The correct relationship between the two current gains `alpha and beta` in a transistor is

Options

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

Answer

To find the correct relationship between the two current gains, alpha (α) and beta (β), in a transistor, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The current gain α (alpha) is defined as the ratio of the collector current (Ic) to the emitter current (Ie):
\[ \alpha = \frac{I_c}{I_e} \]
- The current gain β (beta) is defined as the ratio of the collector current (Ic) to the base current (Ib):
\[ \beta = \frac{I_c}{I_b} \]

2. **Write the Relationship Between Currents**:
- The emitter current (Ie) is the sum of the base current (Ib) and the collector current (Ic):
\[ I_e = I_b + I_c \]

3. **Substitute for Ic**:
- Rearranging the equation for Ie gives:
\[ I_c = I_e - I_b \]

4. **Express Ib in Terms of Ic**:
- From the definitions of α and β, we can express Ib in terms of Ic:
\[ I_b = \frac{I_c}{\beta} \]

5. **Substituting Ib in the Emitter Current Equation**:
- Substitute the expression for Ib into the emitter current equation:
\[ I_e = \frac{I_c}{\beta} + I_c \]
- Factor out Ic:
\[ I_e = I_c \left(\frac{1}{\beta} + 1\right) \]

6. **Express Ic in Terms of Ie**:
- Rearranging gives:
\[ I_c = \frac{I_e}{\left(\frac{1}{\beta} + 1\right)} = \frac{I_e \beta}{1 + \beta} \]

7. **Substituting Ic into the Definition of α**:
- Now substitute this expression for Ic back into the definition of α:
\[ \alpha = \frac{I_c}{I_e} = \frac{\frac{I_e \beta}{1 + \beta}}{I_e} = \frac{\beta}{1 + \beta} \]

8. **Rearranging to Find β in Terms of α**:
- Rearranging the equation gives:
\[ \alpha (1 + \beta) = \beta \]
- This simplifies to:
\[ \alpha + \alpha \beta = \beta \]
- Rearranging further gives:
\[ \alpha = \beta - \alpha \beta \]
- Thus:
\[ \beta = \frac{\alpha}{1 - \alpha} \]

### Final Relationship:
The correct relationship between the two current gains α and β is:
\[ \beta = \frac{\alpha}{1 - \alpha} \]