Resistance and Resistivity
Factors On Which The Resistance Of A Conductor Depends ($ R = \rho L/A $)
As we learned from Ohm's Law, resistance ($R$) is the property of a conductor that opposes the flow of electric current. The magnitude of this opposition is not arbitrary; it depends on specific characteristics of the conductor. The resistance of a uniform conductor is found to depend on the following factors:
- Length of the conductor ($L$)
- Cross-sectional area of the conductor ($A$)
- Nature of the material
- Temperature of the conductor
Relationship with Length and Area
Experimentally, it is observed that:
- The resistance ($R$) of a uniform conductor is directly proportional to its length ($L$). This makes intuitive sense: a longer path means the charge carriers will encounter more obstructions (collisions) along the way.
- The resistance ($R$) of a uniform conductor is inversely proportional to its cross-sectional area ($A$). A larger area means more space for charge carriers to flow, effectively providing more parallel paths for the current, thus reducing the overall opposition.
$ R \propto L $ (when $A$, material, and temperature are constant)
$ R \propto \frac{1}{A} $ (when $L$, material, and temperature are constant)
Combining these two proportionalities, we get:
$ R \propto \frac{L}{A} $
To convert this proportionality into an equation, we introduce a constant of proportionality that depends on the nature of the material and its temperature. This constant is called the resistivity ($\rho$) of the material.
$ R = \rho \frac{L}{A} $
This fundamental formula relates the resistance of a conductor to its dimensions and the properties of its material.
Resistivity ($\rho$)
The constant $\rho$ is known as the electrical resistivity (or specific resistance) of the material of the conductor. It is an intrinsic property of the material itself, unlike resistance, which depends on the dimensions.
From the formula $R = \rho L/A$, we can define resistivity as:
$ \rho = R \frac{A}{L} $
If we consider a conductor of unit length ($L=1$ meter) and unit cross-sectional area ($A=1$ square meter), its resistance is equal to its resistivity ($\rho = R \times 1/1 = R$).
Thus, resistivity is numerically defined as the resistance offered by a wire of the material of unit length and unit cross-sectional area.
Units of Resistivity
The SI unit of resistance is Ohm ($\Omega$), length is meter (m), and area is square meter ($m^2$). Using the definition $\rho = R A/L$, the unit of resistivity is:
$ \text{Unit of } \rho = \frac{\Omega \times m^2}{m} = \Omega \cdot m $
The SI unit of resistivity is ohm-meter ($\Omega \cdot m$).
Dependence on Material and Temperature
The value of resistivity is characteristic of the material. Different materials have vast differences in their resistivities. Materials are broadly classified into conductors, semiconductors, and insulators based on their resistivity values (discussed in detail in Section I4).
The resistivity (and hence resistance) also depends significantly on the temperature of the material. For most metals, resistivity increases with increasing temperature (discussed in detail in Section I5). For semiconductors, resistivity generally decreases with increasing temperature.
Example 1. A metal wire of length 50 cm and radius 0.5 mm has a resistance of 2.5 $\Omega$. Calculate the resistivity of the metal.
Answer:
Given:
Length, $L = 50 \, cm = 0.50 \, m$
Radius, $r = 0.5 \, mm = 0.5 \times 10^{-3} \, m$
Resistance, $R = 2.5 \, \Omega$
First, calculate the cross-sectional area $A$. Since it is a wire, the cross-section is circular:
$ A = \pi r^2 = \pi (0.5 \times 10^{-3} \, m)^2 = \pi (0.25 \times 10^{-6}) \, m^2 $
$ A \approx 3.14159 \times 0.25 \times 10^{-6} \, m^2 \approx 0.785 \times 10^{-6} \, m^2 $
Using the formula $R = \rho \frac{L}{A}$, we rearrange to find resistivity $\rho = R \frac{A}{L}$.
$ \rho = (2.5 \, \Omega) \times \frac{(0.785 \times 10^{-6} \, m^2)}{(0.50 \, m)} $
$ \rho = 2.5 \times \frac{0.785}{0.50} \times 10^{-6} \, \Omega \cdot m $
$ \rho = 2.5 \times 1.57 \times 10^{-6} \, \Omega \cdot m $
$ \rho \approx 3.925 \times 10^{-6} \, \Omega \cdot m $
The resistivity of the metal is approximately $3.925 \times 10^{-6} \, \Omega \cdot m$.
