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Chapter 9 - Direct Current Circuit

Metallic Conduction

In conductor there is a presence of free electron which are in random or zig-zag 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 cross-sectional 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 vd. 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= vdenA
vd= 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= vdenA / A
J= vden

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 R1, R2 and R3 connected in series with the external source of potential difference V. V1, V2,V3 are the potential drop accross the respective resistor. Let, I be the current through the circuit.

We have,
V= V1+V2+V3 <-------- (i)
For a resistor 'R1' using ohm's law,
V1= IR1
V2= IR2
V3= IR3
From equation (i) we get,
V= IR1+IR2+IR3
V= I(R1+R2+R3)
V/I= R1+R2+R3 <------- (ii)
Let, Rs be the equivalent resistance of the series combination. Then from ohm's law,
V= IRs
V/I= Rs <-----(iii)

From equation (ii) and (iii) we get,
Rs= R1+R2+R3
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
Rs= nR
Rmax= 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 R1, R2 and R3 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 I1, I2, I3 are current through the respective reisitance.

Now, we have
I= I1+I2+I3 <------- (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,
Putting these value in equation (i)
therefore, I=V[(1/R1)+(1/R2)+(1/R3)] <------- (ii)
If parallel combination of resistors R1, R2, R3 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 R1, R2, R3.

From equation (ii) and (iii) we have,
V/R = V[(1/R1)+(1/R2)+(1/R3)]
1/R= (1/R1)+(1/R2)+(1/R3)
In general, for 'n' resistors in parallel, we have,
1/R= (1/R1)+(1/R2)+(1/R3)+.......+1/Rn <------- (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/Req= (1/R)+(1/R)+(1/R)+....... upto 'n' number
or, 1/Req= n/R
therefore, Req= R/n (minimum)

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