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Electricity and Magnetism
Electrical Conductivity
Electrical conductivity is the measure of a material's ability to conduct an electric current, defined as the reciprocal of electrical resistivity. In microscopic terms, it is determined by the density of mobile charge carriers, their elementary charge, and their drift mobility within the medium.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of conductivity and resistivity, Ohm's law in vector form (J = sigma E), temperature dependence of resistance and resistivity for metals and semiconductors, color coding of carbon resistors, and grouping of resistances.
PGT extension
Drude model derivation of conductivity, relaxation time of electrons, microscopic interpretation of mobility, tensor conductivity in anisotropic media, and AC response involving complex permittivity and conductivity.
Prerequisites
Ohm's Law and macroscopic resistance
Current density and drift velocity of electrons
Basic thermal physics and carrier statistics
Microscopic Definition and Vector Form of Ohm's Law
Conductivity (sigma) is defined as the inverse of electrical resistivity (rho), such that sigm$a = 1 / \rho$.
Ohm's law in microscopic form relates the current density J to the electric field E via the relation J = sigma E.
The SI unit of electrical conductivity is Siemens per meter (S/m) or ohm-1m-1.
Drude Model of Conduction in Metals
Metals possess a high density of free conduction electrons that move randomly and collide periodically with fixed ionic cores.
Under an applied electric field E, electrons acquire a net directional drift velocity vd superimposed on their random thermal motion.
The average time interval between successive collisions is known as the relaxation time (tau).
Temperature Dependence of Conductivity
For metallic conductors, conductivity decreases with an increase in temperature due to enhanced thermal vibrations of ions reducing the relaxation time tau.
For intrinsic semiconductors and insulators, conductivity increases exponentially with temperature as more covalent bonds are thermally broken to release charge carriers.
The temperature coefficient of resistivity (alpha) is positive for pure metals and negative for semiconductors and carbon.
Carrier Mobility and Conductivity
Mobility (mu) is defined as the magnitude of drift velocity acquired per unit electric field, m$u = |vd| / E$.
Total conductivity in semiconductors involves contributions from both negatively charged electrons and positively charged holes: sigm$a = e (n \mu_n + p \mu_p)$.
In pure (intrinsic) semiconductors, the electron concentration n equals the hole concentration p, equaling the intrinsic carrier concentration ni.
Derivations
Drude Model Derivation of Electrical Conductivity
Conditions: Steady uniform electric field applied to a metal containing free electrons of mass m and charge e
Write the equation of motion for a conduction electron between collisions: m(dvd/dt) = -eE - m(vd/tau), where tau is the relaxation time.
Under steady-state conditions, the acceleration is zero (dvd/dt = 0), yielding the drift velocity: v$d = -(e \tau / m) E$.
Relate current density J to drift velocity and carrier density n: $J = -n e vd = (ne^{2} \tau / m) E$.
Compare with the microscopic Ohm's law J = sigma E to identify the expression for conductivity.
Result: sigm$a = \frac{n e^2 \tau}{m}$
Temperature Dependence of Resistivity
Conditions: Small temperature variations where linear approximation holds
Express resistivity rho at temperature T as a linear perturbation from a reference temperature T_0: rho(T) = rho_0 [1 + alpha (T - T_0)].
Define the temperature coefficient of resistivity alpha as the fractional change in resistivity per unit change in temperature: alph$a = \frac{1}{\rho_0} \frac{d\rho}{dT}$.
Substitute rh$o = m / (n e^2 \tau) into the definition$ to observe how collision frequency and thermal expansion affect alpha in metals.
