Question

L,C and R denote inductance, capacitance and resistance respectively. Pick out the combination which does not have the dimensions of frequency

• A. $\frac{1}{RC}$
• B. $\frac{R}{L}$
• C. $\frac{1}{\sqrt{LC}}$
• D. $\frac{C}{L}$

Answer 1

The correct option is D $\frac{C}{L}$

Dimension of Resistance = Potential Difference / Current = $M{L}^2{T}^{-3}{A}^{-1}/A=M{L}^2{T}^{-3}{A}^{-2}$

Dimension of Capacitance = Charge / Potential Difference =$AT/M{L}^2{T}^{-3}{A}^{-1}=M^{-1}{L}^{-2}{T}^4{A}^2$

Dimension of self Inductance $\left[L\right]=\left[\varphi \right]/\left[I\right]=M{T}^{-2}{L}^2{A}^{-1}/A=M{L}^2{T}^{-2}{A}^{-2}$

c) Dimension of $LC=M{L}^2{T}^{-2}{A}^{-2}\times M^{-1}{L}^{-2}{T}^4{A}^2=T^2$

so Dimension of $\frac{1}{\sqrt{LC}}=T^{-1}$

a) Dimension of $1/RC=1/(M{L}^2{T}^{-3}{A}^{-2}\times M^{-1}{L}^{-2}{T}^4{A}^2)=T^{-1}$

b) Dimension of $R/L=M{L}^2{T}^{-3}{A}^{-2}/M{L}^2{T}^{-2}{A}^{-2}=T^{-1}$

d) Dimension of $C/L=M^{-1}{L}^{-2}{T}^4{A}^2/M{L}^2{T}^{-3}{A}^{-2}=M^{-2}{L}^{-4}{T}^7{A}^{-4}$

So answer is (d)