If `beta, R_L and r` are the ac current gain, load resistance and the input resistance of a transistor respectively in CE configuration, the voltage and the power gains respectively are

Options

A. `betaR_L/r and beta^2 R_L/r`
B. `beta r/R_L and beta^2 r/R_L`
C. `beta R_L/r and beta(R_L/r)^2`
D. `beta r/R_L and beta(r/R_L)^2`

Explanation

To find the voltage and power gains in a common emitter (CE) configuration of a transistor, we can follow these steps:

### Step 1: Understand the Definitions
- **Current Gain (β)**: This is the ratio of the output current to the input current in a transistor. In CE configuration, it is denoted as β.
- **Load Resistance (R_L)**: This is the resistance connected to the output of the transistor.
- **Input Resistance (r)**: This is the resistance seen by the input signal at the base of the transistor.

### Step 2: Calculate the Resistance Gain
The resistance gain (R_gain) can be calculated using the formula:
\[ R_{\text{gain}} = \frac{R_L}{r} \]
This represents the ratio of the load resistance to the input resistance.

### Step 3: Calculate the Voltage Gain
The voltage gain (V_gain) in a CE configuration can be expressed as:
\[ V_{\text{gain}} = \beta \times R_{\text{gain}} \]
Substituting the expression for resistance gain, we get:
\[ V_{\text{gain}} = \beta \times \frac{R_L}{r} \]
Thus, the voltage gain can be written as:
\[ V_{\text{gain}} = \frac{\beta R_L}{r} \]

### Step 4: Calculate the Power Gain
The power gain (P_gain) can be calculated using the formula:
\[ P_{\text{gain}} = \text{(Current Gain)}^2 \times \text{(Resistance Gain)} \]
Substituting the values, we have:
\[ P_{\text{gain}} = \beta^2 \times \frac{R_L}{r} \]

### Final Results
- **Voltage Gain**:
\[ V_{\text{gain}} = \frac{\beta R_L}{r} \]
- **Power Gain**:
\[ P_{\text{gain}} = \frac{\beta^2 R_L}{r} \]