Conducting Materials - Questions and Answers

1. What is a conducting material?

The material possessing higher electrical and thermal conductivities are called conducting materials. The resistance of such materials are very low. The conducting property of a material depends on the number of free electrons.

2. What is meant by a free electron?

The electron moves freely in all directions in the absence of electric field is called free electron or valance electron. These electrons collide with each other, the collisions are perfectly elastic collisions hence, there is no loss of energy. Since the free electron is in random motion.

3. Mention any four postulates of classical free electron theory.

The free electrons in the metal moves freely, similar to the gas molecules moving in a vessel and it obeys the classical kinetic theory of gasses.
These free electrons moves in a constant potential field due to ions fixed in the lattice.
When field is applied the free electron moves in the direction opposite to that of the field direction.
Due to field applied, they acquire a velocity called drift velocity and the electron velocities in the metal obeys the Maxwell Boltzmann statistics.

4. Define Drift velocity of electrons

The average velocity acquired by the free electron in a particular direction, due to the application of electric field is called drift velocity.

\begin{aligned} \text { Drift velocity } & =\frac{\text { average dis tan ce travelled by the electron }}{\text { time taken }} \\ \mathrm{V}_{\mathrm{d}} & =\frac{1}{\mathrm{t}} \dot{\mathrm{ms}}^{-1} \end{aligned}

5. Define relaxation time.

It is the time taken by the free electron to reach its equilibrium position from its disturbed position, in the presence of applied field.

\tau=\frac{1}{\mathrm{~V}_{\mathrm{d}}}

Where $1$ is the distance travelled by the electron.

The value of relaxation time is of the order of 10-14 seconds.

6. Define Mean Collision time (τ_c).

It is the average time taken by a free electron between two successive collision

\tau_{\mathrm{c}}=\frac{\lambda}{\mathrm{V}_{\mathrm{d}}}

Where λ is mean free path.

For a isotropic solid like metals τ = τ_c

7. Define mean free path.

The average distance travelled between two successive collision is called mean free path.

\lambda=\bar{c} \tau_c

Where $\bar{c}$ is the root mean square velocity of the electron.

8. Define mobility of electrons.

Mobility of the electron is defined as the drift velocity $(v d)$ acquired by the electron per unit electric field (E) applied to it.

Mobility $\mu=\frac{\mathrm{V}_{\mathrm{d}}}{\mathrm{E}} \mathrm{mV}^{-1} \mathrm{~s}^{-1}$

9. Define electrical conductivity.

The electrical conductivity is defined as the quantity of electrical charges flowing per unit area per unit time at unit potential gradient.

\sigma=\frac{\mathrm{Q}}{\mathrm{AtE}} \Omega^{-1} \mathrm{~m}^{-1}

10. Define Thermal Conductivity.

Thermal conductivity of a material is equal to the amount of heat flowing through the material of unit cross sectional area under unit temperature gradient. It’s unit is Wm^-1 K^-1

\mathrm{K}=\frac{\mathrm{Q}}{\left(\frac{\mathrm{dT}}{\mathrm{dx}}\right)} \mathrm{Wm}^{-1} \mathrm{~K}^{-1}

11. Distinguish between electrical conductivity and thermal conductivity.

S.No	Electrical conductivity	Thermal conductivity
1.	The quantity of electric charges flows in unit time per unit area of cross section of the conductor per unit potential gradient is called as Electrical conductivity.	The amount of heat flowing through the material of unit cross sectional area under unit temperature gradient is called as Thermal conductivity.
2.	Electrical conductivity is purely due to number of free electrons	Thermal conductivity is due to both free electrons and protons.
3.	Conduction of electricity takes place from higher potential end to the lower potential end	Conduction of heat takes place from hot end to cold end
4.	Unit is ohm^-1 m^-1	Unit is Wm^-1 K^-1

12. State Widemann-franz law.

The ratio between the thermal conductivity and electrical conductivity of a metal is directly proportional to the absolute temperature of the metal.