Resistance Of A System Of Resistors
In many electric circuits, multiple resistors are connected together. To simplify the analysis of such circuits, it is often useful to calculate the equivalent resistance of the combination. The equivalent resistance is a single resistance that would replace the entire combination without changing the total current or potential difference in the rest of the circuit. Resistors can be combined in various ways, but the two basic configurations are series and parallel.
Resistors In Series ($ R_{eq} = \sum R_i $)
When two or more resistors are connected end-to-end such that the same current flows through each resistor, they are said to be connected in series.
Three resistors $R_1$, $R_2$, and $R_3$ connected in series.
Consider $n$ resistors with resistances $R_1, R_2, R_3, ..., R_n$ connected in series across a potential difference $V$.
Key characteristics of series connection:
- The same current ($I$) flows through each resistor in the series combination.
- The total potential difference ($V$) across the combination is the sum of the potential differences across each individual resistor. Let $V_1, V_2, V_3, ..., V_n$ be the potential differences across $R_1, R_2, R_3, ..., R_n$ respectively. Then, $V = V_1 + V_2 + V_3 + ... + V_n$.
Derivation of Equivalent Resistance in Series
According to Ohm's Law, the potential difference across each resistor is given by:
$ V_1 = I R_1 $
$ V_2 = I R_2 $
...
$ V_n = I R_n $
The total potential difference is $V = V_1 + V_2 + ... + V_n$. Substituting the expressions for $V_i$:
$ V = I R_1 + I R_2 + ... + I R_n $
$ V = I (R_1 + R_2 + ... + R_n) $
Let $R_{eq}$ be the equivalent resistance of the series combination. If this equivalent resistance were connected across the same potential difference $V$, the same total current $I$ would flow through it. So, according to Ohm's Law for the equivalent resistance:
$ V = I R_{eq} $
Comparing the two expressions for $V$:
$ I R_{eq} = I (R_1 + R_2 + ... + R_n) $
Since $I$ is the same and non-zero, we can cancel $I$ from both sides:
$ R_{eq} = R_1 + R_2 + ... + R_n $
$ R_{eq} = \sum_{i=1}^{n} R_i $
Thus, the equivalent resistance of resistors connected in series is equal to the sum of their individual resistances. The equivalent resistance in series is always greater than the largest individual resistance in the combination.
Resistors In Parallel ($ \frac{1}{R_{eq}} = \sum \frac{1}{R_i} $)
When two or more resistors are connected across the same two points, they are said to be connected in parallel.
Three resistors $R_1$, $R_2$, and $R_3$ connected in parallel.
Consider $n$ resistors with resistances $R_1, R_2, R_3, ..., R_n$ connected in parallel across a potential difference $V$.
Key characteristics of parallel connection:
- The same potential difference ($V$) exists across each resistor in the parallel combination. This potential difference is equal to the potential difference of the source.
- The total current ($I$) from the source is the sum of the currents flowing through each individual resistor. Let $I_1, I_2, I_3, ..., I_n$ be the currents flowing through $R_1, R_2, R_3, ..., R_n$ respectively. Then, $I = I_1 + I_2 + I_3 + ... + I_n$. This is a consequence of Kirchhoff's Junction Rule (or conservation of charge).
Derivation of Equivalent Resistance in Parallel
According to Ohm's Law, the current through each resistor is given by:
$ I_1 = \frac{V}{R_1} $
$ I_2 = \frac{V}{R_2} $
...
$ I_n = \frac{V}{R_n} $
The total current is $I = I_1 + I_2 + ... + I_n$. Substituting the expressions for $I_i$:
$ I = \frac{V}{R_1} + \frac{V}{R_2} + ... + \frac{V}{R_n} $
$ I = V \left( \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \right) $
Let $R_{eq}$ be the equivalent resistance of the parallel combination. If this equivalent resistance were connected across the same potential difference $V$, the same total current $I$ would flow from the source. So, according to Ohm's Law for the equivalent resistance:
$ I = \frac{V}{R_{eq}} $
Comparing the two expressions for $I$:
$ \frac{V}{R_{eq}} = V \left( \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \right) $
Since $V$ is the same and non-zero, we can cancel $V$ from both sides:
$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} $
$ \frac{1}{R_{eq}} = \sum_{i=1}^{n} \frac{1}{R_i} $
Thus, the reciprocal of the equivalent resistance of resistors connected in parallel is equal to the sum of the reciprocals of their individual resistances. The equivalent resistance in parallel is always smaller than the smallest individual resistance in the combination. This is because adding more resistors in parallel provides more paths for the current, reducing the overall opposition.
