L,C and R represent the physical quantities inductance, capacitance and resistance respectively. Which of the following combinations have dimensions of frequency?

Options

A. (1)/(RC)
B. (R )/(L)
C. (1)/(sqrtLC)
D. C/L

Answer

To determine which combinations of inductance (L), capacitance (C), and resistance (R) have dimensions of frequency, we can analyze the relationships between these quantities and frequency.

### Step-by-Step Solution:

1. **Understanding Frequency**:
Frequency (f) is defined as the reciprocal of the time period (T), which has the dimension of time. Therefore, the dimension of frequency can be expressed as:
\[ [f] = [T]^{-1} \]

2. **Analyzing the LR Circuit**:
In an LR circuit, the time constant (τ) is given by:
\[ τ = \frac{L}{R} \]
The frequency can be expressed as the reciprocal of the time constant:
\[ f = \frac{1}{τ} = \frac{R}{L} \]
Here, the dimensions of R and L are:
- Resistance (R): \([R] = [M][L^2][T^{-3}][I^{-2}]\)
- Inductance (L): \([L] = [M][L^2][T^{-2}][I^{-2}]\)

Thus, the dimensions of \( \frac{R}{L} \) can be calculated as:
\[ [f] = \frac{[R]}{[L]} = \frac{[M][L^2][T^{-3}][I^{-2}]}{[M][L^2][T^{-2}][I^{-2}]} = [T]^{-1} \]
Therefore, \( \frac{R}{L} \) has the dimensions of frequency.

3. **Analyzing the RC Circuit**:
In an RC circuit, the time constant (τ) is given by:
\[ τ = R \cdot C \]
The frequency can be expressed as:
\[ f = \frac{1}{τ} = \frac{1}{R \cdot C} \]
The dimensions of capacitance (C) are:
- Capacitance (C): \([C] = [M^{-1}][L^{-2}][T^4][I^2]\)

Therefore, the dimensions of \( \frac{1}{R \cdot C} \) can be calculated as:
\[ [f] = \frac{1}{[R][C]} = \frac{1}{[M][L^2][T^{-3}][I^{-2}][M^{-1}][L^{-2}][T^4][I^2]} = [T]^{-1} \]
Thus, \( \frac{1}{R \cdot C} \) also has the dimensions of frequency.

4. **Analyzing the LC Circuit**:
In an LC circuit, the angular frequency (ω) is given by:
\[ ω = \frac{1}{\sqrt{LC}} \]
Since angular frequency is related to frequency by \( ω = 2πf \), we can express frequency as:
\[ f = \frac{1}{2π} \cdot \frac{1}{\sqrt{LC}} \]
The dimensions of \( \frac{1}{\sqrt{LC}} \) can be calculated as:
\[ [f] = \frac{1}{\sqrt{[L][C]}} = \frac{1}{\sqrt{[M][L^2][T^{-2}][I^{-2}][M^{-1}][L^{-2}][T^4][I^2]}} = [T]^{-1} \]
Hence, \( \frac{1}{\sqrt{LC}} \) also has the dimensions of frequency.

### Conclusion:
Thus, the combinations \( \frac{R}{L} \), \( \frac{1}{RC} \), and \( \frac{1}{\sqrt{LC}} \) all have the dimensions of frequency. Therefore, all three options are correct.