Resistivity — Revision Notes

⚡ 30-Second Revision

Definition: — Intrinsic material property opposing current flow.
Formula: — $R = \rho \frac{L}{A} \implies \rho = \frac{RA}{L}$
Microscopic Formula: — $\rho = \frac{m}{ne^2\tau}$
SI Unit: — Ohm-meter ($\Omega \cdot \text{m}$)
Factors: — Material nature, Temperature, Impurities.
Temperature Effect:

- Metals: $\rho \uparrow$ with $T$ (due to $\tau \downarrow$) - Semiconductors: $\rho \downarrow$ with $T$ (due to $n \uparrow\uparrow$) - Alloys: $\rho$ changes little with $T$ (low $\alpha$)

Stretching/Compressing: — $\rho$ remains constant. If $L' = xL$, then $A' = A/x$, and $R' = x^2 R$ (for constant volume).

2-Minute Revision

Resistivity ($\rho$) is a fundamental property of a material, indicating its inherent ability to resist electric current, independent of its shape or size. It's distinct from resistance ($R$), which is a property of a specific object and depends on its dimensions ($R = \rho \frac{L}{A}$).

The SI unit for resistivity is ohm-meter ($\Omega \cdot \text{m}$). Microscopically, resistivity is inversely proportional to the number density of free electrons ($n$) and their relaxation time ($\tau$), given by $\rho = \frac{m}{ne^2\tau}$.

Temperature significantly affects resistivity. For most metals, resistivity increases with temperature because increased thermal vibrations reduce the relaxation time. However, for semiconductors, resistivity decreases with increasing temperature due to a substantial increase in the number of charge carriers.

Alloys are often used in resistors because their resistivity changes minimally with temperature. When a wire is stretched or compressed, its resistivity remains constant, but its resistance changes due to alterations in length and cross-sectional area (e.

g., if length doubles, resistance quadruples assuming constant volume).

5-Minute Revision

Resistivity ($\rho$) is an intrinsic characteristic of a material, quantifying its opposition to electric current flow. It's a 'per unit dimension' resistance, independent of the conductor's length ($L$) or cross-sectional area ($A$). The relationship $R = \rho \frac{L}{A}$ is key, where $R$ is resistance. This means $\rho = \frac{RA}{L}$, giving its SI unit as ohm-meter ($\Omega \cdot \text{m}$). Conductivity ($\sigma$) is the reciprocal of resistivity, $\sigma = 1/\rho$.

From a microscopic perspective, resistivity is given by $\rho = \frac{m}{ne^2\tau}$, where $m$ is electron mass, $e$ is electron charge, $n$ is the number density of free electrons, and $\tau$ is the relaxation time (average time between collisions). This formula highlights that resistivity depends on the availability of charge carriers ($n$) and how frequently they collide (governed by $\tau$).

Factors Affecting Resistivity:

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Nature of Material: — This is the primary factor, determining $n$ and $\tau$.
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Temperature:

* Metals: As temperature increases, lattice vibrations become more vigorous, reducing $\tau$ (more collisions). Thus, $\rho$ increases. The relation is $\rho_T = \rho_0 [1 + \alpha (T - T_0)]$. * Semiconductors: As temperature increases, more covalent bonds break, significantly increasing $n$.

This effect dominates the decrease in $\tau$, so $\rho$ decreases. * Alloys (e.g., Nichrome): They have high $\rho$ and very low temperature coefficient ($\alpha$), making their resistance stable with temperature, ideal for resistors and heating elements.

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Impurities: — Adding impurities increases scattering, reducing $\tau$ and thus increasing $\rho$.

Common Problem Type: Stretching/Compressing Wires:

If a wire is stretched or compressed, its volume ($V = AL$) remains constant. If the length changes by a factor $x$ (e.g., $L' = xL$), then the area changes by $1/x$ (i.e., $A' = A/x$). The new resistance $R'$ will be $R' = \rho \frac{L'}{A'} = \rho \frac{xL}{A/x} = x^2 \left(\rho \frac{L}{A}\right) = x^2 R$. Crucially, the resistivity ($\rho$) of the material itself remains unchanged during this process.

