Question

Consider three quantities  \[x = E/B,   y = \sqrt{1/ \mu_0 \epsilon_0}\] 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:

\[qvB = qE\]
\[ \Rightarrow \text{ Dimensions of x }= [v] = \left[ \frac{E}{B} \right] = [ {LT}^{- 1} ]\]

\[y = \frac{1}{\sqrt{\mu_o \epsilon_o}} = \sqrt{\frac{4\pi}{\mu_o} \times \frac{1}{4 \pi\epsilon_o}} = \sqrt{\frac{9 \times {10}^9}{{10}^{- 7}}} = 3 \times {10}^8 = c\]
\[ \Rightarrow \text{ Dimensions of y }= [c] = [ {LT}^{- 1} ]\]

Time constant of RC circuit = RC so dimensionally [RC] = [T]

\[\Rightarrow z = \left[ \frac{l}{RC} \right] \Rightarrow [z] = [ {LT}^{- 1} ]\]

Therefore, x, y and z have the same dimensions.