The current gain of a transistor in common base and common-emitter configurations called `alpha ` and ` beta ` are related as

Options

A. ` alpha = ( beta ) /( 1 + beta )`
B. `beta = ( alpha )/( 1- alpha)`
C. `beta = (1+ alpha)/( alpha)`
D. both 1 & 2

Answer

To find the relationship between the current gain of a transistor in common base (α) and common emitter (β) configurations, we can follow these steps:

### Step 1: Define the Current Gains
- In a common base configuration, the current gain (α) is defined as:
\[ \alpha = \frac{I_C}{I_E} \]
where \(I_C\) is the collector current and \(I_E\) is the emitter current.

- In a common emitter configuration, the current gain (β) is defined as:
\[ \beta = \frac{I_C}{I_B} \]
where \(I_B\) is the base current.

### Step 2: Use the Current Relationship
- The relationship between the emitter current, collector current, and base current is given by:
\[ I_E = I_C + I_B \]

### Step 3: Express \(I_E\) in Terms of \(I_C\)
- From the definition of α, we can express \(I_E\) in terms of \(I_C\):
\[ I_E = \frac{I_C}{\alpha} \]

### Step 4: Substitute \(I_E\) into the Current Relationship
- Substitute \(I_E\) into the equation \(I_E = I_C + I_B\):
\[ \frac{I_C}{\alpha} = I_C + I_B \]

### Step 5: Express \(I_B\) in Terms of \(I_C\)
- Rearranging gives:
\[ I_B = \frac{I_C}{\alpha} - I_C = I_C \left(\frac{1}{\alpha} - 1\right) = I_C \left(\frac{1 - \alpha}{\alpha}\right) \]

### Step 6: Substitute \(I_B\) into the Definition of β
- Now, substitute \(I_B\) into the definition of β:
\[ \beta = \frac{I_C}{I_B} = \frac{I_C}{I_C \left(\frac{1 - \alpha}{\alpha}\right)} = \frac{\alpha}{1 - \alpha} \]

### Step 7: Rearranging the Equation
- Rearranging gives:
\[ \beta = \frac{\alpha}{1 - \alpha} \]
- This can also be expressed as:
\[ \alpha = \frac{\beta}{1 + \beta} \]

### Conclusion
The relationship between α and β can be summarized as:
\[ \alpha = \frac{\beta}{1 + \beta} \]
and
\[ \beta = \frac{\alpha}{1 - \alpha} \]