Resistivity as a Material Property
For a uniform conductor of length $l$ and uniform cross-sectional area $A$, experiment shows that the resistance is directly proportional to $l$ and inversely proportional to $A$. NCERT writes this as
$$ R = \rho\,\frac{l}{A} $$
The constant of proportionality $\rho$ — the resistivity — depends on the material of the conductor but not on its dimensions. This is the heart of the topic. Resistance $R$ is a property of a particular specimen; resistivity $\rho$ is a property of the substance. A thin copper hair and a thick copper bar have very different resistances, yet the same resistivity, because $\rho$ has been factored clear of $l$ and $A$.
Rearranging gives $\rho = RA/l$, from which the SI unit follows directly: $\Omega \cdot \text{m}^2 / \text{m} = \Omega \cdot \text{m}$ (ohm-metre). NIOS frames the same idea operationally — if $l = 1\,\text{m}$ and $A = 1\,\text{m}^2$, then numerically $\rho = R$, so resistivity is the resistance of a unit cube of the material measured between opposite faces.
| Quantity | Symbol | Defining relation | SI unit | Depends on dimensions? |
|---|---|---|---|---|
| Resistance | $R$ | R = ρl/A | ohm (Ω) | Yes — changes with $l$, $A$ |
| Resistivity | $\rho$ | ρ = RA/l | ohm-metre (Ω·m) | No — material property |
| Conductivity | $\sigma$ | σ = 1/ρ | Ω⁻¹·m⁻¹ (S·m⁻¹) | No — material property |
Conductivity and the Microscopic Origin
The reciprocal of resistivity is the conductivity, $\sigma = 1/\rho$, measured in $\Omega^{-1}\text{m}^{-1}$ or $\text{S}\,\text{m}^{-1}$. In terms of the local electric field $\vec{E}$ inside the conductor and the current density $\vec{j}$ (current per unit normal area), Ohm's law takes the compact microscopic form
$$ \vec{E} = \rho\,\vec{j} \qquad\Longleftrightarrow\qquad \vec{j} = \sigma\,\vec{E} $$
This is the field-and-density version of $V = IR$, and it makes the dimension-independence explicit: $\vec{E}$ and $\vec{j}$ are local quantities, so the proportionality between them must also be local — a property of the material at that point, not of the sample's size.
The free-electron model of conduction pins $\rho$ to the material's microscopic constants. NCERT derives
$$ \rho = \frac{m}{n e^{2} \tau} $$
where $m$ is the electron mass, $n$ the number of free electrons per unit volume, $e$ the electronic charge and $\tau$ the average time between collisions (the relaxation time). Resistivity therefore depends inversely on both the carrier density $n$ and the relaxation time $\tau$ — both intrinsic to the substance. This relation is also the bridge to the next topic: temperature changes $\tau$ (and, in semiconductors, $n$), which is exactly why resistivity varies with temperature.
The $\rho = m/ne^{2}\tau$ link drives how $\rho$ changes with heat. See Temperature Dependence of Resistivity for the full picture.
The Range of Resistivities
What makes resistivity so useful as a classifier is its staggering range. Metals sit at the low end, around $10^{-8}\,\Omega\text{m}$; ordinary insulators sit near $10^{16}\,\Omega\text{m}$ and beyond — NCERT notes insulators have resistivities about $10^{18}$ times greater than metals, or more. Semiconductors lie in between. Because the values span more than twenty orders of magnitude, they are only sensibly compared on a logarithmic scale.
Resistivity on a logarithmic axis: conductor → semiconductor → insulator.
Classification by Resistivity
Materials are classified, in increasing order of resistivity, as conductors, semiconductors and insulators. The table below collects representative orders of magnitude in line with NCERT's discussion. The defining behavioural contrast — which NEET tests as often as the values themselves — is the way $\rho$ responds to temperature: conductors increase, semiconductors decrease.
| Class | Examples | Resistivity ρ (Ω·m), order of magnitude | Behaviour with rising temperature |
|---|---|---|---|
| Conductors (metals) | Silver, copper, aluminium | $10^{-8}$ to $10^{-6}$ | $\rho$ increases |
| Alloys | Manganin, constantan, nichrome | $\sim 10^{-6}$ (high, for a metal) | $\rho$ almost unchanged (very small $\alpha$) |
| Semiconductors | Silicon, germanium | Intermediate (≈ $10^{-1}$ to $10^{3}$) | $\rho$ decreases |
| Insulators | Glass, ceramic, rubber, plastics | $\sim 10^{16}$ and above | $\rho$ decreases (weakly) |
Two structural facts from $\rho = m/ne^{2}\tau$ explain the table. In a metal, $n$ is essentially fixed; heating shortens $\tau$, so $\rho$ rises. In a semiconductor, heating frees many more carriers, raising $n$ so steeply that it overwhelms the fall in $\tau$, so $\rho$ drops. NCERT also notes that the resistivity of semiconductors can be lowered by adding small amounts of suitable impurities — the doping that underlies all electronic devices.
