L, C and R represent the physical quantities, inductance, capacitance and resistance respectively. The combination(s) which have the dimensions of frequency are

Options

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

Answer

To solve the problem of determining which combinations of inductance (L), capacitance (C), and resistance (R) have the dimensions of frequency, we will analyze the time constants associated with different circuits and their relationships to frequency.

### Step-by-Step Solution:

1. **Understanding Frequency**:
Frequency (f) is defined as the reciprocal of the time period (T), i.e.,
\[ f = \frac{1}{T} \]
The dimensions of frequency are given by:
\[ [f] = [T]^{-1} \]

2. **RC Circuit**:
- The time constant (τ) for an RC circuit is given by:
\[ τ_{RC} = R \cdot C \]
- Therefore, the frequency can be expressed as:
\[ f_{RC} = \frac{1}{τ_{RC}} = \frac{1}{R \cdot C} \]
- Since this expression is in the form of \( \frac{1}{\text{(some quantity)}} \), it has the dimensions of frequency.

3. **LC Circuit**:
- The time constant (τ) for an LC circuit is given by:
\[ τ_{LC} = \sqrt{L \cdot C} \]
- Thus, the frequency can be expressed as:
\[ f_{LC} = \frac{1}{τ_{LC}} = \frac{1}{\sqrt{L \cdot C}} \]
- This also has the dimensions of frequency.

4. **LR Circuit**:
- The time constant (τ) for an LR circuit is given by:
\[ τ_{LR} = \frac{L}{R} \]
- Therefore, the frequency can be expressed as:
\[ f_{LR} = \frac{1}{τ_{LR}} = \frac{R}{L} \]
- This expression also has the dimensions of frequency.

5. **Conclusion**:
- From the analysis, we find that:
- The RC circuit has dimensions of frequency.
- The LC circuit has dimensions of frequency.
- The LR circuit has dimensions of frequency.
- Therefore, all three combinations (RC, LC, and LR) have dimensions of frequency.

### Final Answer:
All three combinations (A: RC, B: LR, C: LC) have the dimensions of frequency.

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