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

Intuition

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)
\rhoresistivity 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?

Same material ⟹ same $\rho$ ; same thickness ⟹ same $A$ .
$R = \rho L/A$ , so R ∝ L when $\rho$ and $A$ are fixed.
Doubling the length doubles the resistance.

Answer:Wire B has twice the resistance of wire A.

Practice this conceptself-check · 3 quick reps

Try it yourself

A cylindrical resistor's resistance is quoted. Which of these would change it: (i) the current through it, (ii) its length, (iii) its cross-sectional area, (iv) the material?

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
R is directly proportional to which dimension of a wire?
2.
Does the current flowing through a resistor change its resistance?
3.
Doubling a wire's cross-sectional area does what to R?

From the bank · past-year question

Example 1Electricity and MagnetismEASY

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

[Q110 · Apr · 2017]

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 1 more on resistance and what controls it

Concept 2 of 3

Resistivity — the material's own property

Intuition

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?

Resistivity is set by the MATERIAL (and temperature), not the size.
Both pieces are still copper at the same temperature.
So both pieces — and the original — have identical resistivity (only their resistances differ).

Answer:All three have the same resistivity; cutting changes resistance, not resistivity.

Practice this conceptself-check · 3 quick reps

Try it yourself

Which of these statements are correct? (1) Both resistance and resistivity depend on the area of cross-section. (2) Both depend on temperature. (3) Resistance is directly proportional to resistivity. (4) Resistivity is directly proportional to length.

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
SI unit of resistivity?
2.
If a wire's length is doubled, its resistivity becomes…
3.
Resistivity depends on which two things?

From the bank · past-year question

Example 2Electricity and MagnetismEASY

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

[Q138 · Apr · 2025]

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 1 more on resistivity — the material's own property

Concept 3 of 3

Stretching and cutting a wire

Intuition

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?

Stretching keeps volume constant: length ×3 ⟹ area ×1/3.
$R \propto L^2$ , so new R = $3^2 \times 5 = 9 \times 5$ .
New R = 45 Ω.

Answer:45 Ω.

Practice this conceptself-check · 3 quick reps

Try it yourself

A 10 Ω wire is stretched to double its length. If it then stays in the same circuit at the same voltage, what happens to the current through it?

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
A wire stretched to double its length has its resistance multiplied by…
2.
A 50 Ω wire cut into 5 equal pieces — resistance of each piece?
3.
Stretching a wire to 3× its length multiplies R by…

From the bank · past-year question

Example 3Electricity and MagnetismMODERATE

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 :

[Q127 · Apr · 2023]

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

Mastery check — 3 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Electricity and MagnetismHARD

Which of the following statements are correct about the electrical resistance and resistivity of a wire? 1. Both quantities depend on the area of cross-section of the wire 2. Both depend on the temperature 3. Resistance of the wire is directly proportional to the resistivity of the wire 4. Resistivity of the wire is directly proportional to the length of the wire Select the correct answer using the code given below:

[Q86 · Sep · 2023]

Example 2Electricity and MagnetismEASY

Which one of the following correctly represents the SI unit of resistivity?

[Q65 · Apr · 2022]

Example 3Electricity and MagnetismMODERATE

Let us consider a copper wire having radius

r

and length

l

. Let its resistance be

R

. If the radius of another copper wire is

2r

and the length is

l/2

then the resistance of this wire will be

[Q80 · Apr · 2019]

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