Which of the following will have the dimensions of time ?

Options

A. `LC`
B. `(R)/(L)`
C. `(L)/(R)`
D. `(C)/(L)`

Explanation

To determine which of the given options has the dimensions of time, we need to analyze the dimensions of each option step by step.

### Step 1: Identify the dimensions of inductance (L)
Inductance (L) is defined by the formula:
\[ n \Phi = LI \]
Where:
- \( n \) = number of turns (dimensionless)
- \( \Phi \) = magnetic flux
- \( L \) = inductance
- \( I \) = current

The dimension of magnetic flux \( \Phi \) is given by:
\[ [\Phi] = [M][L^2][T^{-2}][I^{-1}] \]
Thus, the dimension of inductance \( L \) can be derived as:
\[ [L] = \frac{[\Phi]}{[I]} = \frac{[M][L^2][T^{-2}][I^{-1}]}{[I]} = [M][L^2][T^{-2}][I^{-2}] \]

### Step 2: Identify the dimensions of capacitance (C)
Capacitance (C) is defined by the formula:
\[ C = \frac{Q}{V} \]
Where:
- \( Q \) = charge
- \( V \) = potential

The dimension of charge \( Q \) can be expressed in terms of current:
\[ [Q] = [I][T] \]
And the dimension of potential \( V \) is:
\[ [V] = \frac{[W]}{[Q]} = \frac{[M][L^2][T^{-2}]}{[Q]} \]
So, the dimension of capacitance becomes:
\[ [C] = \frac{[Q]}{[V]} = \frac{[I][T]}{[\frac{[M][L^2][T^{-2}]}{[Q]}]} = \frac{[I][T][Q]}{[M][L^2][T^{-2}]} \]
Substituting \( [Q] = [I][T] \):
\[ [C] = \frac{[I][T]^2}{[M][L^2][T^{-2}]} = [M^{-1}][L^{-2}][T^4] \]

### Step 3: Combine dimensions of L and C
Now we need to find the dimensions of \( L \times C \):
\[ [L \times C] = [M][L^2][T^{-2}][I^{-2}] \times [M^{-1}][L^{-2}][T^4] \]
Combining these gives:
\[ [L \times C] = [M^{0}][L^{0}][T^{2}][I^{-2}] \]
This indicates that \( L \times C \) does not have the dimension of time.

### Step 4: Identify the dimensions of resistance (R)
Resistance (R) is defined by the formula:
\[ R = \frac{V}{I} \]
Using the dimensions of potential and current:
\[ [R] = \frac{[V]}{[I]} = \frac{[M][L^2][T^{-2}]}{[I]} = [M][L^2][T^{-2}][I^{-1}] \]

### Step 5: Find the dimensions of R/L
Now we check the dimensions of \( R/L \):
\[ [R/L] = \frac{[M][L^2][T^{-2}][I^{-1}]}{[M][L^2][T^{-2}][I^{-2}]} = [I] \]
This does not have the dimension of time.

### Step 6: Find the dimensions of L/R
Now we check the dimensions of \( L/R \):
\[ [L/R] = \frac{[M][L^2][T^{-2}][I^{-2}]}{[M][L^2][T^{-2}][I^{-1}]} = [I^{-1}] \]
This does not have the dimension of time.

### Step 7: Find the dimensions of C/L
Finally, we check the dimensions of \( C/L \):
\[ [C/L] = \frac{[M^{-1}][L^{-2}][T^4]}{[M][L^2][T^{-2}][I^{-2}]} = [M^{-2}][L^{-4}][T^6][I^2] \]
This has the dimension of time.

### Conclusion
After analyzing all the options, we find that the only combination that has the dimensions of time is \( C/L \).