Which of these have dimension, different from others ?

Options

A. `(1)/(RC)`
B. `(R)/(L)`
C. `(1)/(sqrt(LC))`
D. `(C)/(L)`

Answer

To solve the question of which of the given options has a dimension different from the others, we will analyze the dimensions of each option step-by-step.

### Step 1: Identify the dimensions of R, C, and L
- **Resistance (R)** has the dimension of Ohm (Ω).
- **Capacitance (C)** can be expressed in terms of resistance and angular frequency (ω). The capacitive reactance is given by \( X_C = \frac{1}{\omega C} \), which has the same dimension as resistance (Ohm). Therefore, we can derive:
\[ C = \frac{1}{\omega \cdot R} \implies \text{Dimension of } C = \frac{1}{\omega} \cdot \text{Ohm} \]
- **Inductance (L)** is similarly related to resistance through inductive reactance \( X_L = \omega L \). Thus, we can derive:
\[ L = \frac{R}{\omega} \implies \text{Dimension of } L = \frac{\text{Ohm}}{\omega} \]

### Step 2: Analyze each option
Now we will analyze the dimensions of each option provided in the question.

1. **Option A: \( \frac{1}{RC} \)**
- Dimension of \( R \) = Ohm
- Dimension of \( C \) = \( \frac{1}{\omega} \cdot \text{Ohm} \)
- Therefore,
\[ RC = \text{Ohm} \cdot \left(\frac{1}{\omega} \cdot \text{Ohm}\right) = \frac{\text{Ohm}^2}{\omega} \]
- Thus,
\[ \frac{1}{RC} = \frac{\omega}{\text{Ohm}^2} \]

2. **Option B: \( \frac{R}{L} \)**
- Dimension of \( L \) = \( \frac{\text{Ohm}}{\omega} \)
- Therefore,
\[ \frac{R}{L} = \frac{\text{Ohm}}{\frac{\text{Ohm}}{\omega}} = \omega \]

3. **Option C: \( \frac{1}{\sqrt{LC}} \)**
- Dimension of \( L \) = \( \frac{\text{Ohm}}{\omega} \) and \( C \) = \( \frac{1}{\omega} \cdot \text{Ohm} \)
- Therefore,
\[ LC = \left(\frac{\text{Ohm}}{\omega}\right) \cdot \left(\frac{1}{\omega} \cdot \text{Ohm}\right) = \frac{\text{Ohm}^2}{\omega^2} \]
- Thus,
\[ \frac{1}{\sqrt{LC}} = \frac{\omega}{\text{Ohm}} \]

4. **Option D: \( \frac{C}{L} \)**
- Therefore,
\[ \frac{C}{L} = \frac{\left(\frac{1}{\omega} \cdot \text{Ohm}\right)}{\left(\frac{\text{Ohm}}{\omega}\right)} = \frac{1}{\omega} \]

### Step 3: Compare the dimensions
- **Option A:** \( \frac{\omega}{\text{Ohm}^2} \)
- **Option B:** \( \omega \)
- **Option C:** \( \frac{\omega}{\text{Ohm}} \)
- **Option D:** \( \frac{1}{\omega} \)

### Conclusion
From the analysis, we can see that:
- Options A, B, and C have dimensions that are related to \( \omega \).
- Option D has a dimension of \( \frac{1}{\omega} \), which is different from the others.

Thus, the option with a different dimension is **Option D: \( \frac{C}{L} \)**.