"Stretch the wire — does the resistivity change?"
When a wire is melted and redrawn or stretched to $n$ times its length, its resistance changes (it scales as $l^{2}$ at constant volume, giving $R' = n^{2}R$). Its resistivity does not change at all — the material is the same. Examiners hide the trap by asking for "new resistivity" in a stretching problem.
$\rho$ is a material property: independent of length, area and shape. Only $R$ is geometry-dependent.
Alloys: High ρ, Low α
A small family of alloys is singled out by NCERT and NIOS because their behaviour is engineered. Nichrome (an alloy of nickel, iron and chromium), manganin and constantan all have a high resistivity and a very weak dependence of resistivity on temperature — a small temperature coefficient of resistivity $\alpha$. Because their resistance values barely change as they warm, they are widely used in wire-bound standard resistors, where stability matters more than a low value.
The same high resistivity, paired with a high melting point, makes nichrome the standard choice for the heating elements of toasters and heaters: the wire dissipates large power as heat without melting or drifting in value. This is why a heating coil and a precision resistor can be made from the same class of material for opposite-looking reasons.
"High resistivity must mean a large temperature effect"
Resistivity magnitude and temperature coefficient are independent properties. Alloys deliberately combine high $\rho$ with low $\alpha$. A statement claiming nichrome's resistance "rises sharply with temperature" is false — that is precisely what it is designed to avoid.
Alloys for standard resistors: high $\rho$, low $\alpha$ → stable resistance.
Colour Code of Carbon Resistors
Carbon resistors are too small to print numbers on, so their values are marked with coloured bands. On a four-band resistor, NIOS gives the reading as
$$ R = AB \times 10^{C}\ \Omega,\quad \text{tolerance } D $$
The first two bands ($A$, $B$) are the first two significant digits; the third band ($C$) is the power-of-ten multiplier; the fourth band is the tolerance — gold for $\pm5\%$, silver for $\pm10\%$, and no fourth band for $\pm20\%$. Each digit colour follows the standard sequence below.
| Colour | Digit | Multiplier | Colour | Digit | Multiplier |
|---|---|---|---|---|---|
| Black | 0 | $10^{0}$ | Green | 5 | $10^{5}$ |
| Brown | 1 | $10^{1}$ | Blue | 6 | $10^{6}$ |
| Red | 2 | $10^{2}$ | Violet | 7 | $10^{7}$ |
| Orange | 3 | $10^{3}$ | Grey | 8 | $10^{8}$ |
| Yellow | 4 | $10^{4}$ | White | 9 | $10^{9}$ |
A carbon resistor reading Yellow–Violet–Brown–Gold = $47 \times 10^{1}\,\Omega = 470\,\Omega$, $\pm5\%$.
Resistivity of Materials in one glance
- $R = \rho l/A$, so $\rho = RA/l$; resistivity is a material property, independent of $l$, $A$ and shape. SI unit: $\Omega\cdot\text{m}$.
- Conductivity $\sigma = 1/\rho$ (unit $\text{S}\,\text{m}^{-1}$); microscopic Ohm's law $\vec{j} = \sigma\vec{E}$.
- Microscopic origin: $\rho = m/ne^{2}\tau$ — inversely on carrier density $n$ and relaxation time $\tau$.
- Range: metals $10^{-8}$–$10^{-6}\,\Omega\text{m}$; insulators $\sim10^{16}\,\Omega\text{m}$ and above; semiconductors in between (and falling with temperature).
- Manganin, constantan, nichrome: high $\rho$, low $\alpha$ → standard resistors and heating elements.
- Four-band code: $R = AB\times10^{C}\,\Omega$; gold $\pm5\%$, silver $\pm10\%$, none $\pm20\%$.