Example: A wire of resistance $10\,\Omega$ is stretched to three times its original length. Its new resistance will be $R' = (3)^2 \times 10\,\Omega = 90\,\Omega$. Its resistivity remains the same.

Prelims Revision Notes

Resistivity ($\rho$) - NEET Revision Notes

1. Definition & Distinction:

Resistivity ($\rho$): — Intrinsic property of a material, independent of dimensions. It's the resistance offered by a unit cube of the material. Unit: $\Omega \cdot \text{m}$.
Resistance ($R$): — Property of a specific conductor, dependent on material, length ($L$), and cross-sectional area ($A$). Unit: $\Omega$.
Relationship: — $R = \rho \frac{L}{A}$. From this, $\rho = \frac{RA}{L}$.

2. Microscopic Origin:

$\rho = \frac{m}{ne^2\tau}$

* $m$: mass of electron (constant) * $e$: charge of electron (constant) * $n$: number density of free electrons (charge carriers) * $\tau$: relaxation time (average time between collisions)

Higher $n$ or longer $\tau$ leads to lower resistivity.

3. Factors Affecting Resistivity:

Nature of Material: — Different materials have different $n$ and $\tau$.

* Conductors (low $\rho$): High $n$, relatively long $\tau$. * Insulators (high $\rho$): Very low $n$. * Semiconductors (intermediate $\rho$): $n$ can be varied.

Temperature (T):

* Metals: $\rho$ increases with $T$. Increased $T \implies$ increased lattice vibrations $\implies$ decreased $\tau \implies$ increased $\rho$. Formula: $\rho_T = \rho_0 [1 + \alpha (T - T_0)]$. $\alpha$ is positive.

* Semiconductors: $\rho$ decreases with $T$. Increased $T \implies$ more covalent bonds break $\implies$ significantly increased $n$. This effect dominates over decreased $\tau$. $\alpha$ is negative.

* Alloys (e.g., Nichrome, Manganin): High $\rho$, very low $\alpha$. Used for standard resistors as resistance is stable with $T$.

Impurities/Alloying: — Generally increases $\rho$ by introducing scattering centers, reducing $\tau$.

4. Effect of Stretching/Compressing a Wire:

Resistivity ($\rho$): — Remains UNCHANGED (it's a material property).
Resistance ($R$): — Changes significantly.

* If a wire is stretched/compressed, its **volume ($V = AL$) remains constant.** * If new length $L' = xL$, then new area $A' = A/x$. * New resistance $R' = \rho \frac{L'}{A'} = \rho \frac{xL}{A/x} = x^2 \left(\rho \frac{L}{A}\right) = x^2 R$. * Key takeaway: If length becomes $x$ times, resistance becomes $x^2$ times. * If radius becomes $x$ times, area becomes $x^2$ times. If length is constant, $R' = R/x^2$.

5. Conductivity ($\sigma$):

Reciprocal of resistivity: $\sigma = 1/\rho$.
Unit: Siemens per meter ($\text{S/m}$) or $(\Omega \cdot \text{m})^{-1}$.

6. Important Values (Order of Magnitude):

Conductors: $10^{-8} \Omega \cdot \text{m}$ (e.g., Copper: $1.68 \times 10^{-8}$)
Semiconductors: $10^{-5}$ to $10^5 \Omega \cdot \text{m}$ (e.g., Silicon: $2.3 \times 10^3$)
Insulators: $10^{10}$ to $10^{14} \Omega \cdot \text{m}$ (e.g., Glass: $10^{10}$ to $10^{14}$)

Vyyuha Quick Recall

Resistivity Is Not Dependent on Length or Area, Temperature Matters!