If $$L, C$$ and $$R$$ are the self inductance, capacitance and resistance respectively, which of the following does not have the dimension of time?

We need to identify which of the given combinations does not have the dimension of time.

Recall the dimensions of L, C, and R.

Self inductance: $$[L] = ML^2T^{-2}A^{-2}$$

Capacitance: $$[C] = M^{-1}L^{-2}T^4A^2$$

Resistance: $$[R] = ML^2T^{-3}A^{-2}$$

Check each option.

Option A: $$\sqrt{LC}$$

$$[LC] = ML^2T^{-2}A^{-2} \times M^{-1}L^{-2}T^4A^2 = T^2$$

$$[\sqrt{LC}] = T$$ — has the dimension of time.

Option B: $$\frac{L}{R}$$

$$\left[\frac{L}{R}\right] = \frac{ML^2T^{-2}A^{-2}}{ML^2T^{-3}A^{-2}} = T$$ — has the dimension of time.

Option C: $$CR$$

$$[CR] = M^{-1}L^{-2}T^4A^2 \times ML^2T^{-3}A^{-2} = T$$ — has the dimension of time.

Option D: $$\frac{L}{C}$$

$$\left[\frac{L}{C}\right] = \frac{ML^2T^{-2}A^{-2}}{M^{-1}L^{-2}T^4A^2} = M^2L^4T^{-6}A^{-4}$$

This simplifies to $$[R^2]$$ (dimension of resistance squared), which does not have the dimension of time.

The correct answer is Option D.