Resistance (R = V / I)

1. Definition

A. Resistance

Resistance, R (Ω), is a measure of how much a component opposes the flow of current.

B. Resistance equation and Ohm’s law

Resistance at an operating point is defined by:

R = V/I or V = IR

Ohm’s law is the more specific statement that current is directly proportional to potential difference for a metallic conductor when temperature and other physical conditions remain constant. Its resistance is therefore constant and its I–V graph is a straight line through the origin.

2. Key Ideas

• Unit: 1 Ω = 1 V A⁻¹.
• Rearrangements: V = IR, I = V/R.
• For a wire of the same material (and at the same temperature):
  • R propto L (longer wire → larger resistance)
  • R propto 1/A (thicker wire → smaller resistance)
• For a metallic conductor, resistance increases when temperature increases.

3. Detailed Explanations

A. What does resistance do in a circuit?

A larger resistance means it is harder for charges to flow, so for the same voltage you get a smaller current:

I = V/R

B. Factors affecting resistance of a wire (qualitative)

For the same material:

• longer wire → more opposition to current → bigger R
• thicker wire (larger cross-sectional area) → more “paths” for charge → smaller R

C. Temperature effect (metals)

In metals, higher temperature makes the lattice ions vibrate more. Electrons collide more often, so resistance increases.

4. Common Mistakes

• Mixing up the unit: resistance is in Ω, not V or A.
• Calling every use of R = V/I “Ohm’s law”. The equation defines resistance at a point; Ohm’s law requires constant resistance and an Ipropto V relationship.
• Treating “thicker wire” as “bigger diameter → area doubles” (area depends on diameter²).

5. Exam Tips

1. Write the relationship you need: V = IR / R = V/I / I = V/R.
2. Substitute with units (V, A, Ω), then compute and state the unit.
3. For wire comparisons (same material): use R propto L and R propto 1/A.
4. For metals and temperature: “temperature increases → more collisions → R increases”.

6. Worked Examples

Example 1: Finding resistanceCore

A resistor has a p.d. of 6.0 V across it and a current of 0.50 A through it. Find its resistance.

Show Answer

R = V/I = 6.0/0.50 = 12 Ω

Example 2: Finding currentCore

A 12 Ω resistor is connected across a 6.0 V supply. Find the current.

Show Answer

I = V/R = 6.0/12 = 0.50 A

Example 3: Finding voltageCore

A current of 0.20 A flows through a 15 Ω resistor. Find the p.d. across it.

Show Answer

V = IR = 0.20 × 15 = 3.0 V

Example 4: Comparing wires (same material)Core

Wire A and Wire B are made of the same material and have the same cross-sectional area. Wire A is twice as long as Wire B. Compare their resistances.

Show Answer

Since R propto L, doubling the length doubles the resistance. So RA = 2RB.

Example 5: Unit conversion (mA to A)Core

A resistor has a p.d. of 12 V across it and a current of 200 mA through it. Find the resistance.

Show Answer

Convert current: 200 mA = 0.200 A.

R = V/I = 12/0.200 = 60 Ω

7. Mind Stretchers

Mind stretcher 1: Diameter changeExtension

Two wires are the same material and same length. Wire A has twice the diameter of Wire B. Compare their resistances.

Show Answer

Cross-sectional area A propto d², so doubling diameter makes area 4 × bigger. Since R propto 1/A, RA = 1/4RB.

Mind stretcher 2: Stretching a wire (constant volume)Extension

A wire is stretched so its length doubles, but its volume stays constant. Compare the new resistance to the original resistance.

Show Answer

Constant volume means AL is constant, so if L doubles, A halves.

Since R propto L/A, the resistance becomes:

R' propto 2L/A/2 = 4L/A

So the resistance becomes 4× the original.

8. Practice and next step

Complete the resistance and wire-comparison questions in Structured Current Electricity. Then continue to the metallic conductor I–V graph to see how constant resistance appears graphically.