Special Case: Two Resistors in Parallel
For two resistors $R_1$ and $R_2$ in parallel, the formula simplifies to:
$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{R_2 + R_1}{R_1 R_2} $
Taking the reciprocal of both sides:
$ R_{eq} = \frac{R_1 R_2}{R_1 + R_2} $ (Product over Sum)
Example 2. Three resistors of resistances 4 $\Omega$, 6 $\Omega$, and 12 $\Omega$ are connected (a) in series and (b) in parallel. Calculate the equivalent resistance in each case.
Answer:
Given resistances: $R_1 = 4 \, \Omega$, $R_2 = 6 \, \Omega$, $R_3 = 12 \, \Omega$.
(a) When connected in series, the equivalent resistance $R_{eq, series}$ is the sum of individual resistances:
$ R_{eq, series} = R_1 + R_2 + R_3 $
$ R_{eq, series} = 4 \, \Omega + 6 \, \Omega + 12 \, \Omega $
$ R_{eq, series} = 22 \, \Omega $
The equivalent resistance in series is 22 $\Omega$. Note that it is greater than the largest individual resistance (12 $\Omega$).
(b) When connected in parallel, the reciprocal of the equivalent resistance $R_{eq, parallel}$ is the sum of the reciprocals of individual resistances:
$ \frac{1}{R_{eq, parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} $
$ \frac{1}{R_{eq, parallel}} = \frac{1}{4 \, \Omega} + \frac{1}{6 \, \Omega} + \frac{1}{12 \, \Omega} $
Find a common denominator, which is 12:
$ \frac{1}{R_{eq, parallel}} = \frac{3}{12 \, \Omega} + \frac{2}{12 \, \Omega} + \frac{1}{12 \, \Omega} $
$ \frac{1}{R_{eq, parallel}} = \frac{3 + 2 + 1}{12 \, \Omega} = \frac{6}{12 \, \Omega} = \frac{1}{2 \, \Omega} $
Taking the reciprocal:
$ R_{eq, parallel} = 2 \, \Omega $
The equivalent resistance in parallel is 2 $\Omega$. Note that it is smaller than the smallest individual resistance (4 $\Omega$).
Drift Of Electrons And The Origin Of Resistivity
We've seen that resistance is an opposition to current flow, and resistivity is an intrinsic material property. Understanding the origin of resistivity requires looking at the microscopic behaviour of charge carriers within the material, specifically the drift of electrons in metals.
Recap: Free Electron Model
In metals, the conduction electrons are essentially free and move randomly within the crystal lattice formed by positive ions. Their thermal velocities are high, but their random nature means there is no net directional motion in the absence of an electric field.
When an external electric field $\vec{E}$ is applied, it exerts a force $\vec{F} = -e\vec{E}$ on each electron. This force tends to accelerate the electrons in the direction opposite to the field. However, the electrons continuously collide with the lattice ions. These collisions are the primary source of resistance in metals.
Collisions and Relaxation Time
Between collisions, an electron accelerates under the influence of the electric field. If an electron's velocity just after a collision is $\vec{u}$, its velocity after time $t$ (before the next collision) is $\vec{v} = \vec{u} + \vec{a}t = \vec{u} - \frac{e\vec{E}}{m}t$.
Due to the random nature of thermal motion and collisions, the initial velocities $\vec{u}$ average to zero over a large number of electrons: $\langle \vec{u} \rangle = 0$.
The average time between successive collisions is called the relaxation time ($\tau$). It is an average because the time between collisions varies randomly.
The average velocity acquired by the electrons due to the field is the drift velocity $\vec{v}_d$. Averaging the velocity equation over all electrons:
$ \vec{v}_d = \langle \vec{v} \rangle = \langle \vec{u} \rangle - \frac{e\vec{E}}{m}\langle t \rangle = 0 - \frac{e\vec{E}}{m}\tau $
$ \vec{v}_d = -\frac{e\vec{E}}{m}\tau $
The negative sign indicates that the drift velocity is opposite to the electric field direction (for negatively charged electrons). The magnitude of the drift velocity is $v_d = \frac{eE}{m}\tau$.
