Problem

The current flowing into the base of a transistor is $100 \, \mu \mathrm{A}$. Find its collector current $\mathrm{I}_{\mathrm{C}}$, its emitter current $\mathrm{I}_{\mathrm{E}}$, and the ratio $\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}}$ if the current gain $\beta$ is 100.

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Data

• Base current: $\mathrm{I}_{\mathrm{B}} = 100 \, \mu \mathrm{A} = 100 \times 10^{-6} \, \mathrm{A}$
• Current gain: $\beta = 100$

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Prerequisite Concepts

1. The current gain $\beta$ is defined as:

$$\beta = \frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{B}}}$$

Rearranging for $\mathrm{I}_{\mathrm{C}}$:

$$\mathrm{I}_{\mathrm{C}} = \beta \cdot \mathrm{I}_{\mathrm{B}}$$

2. The emitter current $\mathrm{I}_{\mathrm{E}}$ is the sum of the base current and the collector current:

$$\mathrm{I}_{\mathrm{E}} = \mathrm{I}_{\mathrm{B}} + \mathrm{I}_{\mathrm{C}}$$

3. The ratio of collector current to emitter current is:

$$\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}}$$

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Solution

Step 1: Calculate the Collector Current $\mathrm{I}_{\mathrm{C}}$

Using the formula:

$$\mathrm{I}_{\mathrm{C}} = \beta \cdot \mathrm{I}_{\mathrm{B}}$$

Substitute the values:

$$\mathrm{I}_{\mathrm{C}} = 100 \cdot 100 \times 10^{-6} \, \mathrm{A}$$ $$\mathrm{I}_{\mathrm{C}} = 10 \times 10^{-3} \, \mathrm{A} = 10 \, \mathrm{mA}$$

Step 2: Calculate the Emitter Current $\mathrm{I}_{\mathrm{E}}$

Using the formula:

$$\mathrm{I}_{\mathrm{E}} = \mathrm{I}_{\mathrm{B}} + \mathrm{I}_{\mathrm{C}}$$

Substitute the values:

$$\mathrm{I}_{\mathrm{E}} = 100 \times 10^{-6} \, \mathrm{A} + 10 \times 10^{-3} \, \mathrm{A}$$ $$\mathrm{I}_{\mathrm{E}} = 10.1 \times 10^{-3} \, \mathrm{A} = 10.1 \, \mathrm{mA}$$

Step 3: Calculate the Ratio $\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}}$

Using the formula:

$$\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}} = \frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}}$$

Substitute the values:

$$\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}} = \frac{10}{10.1}$$ $$\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}} \approx 0.99$$

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Answer

1. Collector current ($\mathrm{I}_{\mathrm{C}}$): $10 \, \mathrm{mA}$
2. Emitter current ($\mathrm{I}_{\mathrm{E}}$): $10.1 \, \mathrm{mA}$
3. Ratio ($\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{E}}}$): $0.99$