NDA Physics · Electricity and Magnetism

Resistance and Resistivity

Resistance opposes current and depends on the wire's material AND shape (R = ρL/A); resistivity is the material's intrinsic opposition, independent of size — so stretching or cutting a wire changes R but never ρ.

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Why this matters

Six PYQs, and the launch pad for the chapter's hardest networks. The recurring tests are: which quantities affect R (length, area, material, temperature — never the current), the difference between resistance and resistivity, the SI unit Ω·m, and the geometry tricks — stretching a wire (R ∝ L²) and cutting it into equal pieces.

Concept 1 of 3: Resistance and what controls it

A long, thin wire of a poorly conducting material resists current most. Resistance grows with LENGTH (more obstacles to push charge through), falls with cross-section AREA (a wider pipe), and depends on the MATERIAL through its resistivity. It does NOT depend on the current you push through it.

Definition

Resistance $R = \rho L / A$ measures opposition to current (unit: ohm, Ω). It depends on:

Length $L$ — directly (R ∝ L).
Cross-sectional area $A$ — inversely (R ∝ 1/A).
Material — through the resistivity $\rho$ .
Temperature — for metals, R rises with temperature.

It does not depend on the current or voltage (for an ohmic conductor).

Resistance of a uniform wire

R = \rho\,\dfrac{L}{A}

Rresistance (Ω)
$\rho$ resistivity of the material (Ω·m)
Llength of the wire (m)
Across-sectional area (m²)

R = ρL/A — resistance grows with length and falls with cross-sectional area. Stretching keeps volume fixed: longer ⟹ thinner ⟹ R ∝ L².

Worked example

Two wires are made of the same material. Wire B is twice as long as wire A but has the same thickness. How do their resistances compare?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Apr 2017 · Q110Easy

Example 1 · Electricity and Magnetism · Resistance and Resistivity

Which one of the following physical quantities does NOT affect the resistance of a cylindrical resistor ?

Current does not affect resistance

R is a property of the conductor (material + geometry), set before any current flows. The trap option 'the current through it' is exactly what does NOT change R for an ohmic resistor.

Drill 3 more on resistance and what controls it

Concept 2 of 3: Resistivity — the material's own property

Resistivity is the resistance built into the material itself, independent of how you cut or stretch it. Copper has a low resistivity (good conductor); nichrome a high one (good heater). Change the shape all you like — the resistivity stays the same; only the resistance changes.

Definition

Resistivity $\rho$ is an intrinsic property of the material: it depends on the material and its temperature, but NOT on the length, area, or shape of a particular sample. Its SI unit is the ohm-metre (Ω·m) (from $\rho = RA/L$ ). Two wires of the same material at the same temperature have the same $\rho$ even if their resistances differ wildly.

Worked example

A copper wire is cut into two unequal pieces. How do the resistivities of the two pieces compare with each other and with the original?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Apr 2025 · Q138Easy

Example 2 · Electricity and Magnetism · Resistance and Resistivity

If the length of a copper wire is increased by twice, then its resistivity will be

Stretching changes R, not ρ

A wire stretched longer has more resistance, but its resistivity is unchanged — same material, same temperature. The distractors 'doubled/halved' tempt you to treat ρ like R. ρ is intrinsic; it doesn't care about shape.

Drill 2 more on resistivity — the material's own property

Concept 3 of 3: Stretching and cutting a wire

When you stretch a wire its volume stays the same: it gets longer AND thinner together. Length up by a factor k means area down by k, so R = ρL/A jumps by k². Cutting a wire into n equal pieces gives each piece 1/n of the original resistance.

Definition

Stretching (volume $V = LA$ constant): if length becomes $k$ times, area becomes $1/k$ times, so resistance scales as $k^2$ — $R \propto L^2$ (equivalently $R \propto 1/A^2$ ). Doubling the length quadruples R. Cutting into $n$ equal pieces: each piece has length $L/n$ , same area, so each has resistance $R/n$ .

Stretched wire (constant volume)

R' = k^2 R \quad\text{when length}\to kL,\ \text{area}\to A/k

kfactor by which the length increases
R'new resistance after stretching
Roriginal resistance

Worked example

A wire of resistance 5 Ω is stretched until it is three times its original length (volume constant). What is its new resistance?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Apr 2023 · Q127Moderate

Example 3 · Electricity and Magnetism · Resistance and Resistivity

In an electric circuit, a wire of resistance 10

\Omega

is used. If this wire is stretched to a length double of its original value, the current in the circuit would become :

Stretching is R ∝ L², not R ∝ L

Forgetting that the wire also gets THINNER is the classic error. Volume is fixed, so doubling length halves area, and R = ρL/A picks up BOTH factors: ×2 from length and ×2 from area = ×4 overall. Use R ∝ L².

Drill 1 more on stretching and cutting a wire

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Resistance and what controls it
Resistance of a uniform wire
$R = \rho\,\dfrac{L}{A}$
Stretching and cutting a wire
Stretched wire (constant volume)
$R' = k^2 R \quad\text{when length}\to kL,\ \text{area}\to A/k$

Watch out for (3)

Current does not affect resistance
→ Resistance and what controls it
Stretching changes R, not ρ
→ Resistivity — the material's own property
Stretching is R ∝ L², not R ∝ L
→ Stretching and cutting a wire

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