Connecting Drift Velocity to Resistivity
We previously established the relationship between current density $\vec{J}$ and drift velocity $\vec{v}_d$:
$ \vec{J} = ne\vec{v}_d $ (for positive charge carriers)
For electrons (charge $-e$), the current density is in the opposite direction of the drift velocity:
$ \vec{J} = n(-e)\vec{v}_d $
Substituting the expression for $\vec{v}_d$:
$ \vec{J} = n(-e) \left(-\frac{e\vec{E}}{m}\tau\right) = \frac{ne^2\tau}{m}\vec{E} $
Comparing this with the microscopic form of Ohm's Law, $\vec{J} = \sigma \vec{E}$, we identify the conductivity $\sigma$:
$ \sigma = \frac{ne^2\tau}{m} $
Since resistivity $\rho$ is the reciprocal of conductivity ($\rho = 1/\sigma$), we get:
$ \rho = \frac{m}{ne^2\tau} $
This equation provides the microscopic origin of resistivity. It shows that resistivity depends on:
- $m$: Mass of the charge carrier (constant for electrons).
- $e$: Charge of the charge carrier (constant for electrons).
- $n$: Number density of charge carriers (number of free electrons per unit volume). In metals, $n$ is very high and relatively constant with temperature.
- $\tau$: Average relaxation time (average time between collisions). This is influenced by the thermal vibrations of the lattice. Higher vibrations mean more frequent collisions and smaller $\tau$.
In metals, the number density of free electrons ($n$) is very large, leading to low resistivity. The collisions with lattice ions are the main reason for resistance.
Mobility ($\mu = v_d/E $)
Electrical mobility ($\mu$) is a measure of how easily charge carriers move through a material in response to an electric field. It is defined as the magnitude of the drift velocity per unit electric field strength:
$ \mu = \frac{|v_d|}{|E|} $
From the expression for drift velocity magnitude, $v_d = \frac{eE}{m}\tau$, we have $|v_d| = \frac{e|E|}{m}\tau$.
Substituting this into the definition of mobility:
$ \mu = \frac{(e|E|/m)\tau}{|E|} = \frac{e\tau}{m} $
Mobility is thus directly proportional to the relaxation time $\tau$. A longer relaxation time (fewer collisions) means higher mobility. Mobility is related to conductivity by:
$ \sigma = ne\mu $
And since $\rho = 1/\sigma$:
$ \rho = \frac{1}{ne\mu} $
This shows that resistivity is inversely proportional to the number density of charge carriers ($n$) and their mobility ($\mu$). Materials with high $n$ or high $\mu$ have low resistivity (good conductors). Materials with low $n$ or low $\mu$ have high resistivity (insulators or semiconductors).
The unit of mobility is $(m/s) / (V/m) = m^2 / (Vs)$.
Resistivity Of Various Materials
Materials exhibit a vast range of electrical resistivity values. Based on their approximate resistivity, materials are typically classified into three categories: conductors, semiconductors, and insulators. This classification is based on their ability to conduct electric current.
Classification Based on Resistivity
-
Conductors: Materials with very low resistivity, allowing electric current to flow easily. Metals are good conductors. Their resistivities are typically in the range of $10^{-8} \, \Omega \cdot m$ to $10^{-6} \, \Omega \cdot m$. Examples include copper, aluminium, silver, gold.
-
Semiconductors: Materials with resistivity values intermediate between conductors and insulators. Their conductivity can be significantly altered by changing temperature or adding impurities (doping). Their resistivities are typically in the range of $10^{-5} \, \Omega \cdot m$ to $10^{4} \, \Omega \cdot m$. Examples include Silicon, Germanium.
-
Insulators: Materials with very high resistivity, offering extreme opposition to current flow. They effectively block the flow of electric current under normal voltage conditions. Their resistivities are typically in the range of $10^{10} \, \Omega \cdot m$ to $10^{16} \, \Omega \cdot m$ or even higher. Examples include glass, rubber, plastics, wood.
Table of Approximate Resistivity Values (at 20°C)
| Type of Material | Material | Approximate Resistivity ($\Omega \cdot m$) |
|---|---|---|
| Conductors | Silver | $1.59 \times 10^{-8}$ |
| Copper | $1.7 \times 10^{-8}$ | |
| Aluminium | $2.82 \times 10^{-8}$ | |
| Tungsten | $5.6 \times 10^{-8}$ | |
| Alloys (Conductors) | Nichrome | $1.1 \times 10^{-6}$ |
| Manganin | $4.4 \times 10^{-7}$ | |
| Constantan | $4.9 \times 10^{-7}$ | |
| Semiconductors | Germanium | $0.46$ |
| Silicon | $2.3 \times 10^{3}$ | |
| Insulators | Glass | $10^{10} - 10^{14}$ |
| Rubber (Hard) | $10^{13} - 10^{16}$ | |
| Ebonite | $10^{15} - 10^{17}$ | |
| Paper (Dry) | $10^{12}$ |
The table clearly shows the vast difference in resistivity across these classes. Conductors like silver and copper have extremely low resistivity, making them ideal for wiring. Alloys like Nichrome, Manganin, and Constantan have higher resistivity than pure metals and also exhibit very little change in resistance with temperature, making them suitable for heating elements and standard resistors. Semiconductors have resistivities in the middle range, and their unique properties (like doping) are utilised in electronic devices. Insulators have extremely high resistivity, making them effective for preventing unwanted current flow (e.g., sheathing on wires, bases of electrical components).
