Which of the following dimensions will be the same as that of time?

The correct answer is L/R.

To determine which of the given ratios has the same dimensions as time, let us analyze the physical quantities involved:
1. Inductance ($L$)
2. Resistance ($R$)
3. Capacitance ($C$)

Let us find the dimensions of each using standard electrical relationships.

First, let's look at the self-inductance $L$. The induced electromotive force (voltage $V$) across an inductor is given by:
$V=L\frac{dI}{dt}$

From this relationship, the dimension of inductance $L$ can be expressed as:
$\left[L\right]=\left[V\right]·\left[T\right]·{\left[I\right]}^{-1}$

Next, according to Ohm's law, the resistance $R$ is given by:
$R=\frac{V}{I}$

So, the dimension of resistance $R$ is:
$\left[R\right]=\left[V\right]·{\left[I\right]}^{-1}$

Now, let us calculate the dimension of the ratio $\frac{L}{R}$:
$\left[\frac{L}{R}\right]=\frac{\left[V\right]·\left[T\right]·{\left[I\right]}^{-1}}{\left[V\right]·{\left[I\right]}^{-1}}=\left[T\right]$

Thus, the dimension of $\frac{L}{R}$ is equivalent to the dimension of time ($T$). This quantity is commonly known as the time constant of an L-R circuit.