

PHYSICS12 COMPLETE SYLLABUS
Chapter 9  Direct Current Circuit
Metallic Conduction
In conductor there is a presence of free electron which are in random or zigzag motion. So the electron
which are present at one end of the conductor cannot
reached to the other end in the absence of electric field.
So, conductor are neutral. In this case, the free electron are equilibrium in three dimensional
crystallatice unit.
When the electric field is supplied, the free electron get accelerated and collides with the other atom.
Due to this successive collision the free electron
losses their kinetic energy and the electron get allign in the same direction and the free electron have
different kinetic energy. Thus, the average velocity
of the free electron (charge carrier) in the presence of electric field, whose direction is opposite to
the applied electric field is called drift velocity of
the electron.
Relation between drift velocity and electric current
Let us consider a conductor PQ in the form of cylinder of crosssectional area 'A' and length 'l'. When
the given conducor is connected to a battery or source, the free electron drift in the opposite
direction
to the field 'E' with a velocity v_{d}. Let, 'n' be the free electron density (number of free
electron per unit volume). The
current 'I' passed through the conductor 'PQ'.
The total number of free electron present in the conductor (N)= electron density x volume
= n x V
n x A x l
therefore, the total charge present in the conductor,
Q= Ne
n x A x l x e
The electric current (I)= charge (Q)/time (t)
or, Q/t = (n x A x l x e)/t
or, I= v_{d}enA
v_{d}= l/t
I= Q/t
This is the required relation between elecric current and drift velocity.
Current density: ()
The current density on a point in a conductor is defined as the electric current 'I' per unit area
'A'.
i.e. J= I/A , it is a vector quantity
Unit: AM^{2}
Dimensional Formula: [AL^{2}]
* Characteristics of current density:
1. Current density is the characteristics of a point not the whole conductor.
2. If a conductor of a small area 'A' makes an angle θ to the direction of electric current then,
I= .
I= JA cosθ
J= I/Acosθ
Relation between current density and drift velocity:
we have,
J= I/A
J= v_{d}enA / A
J= v_{d}en
Grouping of a resistor
1. Series combination of a resistor
i. resistor connected end to end.
ii. Current through each and every resistor is same but potential difference p.d. across them are
different.
We consider three resistor of resistance R_{1}, R_{2} and R_{3} connected in
series with the external source of potential
difference V. V_{1}, V_{2},V_{3} are the potential drop accross the respective
resistor. Let, I be the current through the circuit.
We have,
V= V_{1}+V_{2}+V_{3}
< (i)
For a resistor 'R_{1}' using ohm's law,
V_{1}= IR_{1}
Similarly,
V_{2}= IR_{2}
V_{3}= IR_{3}
From equation (i) we get,
V= IR_{1}+IR_{2}+IR_{3}
V= I(R_{1}+R_{2}+R_{3})
V/I= R_{1}+R_{2}+R_{3}
< (ii)
Let, R_{s} be the equivalent resistance of the series combination. Then from ohm's law,
V= IR_{s}
V/I= R_{s}
<(iii)
From equation (ii) and (iii) we get,
R_{s}= R_{1}+R_{2}+R_{3}
which gives the equivalent resistance of the series combination.
The equivalent resistance is equal to the sum of the resistance of individual resistor
connected in series.
*** Point 1: If 'n' number of identical resistor of resistance 'R' are connected in series,
then
R_{s}= nR
R_{max}= nR
2. Parallel combination of resistor
i. Common terminal
ii. Potential difference across each and every resistor are same but current through them
are different.
Let us consider three resistor of Resistance R_{1}, R_{2} and R_{3}
are connected to a external source of potential difference p.d. V by making common terminal
A and B
as shown in figure. Let I be the current through the circuit and I_{1},
I_{2}, I_{3}
are current through the respective reisitance.
Now, we have
I= I_{1}+I_{2}+I_{3}
< (i)
But the potential difference across each resistance is the same equal to the voltage V
of battery. So,
from ohm's law we have,
I_{1}=V/R_{1}
I_{2}=V/R_{2}
I_{3}=V/R_{3}
Putting these value in equation (i)
I=V/R_{1}+V/R_{2}+V/R_{3}
therefore, I=V[(1/R_{1})+(1/R_{2})+(1/R_{3})] < (ii)
If parallel combination of resistors R_{1}, R_{2}, R_{3} is
replaced by a resistor of resistance R in
such a way that the same current I flows through it when the same potential
difference V is applied across it, then
we can write,
I=V/R < (iii)
Here, R is called equivalent resistance of R_{1}, R_{2},
R_{3}.
From equation (ii) and (iii) we have,
V/R = V[(1/R_{1})+(1/R_{2})+(1/R_{3})]
1/R= (1/R_{1})+(1/R_{2})+(1/R_{3})
In general, for 'n' resistors in parallel, we have,
1/R=
(1/R_{1})+(1/R_{2})+(1/R_{3})+.......+1/R_{n}
< (iv)
Thus, if resistors are connected in parallel, then reciprocal of equivalent
resistance is equal to the sum of
reciprocal of individual resistances.
From equation (iv), it is clear that equivalent resistance in parallel
combination of resistances is always less than individual
resistance. To decrease the resistancein the circuit, resistors are joined
in parallel. If we consider
'n' number of resistors of equal resistance then,
1/R_{eq}= (1/R)+(1/R)+(1/R)+....... upto 'n' number
or, 1/R_{eq}= n/R
therefore, R_{eq}= R/n (minimum)
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