Question

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

• 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)

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