Distinguish resistance from resistivity
The resistance, R, represents the value of a specific conductor or component, and its unit is Ω. The resistivity, ρ, indicates the property a material possesses at a given temperature, and its unit is Ω·m. Even with the same material, changing the length or diameter of a component will result in a different resistance.
In a uniform cable, the resistance is R = ρL/A. It is proportional to the length, L, and inversely proportional to the cross-sectional area, A. Since the cross-sectional area of a circular cable is πr², if the radius is doubled, then for the same resistance, the resistance becomes ¼. It is important not to confuse the multiples of diameter and cross-sectional area.
How does higher resistance affect current?
When the temperature and other physical conditions are constant, in an ohmic resistor, the resistance is I = V/R. If you double the resistance while maintaining the same voltage, the current is halved. Conversely, if you maintain a constant current, a larger voltage is required as the resistance increases.
The power of a resistor is P = VI = I²R = V²/R. If the current is kept constant, the power increases in a large resistor, but if the voltage is kept constant, the power decreases in a large resistor. We cannot simply say that a large resistor always generates a lot of heat, without specifying the conditions.
Why do temperature and material matter?
Metal resistance generally increases with temperature. Semiconductors and special materials can have different temperature dependence, so the same trend must not be applied to all materials. Read resistivity tables together with their measurement temperature.
The Ohm's Law experiment utilizes an ideal resistor, whose value does not change with temperature. In actual components, such as a light bulb, which heat up, the resistance may not produce a simple proportional graph with voltage and current. Resistance in the connecting wires and measurement devices also contributes to errors in the actual circuit.
Key relationships and symbols
R = ρL/A · I = V/R · P = I²R = V²/R (ohmic resistor)
| Symbol | Meaning | Unit |
|---|---|---|
| R | Electrical resistance | Ω |
| ρ | Resistivity | Ω·m |
| L | Wire length | m |
| A | Wire cross-sectional area | m² |
Worked example
What happens if you double the cross-sectional length and radius of a conductor made of the same material?
Doubling the length and quadrupling the cross-sectional area makes the resistance 2/4 = ½ of its original value. An original resistance of 10 Ω becomes 5 Ω, and the current doubles at the same voltage.
Common misconceptions
- Equal resistivity does not mean that wires of different sizes have equal resistance.
- When the voltage is changed, V/I is not always constant. It is necessary to check whether there is an ohm and the temperature.
Explore this concept through experiments
Compare the current and voltage-current graph slopes by only changing the resistance at the same voltage.
Series and parallel circuitsCheck how the total resistance changes depending on how the resistors are connected.
Wire length, cross-section and resistivityAt a length of 1 m, change only the cross-section or material.