https://file.aiquickdraw.com/mathbot/file/17280589043581u96jhi2.jpg

Let's analyze each of the resistors in the problem to determine if they may have been damaged by overheating.

### Power Dissipation Formula:
The power dissipated by a resistor can be calculated using:
\[
P = \frac{V^2}{R} \quad \text{or} \quad P = I^2 R
\]
Where:
- \( P \) is the power in watts (W),
- \( V \) is the voltage across the resistor in volts (V),
- \( I \) is the current through the resistor in amperes (A),
- \( R \) is the resistance in ohms (Ω).

A resistor will be damaged if the power dissipated exceeds its rated power.

### (a) \( 560 \, \Omega, \, \frac{1}{2} \, \text{W}, \, 75 \, \text{V} \) across it
1. Power dissipated by the resistor:
\[
P = \frac{V^2}{R} = \frac{75^2}{560} = \frac{5625}{560} \approx 10.04 \, \text{W}
\]
2. The rated power is \( \frac{1}{2} \, \text{W} = 0.5 \, \text{W} \).

**Conclusion**: This resistor is dissipating 10.04 W, which far exceeds its rated power of 0.5 W. **It is likely to be damaged.**

---

### (b) \( 3 \, \Omega, \, 20 \, \text{W}, \, 4 \, \text{A} \) through it
1. Power dissipated by the resistor:
\[
P = I^2 R = 4^2 \times 3 = 16 \times 3 = 48 \, \text{W}
\]
2. The rated power is 20 W.

**Conclusion**: This resistor is dissipating 48 W, which exceeds its rated power of 20 W. **It is likely to be damaged.**

---

### (c) \( \frac{1}{4} \, \text{W}, \, 0.25 \, \text{mA} \, \text{through it and} \, 40 \, \text{V} \)
1. Current \( I = 0.25 \, \text{mA} = 0.00025 \, \text{A} \),
2. Voltage across the resistor \( V = 40 \, \text{V} \),
3. Resistance \( R \) can be found using Ohm's law:
\[
R = \frac{V}{I} = \frac{40}{0.00025} = 160,000 \, \Omega \, (\text{or} \, 160 \, \text{k} \Omega)
\]
4. Power dissipated by the resistor:
\[
P = V \times I = 40 \times 0.00025 = 0.01 \, \text{W}
\]
5. The rated power is \( \frac{1}{4} \, \text{W} = 0.25 \, \text{W} \).

**Conclusion**: This resistor is dissipating only 0.01 W, which is well below its rated power of 0.25 W. **It is not likely to be damaged.**

---

### Final Results:
- **(a)** and **(b)** resistors are **likely to be damaged**.
- **(c)** resistor is **not likely to be damaged**.

---

Would you like more details on any part of the explanation?

Here are five additional questions that build on this topic:
1. How does the resistor's physical size affect its power rating?
2. Can you calculate the current through the resistor in part (a)?
3. What would happen if the power rating is exceeded slightly but not drastically?
4. How does temperature impact the performance of resistors?
5. How do wire-wound resistors differ in terms of power handling?

**Tip**: Always ensure that resistors used in high-power applications have a safety margin above their expected power dissipation to avoid overheating issues.

Determine which, if any, of the following resistors may have been damaged by overheating. Justify your answer: 
 a) 560 Ω, ½ W, with 75 V across it. 
 b) 3 Ω, 20 W, with 4 A through it. 
 c) ¼ W, with 0.25 mA through it and 40 V across it.

Analyze which resistors may have been damaged by overheating using power dissipation formulas. Includes calculations of power dissipation using Ohm's law and resistor specifications. Suitable for advanced high school or college-level physics or electrical engineering students.

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