Result: \rho(T) = \rho_0 [1 + alpha (T - T_0)]
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Microscopic Ohm's Law
$$J = \sigma E$$
J = Current density, sigma = Conductivity, E = Electric field J in A/m2, sigma in S/m, E in V/m
Valid for isotropic linear media
Drude Conductivity Formula
$$\sigma = \frac{n e^2 \tau}{m}$$
n = Electron density, e = Elementary charge, tau = Relaxation time, m = Mass of electron n in m-3, e in C, tau in s, m in kg
Free electron gas approximation
Conductivity via Mobility
$$\sigma = n e \mu$$
n = Carrier density, e = Elementary charge, mu = Mobility sigma in S/m, mu in m2/(V·s)
Single type of dominant charge carrier
Semiconductor Total Conductivity
$$\sigma = e (n \mu_n + p \mu_p)$$
n, p = Electron and hole concentrations, mu_n, m$u_p = Their respective mobilities$ S/m
Extrinsic or intrinsic semiconductor
Temperature Coefficient of Resistivity
$$\alpha = \frac{1}{\rho} \frac{d\rho}{dT}$$
rho = Resistivity, T = Absolute temperature, alpha = Temperature coefficient K-1 or °C-1
Small temperature intervals
Resistivity-Conductivity Inverse Relation
$$\rho = \frac{1}{\sigma}$$
rho = Resistivity, sigma = Conductivity ohm·m and S/m
General scalar case
Drift Velocity Expression
$$v_d = \mu E$$
vd = Drift velocity magnitude, mu = Mobility, E = Electric field m/s
Low electric field regime
Resistance from Resistivity and Dimensions
$$R = \rho \frac{l}{A}$$
R = Resistance, l = Length, $A = Cross-sectional area$ R in ohm, l in m, A in m2
Uniform cross-section
Graphs and experiments
Resistivity vs Temperature for Metals
Draw: A plot of resistivity (rho) on the y-axis against absolute temperature (T) on the x-axis, showing a nearly linear increase at high temperatures and a non-linear drop towards zero near absolute zero.
Interpret: At high temperatures, phonon scattering dominates, making resistivity proportional to T. At very low temperatures, impurity scattering dictates residual resistivity.
Exam use: Direct identification questions on metallic behavior vs temperature and intercept interpretations.
Resistivity vs Temperature for Semiconductors
Draw: A steeply decaying exponential curve plotting resistivity versus temperature.
Interpret: Thermal excitation exponentially increases carrier density n, overpowering the decrease in relaxation time, causing rapid drop in resistivity.
Exam use: Distinguishing semiconductor thermal properties from metallic properties in assertion-reasoning items.
Current Density vs Electric Field (Ohmic vs Non-Ohmic)
Draw: A straight line passing through the origin for ohmic conductors, and a curved/saturating profile for non-ohmic devices like diodes or electrolytes.
Interpret: The slope of the J-E curve yields the electrical conductivity sigma for linear media.
Exam use: Graphical determination of conductivity values and breakdown conditions.
Solved numericals
TGT core
A silver wire has a resistivity of 1.6 x 10-8 ohm·m at 20°C. Calculate its electrical conductivity.
Recall the relation between conductivity and resistivity: sigm$a = 1 / \rho$.
Substitute the given value of rh$o = 1.6 x 10^{-8} ohm·m$.
Calculate the reciprocal: sigm$a = 1 / (1.6 x 10^{-8}) = 0.625 x 10^{8} = 6.25 x 10^{7} S/m$.
Answer: 6.25 x 107 S/m
PGT extension
In a doped silicon semiconductor, the concentration of electrons is 2 x 1019 m-3 and holes is 1 x 1018 m-3. Given the electron mobility is 0.13 m2/(V·s) and hole mobility is 0.05 m2/(V·s), find the total electrical conductivity.
Use the formula for semiconductor conductivity: sigm$a = e (n \mu_n + p \mu_p)$.
Substitute values: $e = 1.6 x 10^{-19} C$, $n = 2 x 10^{19}$, m$u_n = 0.13$, $p = 1 x 10^{18}$, m$u_p = 0.05$.
Calculate electron contribution: 1.6 x 10-19 x (2 x 1019 x 0.13) = 1.6 x 2 x $0.13 = 0.416 S/m$.
Calculate hole contribution: 1.6 x 10-19 x (1 x 1018 x 0.05) = 1.6 x 0.1 x $0.05 = 0.008 S/m$.
Sum both contributions: sigm$a = 0.416 + 0.008 = 0.424 S/m$.
Answer: 0.424 S/m
PGT extension
Assuming each copper atom contributes one free electron, calculate the conductivity of copper at room temperature. Given: density of coppe$r = 8960 kg/m^{3}$, atomic mas$s = 63.5 g/mol$, relaxation time ta$u = 2.5 x 10^{-14} s$, mass of electro$n = 9.1 x 10^{-31} kg$, and Avogadro numbe$r = 6.023 x 10^{23} mol^{-1}$.
