For a common emiter configuration if `a` and `beta` have their usualy meaning , the incorrrect relationship between `a and beta` is :

Options

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

Answer

To solve the problem regarding the relationships between alpha (α) and beta (β) in a common emitter configuration, let's go through the steps systematically.

### Step-by-Step Solution:

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

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

3. **Expressing Ib in terms of Ic and β**:
- From the definition of β:
\[ I_b = \frac{I_c}{\beta} \]
- Substituting this into the equation for Ie:
\[ I_e = I_c + \frac{I_c}{\beta} = I_c \left(1 + \frac{1}{\beta}\right) \]

4. **Expressing Ic in terms of α and Ie**:
- From the definition of α:
\[ I_c = \alpha I_e \]

5. **Combining the equations**:
- Substitute \(I_c\) from the α equation into the equation for Ie:
\[ I_e = \alpha I_e + \frac{\alpha I_e}{\beta} \]
- Factor out \(I_e\) (assuming \(I_e \neq 0\)):
\[ 1 = \alpha + \frac{\alpha}{\beta} \]

6. **Rearranging the equation**:
- Rearranging gives:
\[ 1 = \alpha \left(1 + \frac{1}{\beta}\right) \]
- This leads to:
\[ \frac{1}{\alpha} = 1 + \frac{1}{\beta} \]

7. **Finding the relationship between α and β**:
- Rearranging the above equation gives:
\[ \frac{1}{\alpha} - 1 = \frac{1}{\beta} \]
- This can be simplified to:
\[ \beta = \frac{\alpha}{1 - \alpha} \]

8. **Final relationships**:
- We have two important relationships:
- \( \alpha = \frac{\beta}{1 + \beta} \)
- \( \beta = \frac{\alpha}{1 - \alpha} \)

9. **Identifying the incorrect relationship**:
- Now, we need to check which of the given relationships is incorrect. The relationships we derived are:
- \( \alpha = \frac{\beta}{1 + \beta} \) (Correct)
- \( \beta = \frac{\alpha}{1 - \alpha} \) (Correct)
- Any other relationships provided in the question that do not match these derived relationships can be considered incorrect.

### Conclusion:
The incorrect relationships between α and β in the given options are those that do not match the derived equations.