IEMDaily - Video Lecture Notes

Definition

Mean free path

The mean free path is the average distance traveled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modify its direction or energy or other particle properties.

l = (σ n)^{(- 1)}

Where

l

is the mean free path,

n

is the number of target particles per unit volume, and

σ

is the effective cross-sectional area of collision.

Definition

Relaxation time of electrons

The average value of the time elapsed after the last collision of all the electrons in a perturbed system is known as the relaxation time. It is represented by

τ

τ = \frac{σ m}{n e^{2}}

Formula

Relation of drift velocity with electric field

An electron will suffer collisions with the heavy fixed ions, but after collision, it will emerge with the same speed but in random directions. If we consider all the electrons, their average velocity will be zero since their directions are random. Hence, $\frac{1}{N} \sum_{i = 1}^{N} v_{i} = 0$
Now, if an electric field is present. Electrons will be accelerated due to this field by $a = \frac{- e E}{m}$ where $- e$ is the charge and $m$ is the mass of electron.
Consider the $i t h$ electron at a given time $t$ . This electron would have had its last collision some time before $t$ , and let $t_{i}$ be the time elapsed after its last collision. If $v_{i}$ was its velocity immediately after the last collision, then its velocity $V_{i}$ at time t is $V_{i} = v_{i} + (\frac{- e E}{m} t_{i})$ .
The average velocity of the electrons at time t is the average of all the $V_{i} s$ .
Hence $(V_{i})_{a v e r a g e} = (v_{i})_{a v e r a g e} + \frac{e E}{m} (t_{i})_{a v e r a g e}$
The average of $v_{i} s$ is zero as seen above.
The average of $t_{i} s$ is called relaxation time denoted by $τ$ .
The average of $V_{i} s$ is called drift velocity $V_{d}$
Hence $V_{d} = 0 + \frac{- e E}{m} τ$
The last result show that electron move with average velocity which is independent of time.

Formula

Current density in terms of drift velocity

j = n q v_{d}

where

n

is the number density (number per unit volume) of charge carriers each of charge

q

, and

v_{d}

is the drift velocity of the charge carriers.

Definition

Boltzmann constant

The Boltzmann constant

(k_{B}

k)

is a physical constant relating energy at the individual particle level with temperature. It is the gas constant

R

divided by the Avogadro constant

N_{A}

k = \frac{R}{N_{A}}

The Boltzmann constant has the dimension of energy divided by temperature, the same as entropy. The accepted value in SI units is

1.38064852 (79) \times 10^{23} J / K

Formula

Mobility of electrons

Mobility

μ

is defined as the magnitude of drift velocity per unit electric field.

μ = \frac{| v_{d} |}{E}

.
Its SI unit is

m^{2} / V s

Definition

Ohmic and Non-ohmic resistors

S.No.	Ohmic Resistor	Non-ohmic Resistor
1	Obeys Ohm's Law	Does not obey Ohm's Law
2	Graph of V v/s I is a straight line	Graph of V v/s I is not a straight line
3	Slope of V-I graph is constant	Slope of V-I graph changes
4	Examples: Metals	Examples: Semiconductors, inductor, etc

Law

Ohm's Law

According to Ohm's Law, the current flowing in a conductor is directly proportional to the potential difference across its ends provided the physical conditions and temperature of conductor remains constant.

I \propto V

V = I R

Definition

Limitations of Ohm's Law

The limitations are:

V ceases to be proportional to I.
The relation between V and I depends on the sign of V. In other words, if I is the current for a certain V, then reversing the direction of V keeping its magnitude fixed, does not produce a current of the same magnitude as I in the opposite direction. This happens for example in case of a diode.
The relation between V and I is not unique. i.e. there is more than one value of V for the same current I. A material exhibiting such behaviour is GaAs.

Definition

Conductors and insulators

Substances which allow electric charges to flow through them easily are called good conductors of electricity. They have a large number of free electrons.
Substances which do not allow any electric charges to flow through are called bad conductors of electricity or insulators. They have a very small number of free electrons or no free electrons.