As per the Drude model of metals, the electrical resistance of a metallic wire of length $L$ and cross-section area $A$ is
(Consider $\tau$ as the relaxation time, $m$ as electron mass, $n$ as carrier concentration and $e$ as electronic charge)
Drude Model Conductivity Derivation
The Drude model provides a simple way to understand electrical conductivity in metals. It treats electrons as classical particles subject to collisions.
- The average drift velocity ($v_d$) of electrons in response to an electric field ($E$) is given by:
$v_d = \frac{eE\tau}{m}$
- The current density ($J$) is related to drift velocity by $J = n e v_d$, where $n$ is the electron concentration. Substituting $v_d$:
$J = n e \left(\frac{eE\tau}{m}\right) = \frac{ne^2\tau}{m} E$
- Conductivity ($\sigma$) is defined as $J = \sigma E$. Therefore, from the above equation:
$\sigma = \frac{ne^2\tau}{m}$
Calculating Electrical Resistance
Electrical resistance ($R$) is related to resistivity ($\rho$) and the physical dimensions of the conductor (length $L$ and area $A$) by the formula $R = \rho \frac{L}{A}$.
- Resistivity ($\rho$) is the reciprocal of conductivity ($\sigma$):
$\rho = \frac{1}{\sigma} = \frac{m}{ne^2\tau}$
- Substituting this expression for resistivity into the resistance formula:
$R = \left(\frac{m}{ne^2\tau}\right) \frac{L}{A}$
- Simplifying the expression gives the resistance of the metallic wire:
$R = \frac{mL}{ne^2 A\tau}$
This matches the first option.
Important Questions from Free Electron Theory Fermi Energy Velocity
- Copper has an electron number density of $8.3 \times 10^{28} \text{ m}^{-3}$. Its Fermi energy in eV (rounded off to one decimal place) is _____
($\hbar = 1.06 \times 10^{-34} \text{ J.s}$, mass of electron $\text{m}_e = 9.10 \times 10^{-31} \text{ kg}$, charge of electron $= 1.60 \times 10^{-19} \text{ C}$) - Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is $E_F$. The energy corresponding to the filling of $\frac{N_A}{2}$ electrons, where $N_A$ is the Avogadro number, is $2^n E_F$. The value of $n$ is
Crystal structures of two metals A and B are two-dimensional square lattices with same lattice constant $a$. Electrons in metals behave as free electrons. The Fermi surfaces corresponding to A and B are shown by solid circles in figures.
The electron concentrations in A and B are $n_A$ and $n_B$, respectively. The value of $(\frac{n_B}{n_A})$ isIf $X$ is the dimensionality of a free electron gas, the energy ($E$) dependence of density of states is given by $E^{½X-Y}$, where $Y$ is ________.
- Potassium metal has electron concentration of $1.4 \times 10^{28}m^{-3}$ and the corresponding density of states at Fermi level is $6.2 \times 10^{46}$ Joule$^{-1} m^{-3}$. If the Pauli paramagnetic susceptibility of Potassium is $n \times 10^{-k}$ in standard scientific form, then the value of $k$ (an integer) is __________ (Magnetic moment of electron is $9.3 \times 10^{-24}$ Joule T$^{-1}$; permeability of free space is $4\pi \times 10^{-7}$ T m A$^{-1}$)
