Factors affecting of resistances, Combination Of Series and Paralell

Electric resistance of a conductor (or a wire) depends on the following factors

Length of the conductor: -
Resistance of a conductor is directly proportional to its length. So, when length of the wire is doubled, its resistance also gets doubled; and if length of the wire is halved its resistance also gets halved.
Thus a long wire has more resistance then a short wire.
Area of cross-section:-
Resistance of a conductor is inversely proportional to its area of cross-section. So, when the area of cross-section of a wire is doubled, its resistance gets halved; and if the area of cross-section of wire is halved then its resistance will get doubled.
Thus a thick wire has less resistance and a thin wire has more resistance.
3. Nature of material of conductor:-
Electrical resistance of a conductor also depends on the nature of the material of which it is made. For example a copper wire has less resistance then a nichrome wire of same length and area of cross-section.
4. Effect of temperature:-
It has been found that the resistance of all pure metals increases on raising the temperature and decreases on lowering the temperature.
Resistance of alloys like manganin, nichrome and constantan remains unaffected by temperature.

Note ** Resistance is simply a difficulty faced by an electron while it jumps from one atom to another in order to reach from end of wire to another end. That difficulty is known as Resistance or (opposition)

Resistance of a system of resistors

We know that current through a conductor depends upon its resistance and potential difference across its ends.
In various electrical instruments resistors are often used in various combinations and Ohm’s Law can be applied to combination of resistors to find the equivalent resistance of the combination.
The resistances can be combined in two ways
1. In series
2. In parallel
To increase the resistance individual resistances are connected in series combination and to decrease the resistance individual resistances are connected in parallel combination.

(a) Resistors in Series

When two or more resistances are connected end to end then they are said to be connected in series combination.
Figure below shows a circuit diagram where two resistors are connected in series combination.
Now value of current in the ammeter is the same irrespective of its position in the circuit. So we conclude that " For a series combination of resistors the current is same in every part of the circuit or same current flows through each resistor "
Again if we connect three voltmeters one across each resistor as shown below in the figure 4.The potential difference measured by voltmeter across each one of resistors R₁ , R₂ and R₃ is V₁ , V₂ and V₃ respectively and if we add all these potential differences then we get

This total potential difference V in equation 6 is measured to be equal to potential difference measured across points X and Y that is across all the three resistors in figure 3. So, we conclude that <"the total potential difference across a combination of resistors in series is equal to the sum of potential differences across the individual resistors."
Again consider figure 4 where I is the current flowing through the circuit which is also the current through each resistor. If we replace three resistors joined in series by an equivalent single resistor of resistance R such that, the potential difference V across it, and the current I through the circuit remains same.
Now applying Ohm’s law to entire circuit we get<

On applying Ohm's law to the three resistors separately we have,<

From equation 6

So here from above equation 9 we conclude that when several resistances are connected in series combination, the equivalent resistance equals the sum of their individual resistances and is thus greater than any individual resistance.

(b) Resistors in parallel

When two or more resistances are connected between the same two points they are said to be connected in parallel combination.
Figure below shows a circuit diagram where two resistors are connected in parallel combination.

IMPORTANT NOTE
1. When a number of resistors are connected in parallel, then the potential difference across each resistance is equal to the voltage of the battery applied.
2. When a number of resistances are connected in parallel, then the sum of the currents flowing through all the resistances is equal to total current flowing in the circuit.
3. When numbers of resistances are connected in parallel then their combined resistance is less than the smallest individual resistance. This happens because the same current gets additional paths to flow resulting decrease in overall resistance of the circuit
To calculate the equivalent resistance of the circuit shown in figure 5 consider a battery B which is connected across parallel combination of resistors so as to maintain potential difference V across each resistor. Then total current in the circuit would be

Since potential difference across each resistors is V. Therefore, on applying Ohm's Law

Putting these values of current in equation 10 we have

If R is the equivalent resistance of parallel combination of three resistors heaving resistances R₁, R₂ and R₃ then from Ohm's Law

Or,

Comparing equation (10) and (11) we get
For resistors connected in parallel combination reciprocal of equivalent resistance is equal to the sum of reciprocal of individual resistances.
Value of equivalent resistances for capacitors connected in parallel combination is always less than the value of the smallest resistance in circuit.