Which of the following does not have the dimensions of time ?

Options

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

Answer

To solve the question "Which of the following does not have the dimensions of time?", we need to analyze the dimensions of various physical quantities. Here’s a step-by-step breakdown:

### Step 1: Identify the Physical Quantities
We need to identify the physical quantities provided in the options. Let's assume the options are:
1. Resistance (R)
2. Capacitance (C)
3. Inductance (L)
4. Frequency (f)

### Step 2: Determine the Dimensions of Each Quantity
We will calculate the dimensions of each of these quantities.

1. **Resistance (R)**:
- Formula: \( R = \frac{V}{I} \)
- Voltage (V) has dimensions \( [M^1 L^2 T^{-3} I^{-1}] \) (from \( V = W/Q \) and \( W = F \cdot d \))
- Current (I) has dimensions \( [I^1] \)
- Thus, \( R = \frac{M^1 L^2 T^{-3} I^{-1}}{I^1} = M^1 L^2 T^{-3} I^{-2} \)

2. **Capacitance (C)**:
- Formula: \( C = \frac{Q}{V} \)
- Charge (Q) has dimensions \( [I^1 T^1] \)
- Thus, \( C = \frac{I^1 T^1}{M^1 L^2 T^{-3} I^{-1}} = M^{-1} L^{-2} T^4 I^2 \)

3. **Inductance (L)**:
- Formula: \( L = \frac{V}{\frac{dI}{dt}} \)
- Thus, \( L = \frac{M^1 L^2 T^{-3} I^{-1}}{I^1 T^{-1}} = M^1 L^2 T^{-2} I^{-2} \)

4. **Frequency (f)**:
- Formula: \( f = \frac{1}{T} \)
- Thus, \( f \) has dimensions \( [T^{-1}] \)

### Step 3: Analyze the Dimensions
Now, we will check which of the above quantities does not have the dimension of time:

- **Resistance (R)**: \( [M^1 L^2 T^{-3} I^{-2}] \) - has dimensions of mass, length, and current, but not time.
- **Capacitance (C)**: \( [M^{-1} L^{-2} T^4 I^2] \) - has dimensions of time but not in a simple form.
- **Inductance (L)**: \( [M^1 L^2 T^{-2} I^{-2}] \) - has dimensions of time but not in a simple form.
- **Frequency (f)**: \( [T^{-1}] \) - has dimensions of time.

### Conclusion
From the analysis, **Resistance (R)** does not have the dimensions of time.