Question

If $C, \, R, \, L \, and \, I$ denote capacity, resistance, inductance and electric current respectively, the quantities having the same dimensions of time among the following are
(1) $CR$
(2) $ \, \frac{L}{R}$
(3) $\sqrt{L C}$
(4) $LI^{2}$

• A. (1) and (2) only
• B. (1) and (3) only
• C. (1) and (4) only
• D. (1), (2) and (3) only

Answer 1

Answer: (D)

$\left[C\right]=\left[M^{- 1} L^{- 2} T^{4} A^{2}\right],\left[R\right]=\left[M L^{2} T^{- 3} A^{- 2}\right]$
$\left[L\right]=\left[M L^{2} T^{- 2} A^{- 2}\right] \, and \, \left[I\right]=\left[M^{0} L^{0} T^{0} A\right] \, $
1. $\left[C R\right]=\left[M^{- 1} L^{- 2} T^{4} A^{2}\right] \, \left[M L^{2} T^{- 3} A^{- 2}\right]$
$=\left[M^{0} L^{0} T A^{0}\right]$
2. $\frac{\left[L\right]}{\left[R\right]}=\frac{\left[M L^{2} T^{- 2} A^{- 2}\right]}{\left[M L^{2} T^{- 3} A^{- 2}\right]}=\left[M^{0} L^{0} T A^{0}\right]$
3. $\left(\sqrt{\textit{LC}}\right) = \left(\left[\left(\text{ML}\right)^{2} \left(\text{T}\right)^{- 2} \left(\text{A}\right)^{- 2}\right] \times \left(\text{M}\right)^{- 1} \left(\text{L}\right)^{- 2} \left(\text{A}\right)^{2}\right)^{1 / 2}$
$ \, =\left[M^{0} L^{0} T A^{0}\right]$
4. $\left[L I^{2}\right]=\left[M L^{2} T^{- 2} A^{- 2}\right] \, \left[M^{0} L^{0} T^{0} A^{1}\right]^{2}$
$ \, \, =\left[M L^{2} T^{- 2} A^{0}\right]$