For a common emitter configuration, if ' $\alpha$ ' and ' $\beta$ ' have their usual meanings, the incorrect…

For a common emitter configuration, if ' $\alpha$ ' and ' $\beta$ ' have their usual meanings, the incorrect relation between ' $\alpha$ ' and ' $\beta$ ' is
  1. $\frac{1}{\alpha}=\frac{1}{\beta}+1$
  2. $\alpha=\frac{\beta}{1-\beta}$
  3. $\alpha=\frac{\beta}{1+\beta}$
  4. $\frac{1}{\beta}=\frac{1}{\alpha}-1$

Solution

$\beta=\frac{\mathrm{I}_{\mathrm{B}}}{\mathrm{I}_{\mathrm{C}}} \text { and } \alpha=\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}}$ $\begin{aligned} & \therefore \quad \frac{1}{\alpha}=\frac{\mathrm{I}_{\mathrm{B}}+\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{C}}} \quad \Rightarrow \quad \frac{1}{\alpha}=\frac{1}{\beta}+1 \\ & \therefore \quad \frac{1}{\alpha}-1=\frac{1}{\beta} \quad \Rightarrow \quad \alpha=\frac{\beta}{\beta+1} \\ & \end{aligned}$ ~

Asked in: MHT CET 2023 (14 May Shift 1)

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