The time rate of flow of charge through any cross-section is called current
If $\Delta$Q be the net charge flowing across a cross section of a conductor during the time interval $\Delta$t, the current across the cross-section of the conductor is defined as the value of the ratio of $\Delta$Q to $\Delta$t in the limit of $\Delta$t tending to zero.
\[I=\lim_{\Delta t\to 0}\frac{\Delta Q}{\Delta t}\]
S.I. of current is Ampere. There are two types of Currents. One Alternating Current (AC) and Direct Current (DC).
Direct Current (DC): Its' magnitude and direction does not change with time. A Cell, Battery and DC Dynamo are sources of DC.
Alternating Current (AC): Its' magnitude and direction changes periodically. AC Dynamo is a source of Alternating Current.
Alternating Current (AC): Its' magnitude and direction changes periodically. AC Dynamo is a source of Alternating Current.
Ohm's Law
Ohm's Law deals with the relationship between voltage
and current in an ideal conductor. This relationship states that:
The potential difference (voltage) across an ideal conductor
is proportional to the current through it.
If V is the potential difference between two points of a
conductor and I is the current flowing through it then
Vα I
⇒V=IR ...... (1)
Where R is the proportionality constant and resistance of
the conductor.
The resistance (R) of a conductor is proportional to its’ length (L).
The resistance (R) of a conductor is inversely proportional to its’ cross sectional area (A).
The resistance (R) of a conductor is proportional to its’ length (L).
R α L …… (2)
The resistance (R) of a conductor is inversely proportional to its’ cross sectional area (A).
R α 1/A …… (3)
By combining (2) and (3),
From Ohm’s Law,
\[R\alpha \frac{L}{A}\]
\[\therefore R=\rho \frac{L}{A} \cdot \cdot \cdot \cdot \cdot \cdot (4)\]
\[\therefore R=\rho \frac{L}{A} \cdot \cdot \cdot \cdot \cdot \cdot (4)\]
Where ρ is the proportionality constant which depends on the material of the conductor but not on its dimensions and ρ is called resistivity.
From Ohm’s Law,
\[V=IR=\frac{I\rho L}{A}\cdot \cdot \cdot \cdot \cdot \cdot (5)\]
If $\vec{J}$ be Current density (where $\vec{J}\cdot \vec{A}=I$ ) and $\vec{E}$ (where $\vec{E}\cdot \vec{L}=V$ ) be the electric field then equation (5) can be rewritten as,
\[\vec{E}= \vec{J}\rho \]
or $\vec{J}=\sigma \vec{E}$ ..... (6)
σ (= 1/ρ ) is the conductivity.
Drift Velocity and Origin of Resistivity
The drift velocity is the average velocity that
a particle, such as an electron, attains in a material due to an electric
field.
A free
electron can move in all directions. Its direction gets change after collision with
heavily fixed electrons but its speed remains same. If we consider all the
electrons, their average velocity will be zero since their directions are
random. Thus, if there are n electrons and the velocity of the ith electron
(i = 1, 2, 3, ... n) at a given time is ui, then
\[\frac{1}{n}\sum_{i=1}^{n}\vec{u_{i}}=0\cdot \cdot \cdot \cdot \cdot \cdot (7)\]
If an electric field (E) is present the electron will be
accelerated due to this field, i.e. ma=-e| $\vec{E}$|
\[\Rightarrow a=-\frac{e\left | \vec {E} \right |}{m}\cdot \cdot \cdot \cdot \cdot \cdot (8)\]
Where –e is the
charge and m is the mass of an electron. Let us consider ti be elapse
time after the last collision of ith electron and ui was
the velocity of it just after the last collision. Thus its velocity Vi
is,
Vi=ui + ati
Averaging Eq. (9) over the n-electrons at any given time t gives
us for the average velocity with vector notation $\vec{v_{d}}$
\[\vec{v_{d}}=\frac{1}{n}\sum_{i=1}^{n}\vec{u_{i}}-\frac{e\vec{E}}{m}\frac{1}{n}\sum_{i=1}^{n}t_{i}\]
\[\vec{v_{d}}=-\frac{e\vec{E}}{m}\tau \cdot \cdot \cdot \cdot \cdot \cdot (10)\]
\[\vec{v_{d}}=-\frac{e\vec{E}}{m}\tau \cdot \cdot \cdot \cdot \cdot \cdot (10)\]
where  $\frac{1}{n}\sum_{i=1}^{n}\vec{u_{i}}=0$ and $\frac{1}{n}\sum_{i=1}^{n}t_{i}=\tau$;
The velocity $\vec{v_{d}}$ in Eq. (10) is called the drift velocity.
Now Consider a planar area A,
located inside the conductor such that the normal to the area is parallel to $\vec{E}$.
In an infinitesimal amount of
time $\Delta t$ , all free electrons to
the left of the area at distances upto $\left| \vec{v_{d}}\right |\Delta t$ . If n is the number of free
electrons per unit volume in the metal, then there are $nA\left | \vec{v_{d}}\right |\Delta t$ such electrons.
Since each electron carries a
charge –e, the total charge transported across this area A in time $\Delta t$ is – $neA\left | \vec{v_{d}}\right |\Delta t$ . If I be the current
then by definition,
\[I\Delta t=+neA\left | v_{d} \right |\Delta t \cdot \cdot \cdot \cdot \cdot \cdot (11)\]
Substituting the value of | $\vec{v_{d}}$| from Eq. (10)
\[I\Delta t=\frac{e^{2}A}{m}\tau n\Delta t\left | \vec{E} \right |\cdot \cdot \cdot \cdot \cdot (12)\]
Using Current density | $\vec{J}$|=I/A
\[\left |\vec{J} \right |=\frac{ne^{2}}{m}\tau \left | \vec{E} \right |\cdot \cdot \cdot \cdot \cdot (13)\]
$\vec{J}$ and $\vec{E}$ are parallel, thus
\[\vec{J} =\frac{ne^{2}}{m}\tau \vec{E} \cdot \cdot \cdot \cdot \cdot (14)\]
Comparing Eqn (6) and (14) we get $\sigma =\frac{ne^{2}}{m}\tau$ . Thus we reproduce Ohm's Law.
Mobility: Drift velocity per unit electric field is called mobility of electron.
\[i.e. \mu =\frac{\left | \vec{v_{d}}\right |}{\left |\vec{E} \right |}\]
\[\Rightarrow \mu =\frac{e\tau }{m}\]
SI unit of mobility is Sq. Metre/ Volt-Sec.
