Question

The correct relation between $\alpha$ and $\beta$ in a transistor is?
A. $$\beta ={\displaystyle \frac{\alpha }{1-\alpha }}$$
B. $$\beta ={\displaystyle \frac{\alpha }{1+\alpha }}$$
C. $$\beta ={\displaystyle \frac{1+\alpha }{\alpha }}$$
D. $$\beta =1-\alpha$$

Answer 1

Hint: A transistor has three kinds of currents flowing through it which are collector current, base current and emitter current. The alpha and beta values are the ratio of the two of these as given below. We can solve this question by writing the values of and in a transistor separately and then comparing them. The values of both of them is given below:
$$\alpha ={\displaystyle \frac{\mathrm{\Delta }{i}_C}{\mathrm{\Delta }{i}_E}}$$
$$\beta ={\displaystyle \frac{\mathrm{\Delta }{i}_C}{\mathrm{\Delta }{i}_B}}$$

Complete step-by-step answer:
Alpha of a transistor is defined as the current gain in the common base configuration which is in turn defined as the ratio of change in the collector current to change in the emitter current. It can possess a maximum value of one. Also, beta is the current gain in Common Emitter configuration. It can be defined as the ratio of the change in collector current to the change in base current. $$\mathrm{\Delta }{i}_C$$is the collector current. It can possess any value.
$$\mathrm{\Delta }{i}_E$$ is the emitter current
$$\mathrm{\Delta }{i}_B$$ is the base current
Now,
$$\alpha ={\displaystyle \frac{\mathrm{\Delta }{i}_C}{\mathrm{\Delta }{i}_E}}$$
$$\beta ={\displaystyle \frac{\mathrm{\Delta }{i}_C}{\mathrm{\Delta }{i}_B}}$$$$={\displaystyle \frac{\mathrm{\Delta }{i}_C}{\mathrm{\Delta }{i}_E}}\times {\displaystyle \frac{\mathrm{\Delta }{i}_E}{\mathrm{\Delta }{i}_B}}$$
$$=\alpha \times {\displaystyle \frac{\mathrm{\Delta }{i}_E}{\mathrm{\Delta }{i}_B}}$$…(1)
$$\mathrm{\Delta }{i}_B=\mathrm{\Delta }{i}_E-\mathrm{\Delta }{i}_C$$…(2)
Substituting the equation (2) in equation (1) we get,
$$\beta =\alpha \times {\displaystyle \frac{\mathrm{\Delta }{i}_E}{\mathrm{\Delta }{i}_E-\mathrm{\Delta }{i}_C}}$$
$$\beta =\alpha \times {\displaystyle \frac{1}{1-{\displaystyle \frac{\mathrm{\Delta }{i}_C}{\mathrm{\Delta }{i}_E}}}}$$
Now, we know $$\alpha ={\displaystyle \frac{\mathrm{\Delta }{i}_C}{\mathrm{\Delta }{i}_E}}$$ therefore we get,
$$\beta ={\displaystyle \frac{\alpha }{1-\alpha }}$$

So, the correct answer is “Option A”.

Note: A transistor is defined as a semiconductor device which is used to amplify or switch electronic signals and electrical power. It is an electronic device that works by controlling the flow of the electrical current through the instrument. It is composed of semiconductor material which usually has at least three terminals for connection to an external circuit. We generally use transistors in computers and other technological devices.