Write down the relationship between $\alpha $ and $\beta $ related to an amplifier.

Hint: In this question use the concept that the current gain in CB configuration that is $\alpha $ can be represented in terms of change in collector current to the change in emitter current, whereas current gain in CE configuration can be represented in terms of the change in collector current and change in base current. This will help approaching the problem.

Complete step-by-step answer:
In a transistor $\alpha $ is known as the current gain in the common base configuration and it is the ratio of the change in collector current to change in the emitter current.
$ \Rightarrow \alpha = \dfrac{{{I_C}}}{{{I_E}}}$.............. (1)
Where, ${I_C}$ = change in the collector current and ${I_E}$= change in the emitter current.
In a transistor $\beta $ is known as the current gain in the common emitter configuration and it is the ratio of the change in collector current to change in the base current.
$ \Rightarrow \beta = \dfrac{{{I_C}}}{{{I_B}}}$................. (2)
Where, ${I_C}$ = change in the collector current and ${I_B}$= change in the base current.
Now as we all know that the emitter current is the sum of the collector current and the base current so we have,
$ \Rightarrow {I_E} = {I_C} + {I_B}$.................... (3)
Now from equation (1), ${I_E} = \dfrac{{{I_C}}}{\alpha }$ so substitute this value in equation (3) we have,
$ \Rightarrow \dfrac{{{I_C}}}{\alpha } = {I_C} + {I_B}$................... (4)
Now from equation (2), ${I_B} = \dfrac{{{I_C}}}{\beta }$ so substitute this value in equation (4) we have,
$ \Rightarrow \dfrac{{{I_C}}}{\alpha } = {I_C} + \dfrac{{{I_C}}}{\beta }$
Now cancel the term ${I_C}$ from both L.H.S and R.H.S we have,
$ \Rightarrow \dfrac{1}{\alpha } = 1 + \dfrac{1}{\beta }$................ (5)
Now simplify this we have,
$ \Rightarrow \dfrac{1}{\alpha } = \dfrac{{\beta + 1}}{\beta }$
$ \Rightarrow \alpha = \dfrac{\beta }{{\beta + 1}}$
So this is the relation of $\alpha $in terms of $\beta $.
Now again from equation (5) we have,
$ \Rightarrow \dfrac{1}{\alpha } - 1 = \dfrac{1}{\beta }$
$ \Rightarrow \dfrac{{1 - \alpha }}{\alpha } = \dfrac{1}{\beta }$
$ \Rightarrow \beta = \dfrac{\alpha }{{1 - \alpha }}$
So this is the relation of the $\beta $ in terms of $\alpha $.
So this is the required relation between $\alpha $ and $\beta $ related to the amplifier.

Note – The basic functioning of an amplifier is to enhance or to amplify the input fed. Amplifier affects the amplitude of the signal fed into the input signal circuit and enhances this amplitude. This enhanced input signal is then being collected from the output end of the circuit via connecting the output circuit to a load.