times \frac{1}{4{\mathrm{\pi \epsilon }}_{\mathrm{o}}}}=\sqrt{\frac{9\times {10}^9}{{10}^{-7}}}=3\times {10}^8=\mathrm{c} \\ \Rightarrow \mathrm{Dimensions}\mathrm{of}\mathrm{y}=\left[\mathrm{c}\right]=\left[{\mathrm{LT}}^{-1}\right] \end{gathered}$$
Time constant of RC circuit = RC so dimensionally [RC] = [T]
$\Rightarrow z=\left[\frac{l}{\mathrm{RC}}\right]\Rightarrow \left[z\right]=\left[{\mathrm{LT}}^{-1}\right]$
Therefore, x, y and z have the same dimensions.