\frac{\mathrm{K}}{\sigma} \infty \mathrm{T}

13. What is Lorentz number?

The ratio between thermal conductivity $(K)$ of a metal o the product of electrical conductivity $(σ)$ of a metal and absolute temperature $(T)$ of the metal is constant. It is called Lorentz number and it is given by

\mathrm{L}=\frac{\mathrm{K}}{\sigma \mathrm{T}}

14. What are the sources of resistance in metals?

The resistance in metal is due to

i) Presence of impurities in the metals
ii) Temperature of the metal.
iii) Number of free electrons.

15. List out the effect of temperature on conducting materials?

When temperature of the metal increases, the mobility of the electron decreases and hence the electrical conductivity decreases.
The addition of impurities in the metal decreases the electrical conductivity.

16. What are the uses (or) success of classical free electron theory?

It is used to verify the Ohm’s law.
It is used to explain electrical conductivity $(σ)$ and thermal conductivity $(K)$ of metals
It is used to derive Widemann-Franz law.
It is used to explain the optical properties of metal.

17. What are the drawbacks of classical free electron theory?

Classical theory states that all free electrons will absorb energy, but quantum theory states that only few electrons will absorb energy.
It cannot explain the electrical conductivity $(σ)$ of semiconductors and insulators.
As per classical theory $\frac{\mathrm{K}}{\sigma \mathrm{T}}$ is constant at all temperatures. But at low temperature $\frac{\mathrm{K}}{\sigma \mathrm{T}}$ is not a constant.
There is a difference in theoretical and experimental values of specific heat of metals and electronic specific heat of metals.
Photoelectric effect, Compton effect and Black body radiation phenomenons cannot be explained by this theory.

18. How classical free electron theory failed to account for specific heat of solid?

According to the classical free electron theory the value if specific heat of metals is given by $4.5 R u$ where $R u$ is the universal gas constant whereas the experimental value is nearly equal to $3R u$ . Also according to this theory the value of electronic specific heat is equal to $\frac{3}{2} \mathrm{R}_{\mathrm{u}}$ while the actual is about $0.01R u$ only.

19. Mention any two important features of quantum free electron theory of metals.

It shows that the energy levels of an electron are discrete.
The Maximum energy level upto which the electrons can be filled is denoted by Fermi energy level.

20. What is the basic assumption of Zone theory or Band theory of solids?

According to quantum free electron theory, the electrons in a metal were assumed to be moving in a region of constant potential but it fails to explain why some solids behave as conductors, some as insulators and some as semiconductors.

Therefore instead of considering an electron to move in a constant potential, the Zone theory of solids tells that the electrons are assumed to move in a field of periodic potential.

21. Define Fermi distribution function.

Fermi distribution function $F(E)$ is used to calculate the probability of an electron occupying a certain energy level. The distribution of electrons among the energy levels as a function of temperature is known as Fermi Dirac distribution function.

\mathrm{F}(\mathrm{E})=\frac{1}{1+\exp \left(\frac{\mathrm{E}-\mathrm{E}_{\mathrm{F}}}{\mathrm{kT}}\right)}

Where

E_F is called Fermi energy

K is the Boltzmann constant

22. Define Fermi level

The Fermi level is that state at which the probability of electron occupation is $1/2$ at any temperature above $0K$ and also it is the level of maximum energy of the filled states at $0K$ .

23. Define the term Fermi energy of metals

Fermi energy is the energy of the state at which the probability of electron occupation is $1/2$ at any temperature above $0K$ . It is also the maximum energy of filled states at 0k. At 0K all states below Fermi energy level are filled and those above are empty.

24. Draw the Fermi distribution curve at $0K$ and at any temperature T K [or] How does the Fermi function varies with temperature.

Fermi distribution

The Fermi function varies with respect to the temperature. At $0K$ all the energy states below $EF$ are filled and all those above it are empty. When the temperature is increased, the electron takes an energy KT and hence the Fermi function falls to zero.

25. Define density of states.

Density of states is defined as the number of energy states per unit volume in an energy interval of a metal. It is used to calculate the number of charge carriers per unit volume of any solid.