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 option has the dimensions of time, we will analyze each option step by step.

### Step 1: Understand the dimensions of time
The dimension of time is represented as [T].

### Step 2: Analyze Option A (LC)
1. **Inductance (L)**:
- Inductance is defined as the ratio of magnetic flux to current.
- The dimension of magnetic flux is [ML²T⁻²].
- The dimension of current (I) is [I].
- Therefore, the dimension of inductance (L) is:
\[ [L] = \frac{[ML²T^{-2}]}{[I]} = [ML²T^{-2}I^{-1}] \]

2. **Capacitance (C)**:
- Capacitance is defined as charge per unit voltage.
- Charge (Q) is current multiplied by time: \( Q = I \cdot T \).
- Voltage (V) is work done per unit charge: \( V = \frac{W}{Q} \).
- Work done has dimensions [ML²T⁻²], so:
\[ [V] = \frac{[ML²T^{-2}]}{[I \cdot T]} = [ML²T^{-3}I^{-1}] \]
- Therefore, the dimension of capacitance (C) is:
\[ [C] = \frac{[Q]}{[V]} = \frac{[IT]}{[ML²T^{-3}I^{-1}]} = [M^{-1}L^{-2}T^{4}I^{2}] \]

3. **Combine dimensions of L and C**:
\[ [LC] = [ML²T^{-2}I^{-1}] \cdot [M^{-1}L^{-2}T^{4}I^{2}] = [T^{2}] \]
- This is not the dimension of time [T].

### Step 3: Analyze Option B (R/L)
1. **Resistance (R)**:
- Resistance is defined as voltage per unit current.
- Therefore, the dimension of resistance is:
\[ [R] = \frac{[V]}{[I]} = \frac{[ML²T^{-3}I^{-1}]}{[I]} = [ML²T^{-3}I^{-2}] \]

2. **Inductance (L)**:
- We already calculated this as [ML²T^{-2}I^{-1}].

3. **Combine dimensions of R and L**:
\[ \left[\frac{R}{L}\right] = \frac{[ML²T^{-3}I^{-2}]}{[ML²T^{-2}I^{-1}]} = [T^{-1}] \]
- This is not the dimension of time [T].

### Step 4: Analyze Option C (L/R)
1. **Combine dimensions of L and R**:
\[ \left[\frac{L}{R}\right] = \frac{[ML²T^{-2}I^{-1}]}{[ML²T^{-3}I^{-2}]} = [T] \]
- This is the dimension of time [T].

### Step 5: Analyze Option D (C/L)
1. **Combine dimensions of C and L**:
\[ \left[\frac{C}{L}\right] = \frac{[M^{-1}L^{-2}T^{4}I^{2}]}{[ML²T^{-2}I^{-1}]} = [M^{-2}L^{-4}T^{6}I^{3}] \]
- This is not the dimension of time [T].

### Conclusion
The option that has the dimensions of time is **Option C (L/R)**.