Question

Derived the relation between alpha and beta for transistor

Answer 1

Let the amplification factors be $\alpha$ and $\beta$.

Now, we know that:

$\beta = \dfrac{I_{C}}{I_{B}} \: and \: \alpha = \dfrac{I_{C}}{I_{E}}$

Again, we know that:

$I_{E} = I_{C} + I_{B}$

$\implies \: \dfrac{I_{C}}{ \alpha } = I_{C} + \dfrac{I_{C}}{ \beta }$

$\implies \: \dfrac{1}{ \alpha } = 1 + \dfrac{1}{ \beta }$

$\implies \: \dfrac{1}{ \alpha } = \dfrac{ \beta + 1}{ \beta }$

$\implies \: \alpha = \dfrac{ \beta }{ \beta + 1}$

So, required relationship is:

$\boxed{ \bold{\: \alpha = \dfrac{ \beta }{ \beta + 1} }}$