A CE amplifier, has a power gain of 40 dB. The input resistance and output load resistance are `500Omega and 2kOmega` respectively. Find the value of common - emitter current gain.
`["take gain in dB "=10log((P_("out"))/(P_("in")))]`

Answer

To solve the problem step by step, we will follow the logical flow of the information given and apply the relevant formulas.

### Step 1: Understand the power gain in decibels
The power gain in decibels (dB) is given by the formula:
\[ \text{Gain (dB)} = 10 \log_{10} \left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) \]
Given that the power gain is 40 dB, we can express this as:
\[ 40 = 10 \log_{10} \left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) \]

### Step 2: Convert the gain from dB to a power ratio
Dividing both sides by 10, we have:
\[ 4 = \log_{10} \left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) \]
To eliminate the logarithm, we exponentiate both sides:
\[ \frac{P_{\text{out}}}{P_{\text{in}}} = 10^4 = 10000 \]
Thus, the power gain \( P_{\text{gain}} \) is 10000.

### Step 3: Relate power gain to voltage gain and current gain
The power gain can also be expressed in terms of voltage gain (\( V_{\text{gain}} \)) and current gain (\( I_{\text{gain}} \)):
\[ P_{\text{gain}} = V_{\text{gain}} \times I_{\text{gain}} \]
For a common emitter amplifier, the voltage gain is given by:
\[ V_{\text{gain}} = I_{\text{gain}} \times \frac{R_L}{R_I} \]
where \( R_L \) is the load resistance and \( R_I \) is the input resistance.

### Step 4: Substitute the voltage gain into the power gain formula
Substituting for \( V_{\text{gain}} \) in the power gain equation, we have:
\[ P_{\text{gain}} = I_{\text{gain}} \times \left(I_{\text{gain}} \times \frac{R_L}{R_I}\right) = I_{\text{gain}}^2 \times \frac{R_L}{R_I} \]

### Step 5: Substitute the known values
We know:
- \( P_{\text{gain}} = 10000 \)
- \( R_L = 2000 \, \Omega \) (2 kΩ)
- \( R_I = 500 \, \Omega \)

Substituting these values into the equation:
\[ 10000 = I_{\text{gain}}^2 \times \frac{2000}{500} \]
This simplifies to:
\[ 10000 = I_{\text{gain}}^2 \times 4 \]

### Step 6: Solve for the current gain
Rearranging gives:
\[ I_{\text{gain}}^2 = \frac{10000}{4} = 2500 \]
Taking the square root:
\[ I_{\text{gain}} = \sqrt{2500} = 50 \]

### Final Answer
The value of the common-emitter current gain is:
\[ \boxed{50} \]