Question

If $C,R,LandI$ 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) $\sqrt{LC}$
(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)

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