Quality factor and power factor both have the dimensions of

Options

A. time
B. frequency
C. work
D. angle

Explanation

To determine the dimensions of the quality factor and power factor, we can analyze their definitions and the quantities involved.

### Step-by-Step Solution:

1. **Understanding Quality Factor (Q)**:
- The quality factor is defined as the ratio of the resonant frequency (\(f_0\)) to the bandwidth (\(\Delta f\)).
- Mathematically, it is expressed as:
\[ Q = \frac{f_0}{\Delta f} \]
- Both \(f_0\) and \(\Delta f\) represent frequencies.

2. **Dimensions of Frequency**:
- Frequency (\(f\)) is defined as the number of cycles per unit time.
- The dimension of frequency is given by:
\[ [f] = [T]^{-1} \]
- Where \([T]\) represents the dimension of time.

3. **Analyzing Quality Factor**:
- Since both \(f_0\) and \(\Delta f\) have the same dimensions of frequency, the ratio \(Q\) becomes:
\[ Q = \frac{[f]}{[f]} = \frac{[T]^{-1}}{[T]^{-1}} = 1 \]
- Therefore, the quality factor \(Q\) is dimensionless.

4. **Understanding Power Factor (PF)**:
- The power factor is defined as the cosine of the phase angle (\(\phi\)), which can also be expressed as the ratio of resistance (\(R\)) to impedance (\(Z\)):
\[ PF = \cos(\phi) = \frac{R}{Z} \]

5. **Dimensions of Resistance and Impedance**:
- Both resistance (\(R\)) and impedance (\(Z\)) have the same dimensions, which are:
\[ [R] = [Z] = [\text{Resistance}] \]
- Therefore, the ratio \(PF\) becomes:
\[ PF = \frac{[R]}{[Z]} = \frac{[\text{Resistance}]}{[\text{Resistance}]} = 1 \]
- Thus, the power factor \(PF\) is also dimensionless.

6. **Conclusion**:
- Both the quality factor \(Q\) and the power factor \(PF\) are dimensionless quantities. They do not have any physical dimensions associated with them.

### Final Answer:
Both the quality factor and power factor are dimensionless quantities.