Find electron number density $n = (Density x Avogadro number) / (Atomic mass)$.
$n = (8960 x 6.023 x 10^{23}) / (63.5 x 10^{-3}) \approx 8.5 x 10^{28} m^{-3}$.
Apply Drude formula: sigm$a = (n e^2 \tau) / m$.
sigm$a = (8.5 x 10^{28} x (1.6 x 10^{-19})^{2} x 2.5 x 10^{-14}) / (9.1 x 10^{-31})$.
Evaluate numerical result: sigma approx 6.0 x 107 S/m.
Answer: 6.0 x 107 S/m
Assumptions and limitations
The Drude model assumes electrons act as a classical ideal gas, ignoring quantum mechanical Fermi-Dirac statistics and Pauli exclusion principle.
Relaxation time tau is treated as a constant independent of electron velocity and energy, which is an approximation valid only near thermal equilibrium.
Common misconceptions
Common error: High conductivity implies that electrons travel through a wire at very high speeds.
Correct: Drift velocity of electrons is extremely small (on the order of mm/s), but the electric field propagates at nearly the speed of light.
The electrical signal travels via electromagnetic wave propagation rather than individual electron transit.
Common error: Semiconductors have a positive temperature coefficient of resistivity like metals.
Correct: Semiconductors have a negative temperature coefficient of resistivity.
Thermal generation of additional charge carriers outweighs the decrease in relaxation time at higher temperatures.
Quick revision
Conductivity is the reciprocal of resistivity: sigm$a = 1 / \rho with SI unit S/m$.
Microscopic Ohm's law is expressed as J = sigma E.
Drude model formula for metals is sigm$a = ne^{2}\tau / m$.
Metals exhibit a positive temperature coefficient of resistivity, whereas semiconductors exhibit a negative coefficient.
Must remember
Microscopic form of Ohm's law is given by $\vec{J} = \sigma \vec{E}$, where $\vec{J}$ is current density, $\sigma$ is conductivity, and $\vec{E}$ is electric field.
Drude model expression for conductivity is $\sigma = \frac{n e^2 \tau}{m}$, connecting microscopic parameters like relaxation time $\tau$ and electron density $n$.
For metallic conductors, conductivity decreases with increasing temperature due to reduced relaxation time $\tau$ caused by enhanced lattice vibrations.
For semiconductors, conductivity increases exponentially with temperature because thermal generation of charge carriers dominates over the decrease in $\tau$.
Total conductivity in semiconductors includes contributions from both electrons and holes: $\sigma = e (n \mu_n + p \mu_p)$.
Mobility $\mu$ is defined as the magnitude of drift velocity per unit electric field: $\mu = \frac{v_d}{E}$.
Glossary
Electrical Conductivity ($\sigma$)The measure of a material's ability to conduct an electric current, defined as the inverse of resistivity $\rho$.
Current Density ($J$)Electric current passing per unit cross-sectional area perpendicular to the current, measured in $\text{A/m}^2$.
Relaxation Time ($\tau$)The average time interval between successive collisions of conduction electrons with fixed ionic cores in a metal.
Drift Velocity ($v_d$)The average uniform velocity acquired by charge carriers inside a material under the influence of an applied electric field.
Mobility ($\mu$)The magnitude of drift velocity acquired per unit electric field, quantifying how quickly a charge carrier can move through a medium.
Temperature Coefficient of Resistivity ($\alpha$)The fractional change in resistivity per degree change in temperature, positive for metals and negative for semiconductors.
Metals vs Semiconductors in Electrical Conductivity
Property
Metals
Semiconductors
Conductivity Magnitude
Very high ($10^7 \text{ S/m}$)
Intermediate to low, strongly dependent on doping/temperature
Temperature Dependence of $\sigma$
Decreases with temperature increase
Increases exponentially with temperature increase
Temperature Coefficient ($\alpha$)
Positive
Negative
Primary Charge Carriers
Free conduction electrons
Both electrons ($n$) and holes ($p$)
Practice Quiz
Choose an option to reveal the explanation.
