Power dissipated in a resistor
For a passive resistor, Ohm's law and P=VI give three equivalent power formulas.
- Derive P=I²R and P=V²/R from P=VI.
- Interpret resistor power as absorption.
- Check units and safe power margins conceptually.
- Separate resistance value from power rating.
A 100 Ω resistor and a 1 kΩ resistor are each connected to 5 V. Which dissipates more power? Now ask the same question if they instead carry the same current. The answer reverses.
The apparent paradox disappears once we keep track of which quantity is held fixed.
03.1Derive the resistor power formulas
Under receiver convention, a passive resistor obeys P=VI and V=RI. Substitution gives
These are not three independent laws. They are the same power relation expressed using different known variables.
03.2Fixed voltage and fixed current are different experiments
At fixed V,
so lowering R increases power. At fixed I,
so increasing R increases power. There is no contradiction because the external constraint changed.
If instead both carry 10 mA, the powers are 100 mW and 10 mW respectively.
03.3Electrical power becomes heat in the resistive model
The ideal resistor has no energy-storage state in this elementary model. The electrical power it absorbs is dissipated into the material/environment, ultimately appearing as heat.
That is why power is not merely an accounting number: excessive dissipation raises temperature and can change resistance or damage the component.
03.4Resistance value and component ratings are separate specifications
A 1 kΩ resistor may be sold in many physical sizes and technologies with different continuous-power ratings, maximum working voltages, pulse limits, tolerances and temperature coefficients. Knowing R alone is not enough to select a real part.
A safe design therefore computes the operating electrical stress and compares it with the manufacturer's ratings under the actual temperature and mounting conditions.
03.5Worked design checks
Example 1. A 470 Ω resistor carries 20 mA:
That number is the electrical dissipation to compare with the component's derated power capability.
Example 2. A 1 kΩ resistor has 12 V across it:
Independent check: I=12 mA, so VI=12 V×12 mA=144 mW as well.
03.6Practice for mastery
- 3.3 kΩ carries 4 mA. Find P two different ways.
- 10 V is applied to 220 Ω. Find current and power.
- At fixed 5 V, which dissipates more: 1 kΩ or 10 kΩ, and by what factor?
- At fixed 2 mA, which dissipates more: 1 kΩ or 10 kΩ, and by what factor?
- Explain why a resistor marked “1 kΩ” is not fully specified for a real design.
Solutions
1. P=I²R=(4 mA)²·3.3 kΩ=52.8 mW; V=13.2 V and VI gives the same result.
2. I≈45.45 mA; P≈0.455 W.
3. 1 kΩ dissipates 10× more at fixed voltage.
4. 10 kΩ dissipates 10× more at fixed current.
5. Tolerance, power rating, voltage rating, temperature coefficient, technology and environmental limits also matter.