Temperature Dependence Of Resistivity
The resistivity of a material is not constant; it changes with temperature. This temperature dependence is significant and varies depending on the type of material (metal, semiconductor, alloy).
Temperature Dependence in Metals
For most metallic conductors, resistivity increases with increasing temperature.
From the microscopic formula for resistivity $\rho = \frac{m}{ne^2\tau}$, we know that resistivity depends on the number density of charge carriers ($n$) and the relaxation time ($\tau$).
In metals, the number density of free electrons ($n$) is very large and is almost independent of temperature. However, as temperature increases, the thermal vibrations of the positive ions in the crystal lattice become more violent. These increased vibrations make it more likely for the drifting electrons to collide with the lattice ions. This reduces the average time between collisions, i.e., the relaxation time ($\tau$) decreases.
Since $\rho \propto 1/\tau$, a decrease in $\tau$ leads to an increase in $\rho$. Thus, the resistivity of metals increases with temperature primarily due to the decrease in relaxation time caused by increased lattice vibrations.
For most metals, the resistivity ($\rho_T$) at a temperature $T$ is approximately related to its resistivity ($\rho_0$) at a reference temperature $T_0$ by the linear relation:
$ \rho_T = \rho_0 [1 + \alpha(T - T_0)] $
Here, $\alpha$ is called the temperature coefficient of resistivity. For metals, $\alpha$ is positive. Its unit is $^\circ C^{-1}$ or $K^{-1}$. A positive $\alpha$ means that as the temperature $T$ increases (from $T_0$), the resistivity $\rho_T$ also increases.
The resistance ($R_T$) of a conductor of fixed dimensions at temperature $T$ is also related to its resistance ($R_0$) at temperature $T_0$ by the same formula, since $R = \rho L/A$ and $L, A$ are usually assumed constant:
$ R_T = R_0 [1 + \alpha(T - T_0)] $
For temperatures significantly different from $T_0$, or for some materials, a more complex polynomial relation might be needed, but the linear approximation is valid over moderate temperature ranges.
Temperature Dependence in Semiconductors
For semiconductors (like Silicon and Germanium), resistivity generally decreases with increasing temperature.
In semiconductors, the number density of charge carriers ($n$) is much smaller than in metals and is highly dependent on temperature. As temperature increases, more covalent bonds are broken, releasing more free electrons and creating more holes (another type of charge carrier in semiconductors). This leads to a significant increase in the number density of charge carriers ($n$).
While the increase in temperature also causes more lattice vibrations, which reduces the relaxation time $\tau$ (and hence mobility $\mu$), the increase in the number density of carriers ($n$) is usually much more dominant.
From $\rho = \frac{1}{ne\mu}$, the large increase in $n$ with temperature overrides the decrease in $\mu$ (due to decreasing $\tau$), resulting in a net decrease in resistivity.
Semiconductors have a negative temperature coefficient of resistivity. Their resistivity drops sharply as temperature rises.
Temperature Dependence in Insulators
Similar to semiconductors, the resistivity of insulators also decreases very sharply with increasing temperature.
In insulators, the number density of free charge carriers ($n$) is extremely low at room temperature. As temperature increases, a small number of charge carriers might become available due to thermal excitation, causing $n$ to increase. While the increase in $n$ is still small compared to metals, it is a significant relative increase for insulators and causes a noticeable decrease in resistivity.
Temperature Dependence in Alloys
Certain alloys, like Nichrome (used in heating elements), Manganin, and Constantan, exhibit very weak temperature dependence of resistivity over a wide range of temperatures.
In alloys, the lattice structure is already disordered due to the presence of atoms of different elements. Collisions of electrons with this disordered lattice are already frequent and contribute significantly to the resistivity. The additional scattering caused by increased thermal vibrations at higher temperatures is less significant compared to pure metals.
This property makes these alloys suitable for use in standard resistors, resistance boxes, and heating elements where resistance is desired to remain relatively constant despite changes in temperature during operation.