1. What is the microscopic form of Ohm's law relating current density $\vec{J}$, conductivity $\sigma$, and electric field $\vec{E}$?
Explanation: The vector form of Ohm's law is given by $\vec{J} = \sigma \vec{E}$, which relates the flow of charge locally to the driving electric field.
2. According to the Drude model, the electrical conductivity of a metal is given by which expression?
Explanation: In the Drude free electron model, conductivity is derived as $\sigma = \frac{n e^2 \tau}{m}$, where $n$ is electron density, $e$ is charge, $\tau$ is relaxation time, and $m$ is electron mass.
3. What happens to the electrical conductivity of a pure metal when its temperature is increased?
Explanation: As temperature increases, thermal vibrations of ionic cores increase, leading to more frequent collisions and a shorter relaxation time $\tau$, which decreases conductivity $\sigma$.
4. Why does the electrical conductivity of an intrinsic semiconductor increase with temperature?
Explanation: In semiconductors, thermal excitation breaks bonds to create additional electron-hole pairs. This exponential rise in carrier concentration outweighs the reduction in relaxation time.
5. Mobility ($\mu$) of charge carriers is defined as the magnitude of drift velocity per unit:
Explanation: Mobility is defined as $\mu = \frac{v_d}{E}$, representing the drift velocity acquired per unit electric field.
6. The total electrical conductivity of a semiconductor containing both electrons and holes is expressed as:
Explanation: Both electrons and holes contribute independently to current conduction in semiconductors, leading to $\sigma = e(n \mu_n + p \mu_p)$.
7. What is the sign of the temperature coefficient of resistivity ($\alpha$) for carbon and semiconductors?
Explanation: Carbon and semiconductors exhibit a negative temperature coefficient of resistivity ($\alpha < 0$), meaning their resistivity decreases as temperature increases.
8. What is the SI unit of electrical conductivity ($\sigma$)?
Explanation: Conductivity is the inverse of resistivity (measured in $\Omega \cdot \text{m}$), making its SI unit Siemens per meter ($\text{S/m}$ or $\Omega^{-1}\text{m}^{-1}$).
9. Which misconception is corrected by stating that electron drift velocity is very small?
Explanation: Although electrical signals propagate at nearly the speed of light via electromagnetic fields, the actual drift velocity of conduction electrons is extremely small (on the order of $\text{mm/s}$).
10. For an intrinsic semiconductor, how does the electron concentration ($n$) relate to the hole concentration ($p$) and intrinsic carrier concentration ($n_i$)?
Explanation: In an intrinsic (pure) semiconductor, thermal excitation generates equal numbers of conduction electrons and holes, so $n = p = n_i$.
11. What is the relationship between electrical resistivity ($\rho$) and conductivity ($\sigma$)?
Explanation: Electrical resistivity is defined as the reciprocal of electrical conductivity: $\rho = \frac{1}{\sigma}$.
12. Under the Drude model, if the relaxation time $\tau$ decreases due to lattice heating, how does conductivity behave?
Explanation: Since $\sigma = \frac{n e^2 \tau}{m}$, conductivity is directly proportional to the relaxation time $\tau$; hence, a decrease in $\tau$ causes a proportional decrease in $\sigma$.
13. What is the condition for validity of the microscopic Ohm's law formula $J = \sigma E$?
Explanation: The simple scalar relation $J = \sigma E$ holds for isotropic linear media where conductivity is independent of direction.
14. How is drift velocity ($v_d$) related to mobility ($\mu$) and electric field ($E$) in the low electric field regime?
Explanation: By definition of mobility $\mu = \frac{v_d}{E}$, rearranging gives $v_d = \mu E$.
15. Which of the following materials has a positive temperature coefficient of resistivity?
Explanation: Pure metals like copper have a positive temperature coefficient of resistivity, whereas semiconductors and carbon have negative coefficients.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The relaxation time of free electrons in a metal is $\tau$. If the temperature of the metal increases, the electrical conductivity:
Explanation: With an increase in temperature, thermal vibrations of atoms increase, which causes electrons to collide more frequently, reducing the relaxation time $\tau$. Since $\sigma \propto \tau$, conductivity decreases.
2. In an intrinsic semiconductor, the concentrations of electrons and holes are $n$ and $p$ respectively. If the mobility of electrons is $\mu_n$ and of holes is $\mu_p$, the conductivity is given by:
Explanation: Total semiconductor conductivity is the sum of electron and hole contributions: $\sigma = e n \mu_n + e p \mu_p = e(n \mu_n + p \mu_p)$.
3. What is the SI unit of mobility ($\mu$)?
Explanation: Since $\mu = \frac{v_d}{E}$, where $v_d$ is in $\text{m/s}$ and $E$ is in $\text{V/m}$, the unit of mobility is $\frac{\text{m/s}}{\text{V/m}} = \text{m}^2 / (\text{V} \cdot \text{s})$.
4. If the electric field inside a conductor of conductivity $\sigma$ is $\vec{E}$, the magnitude of the current density $\vec{J}$ is:
Explanation: By Ohm's law in microscopic form, $\vec{J} = \sigma \vec{E}$, so the magnitude is $J = \sigma E$.
5. The temperature coefficient of resistivity for semiconductors is:
Explanation: Semiconductors possess a negative temperature coefficient of resistivity because thermal generation of carriers dominates over collision frequency increases.
6. Resistance $R$ of a uniform conductor of length $l$, cross-sectional area $A$, and resistivity $\rho$ is given by:
Explanation: Resistance is directly proportional to length $l$ and inversely proportional to cross-sectional area $A$, with resistivity $\rho$ as the proportionality constant: $R = \rho \frac{l}{A}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. In the Drude expression $\sigma = \frac{n e^2 \tau}{m}$, what physical property does $n$ represent?
Explanation: In the free electron gas approximation, $n$ represents the electron density (number of conduction electrons per unit volume).
2. Calculate the approximate conductivity of a metal if its electron density is $10^{28} \text{ m}^{-3}$, relaxation time is $10^{-14} \text{ s}$, mass of electron is $9.1 \times 10^{-31} \text{ kg}$, and elementary charge is $1.6 \times 10^{-19} \text{ C}$.
3. Which of the following factors causes the electrical conductivity of metals to decrease at higher temperatures?
Explanation: In metals, $n$ remains essentially constant with temperature; the decrease in conductivity is entirely due to enhanced thermal vibrations reducing the relaxation time $\tau$.
4. If an electric field of $5 \text{ V/m}$ is applied across a semiconductor specimen where carrier mobility is $0.4 \text{ m}^2 / (\text{V} \cdot \text{s})$, what is the drift velocity of the carriers?
Explanation: Using $v_d = \mu E = 0.4 \times 5 = 2.0 \text{ m/s}$.
5. What is the primary reason why carbon resistors have a negative temperature coefficient of resistivity?
Explanation: Carbon has a non-metallic crystal structure similar to semiconductors, where thermal agitation increases carrier density, resulting in a negative temperature coefficient.
6. The temperature coefficient of resistivity ($\alpha$) is mathematically defined as:
Explanation: By definition, the temperature coefficient of resistivity is the fractional change in resistivity per unit change in temperature: $\alpha = \frac{1}{\rho} \frac{d\rho}{dT}$.
7. In an extrinsic $n$-type semiconductor, what can be said about the relative concentrations of electrons ($n$) and holes ($p$)?
Explanation: In an $n$-type semiconductor, donor impurities supply a large excess of electrons, making electron concentration much greater than hole concentration ($n \gg p$).
8. How does the relaxation time ($\tau$) of electrons in a typical metal vary with absolute temperature $T$?
Explanation: Since thermal velocity of electrons is roughly constant with temperature while lattice vibration amplitude squared is proportional to $T$, the collision frequency is proportional to $T$, meaning relaxation time $\tau \propto \frac{1}{T}$.
9. If conductivity $\sigma = 2.5 \times 10^7 \text{ S/m}$, what is the resistivity $\rho$ of the material?
10. Which statement correctly contrasts microscopic Ohm's law with macroscopic Ohm's law ($V = IR$)?
Explanation: Microscopic Ohm's law ($\vec{J} = \sigma \vec{E}$) relates field and current density at any microscopic point, whereas macroscopic Ohm's law ($V = IR$) applies to an entire bulk conductor or resistor component.