Resistances connection ( series & parallel ) , Electric energy and Electric power

Electric resistances can be connected in the electric circuit using two methods which are series connection , parallel connection , Home appliances are connected in parallel , Thus every device can work individually on the potential difference of the source & does not affect the other devices when switched off or at mal-function , also their equivalent resistance becomes very small & does not weaken the current intensity .

Resistors connection in series

Its purpose : Obtaining big resistance from a group of small resistance where the resistance are considered as a connected path of the electric current .

Method of connection : The electric resistors are connected with a battery , ammeter , voltmeter , rheostat and a switch in an electric circuit .

The electric current intensity : When measuring the electric current intensity passing in all resistors we find it equal .

I = I1 = I2 = I3

Electric potential difference

When measuring the potential difference between the ends of each resistor , we find that the total voltage equals the sum of potential differences on all the resistors in the circuit .

V = V1 = V 2 = V3

Calculating the equivalent resistance ( R‾ ) :

From Ohm’s law :    V = I R

∴ V1 = I R1  ,  V2 = I R2  ,  V3 = I R3

V = V1 = V 2 = V3

  I R‾ = I R1 = I R2 =  I R3

R‾ = R1 + R2 + R3

That means that the equivalent resistance for a group of resistors equals the sum of these resistances , If the resistors connected in series have the same resistance and its number N then :

R‾ = NR
Resistances connection

Resistances connection

Resistors connection in parallel

Its purpose : Obtaining a small resistance from a group of big resistors where the equivalent resistance is less than the smallest resistance in the group .

Method of connection : The electric resistors are connected with a battery , ammeter , voltmeter , rheostat and a switch in an electric circuit .

Electric potential difference : When measuring the potential difference between the ends of each resistor , we find it equal and equals the potential difference between the terminals of the battery :

V = V1 = V 2 = V3

The electric current intensity : When measuring the electric current intensity passing in each resistor , we find that the total current equals the sum of the current intensities passing in all the resistors :

Itotal = I1 + I2 + I3

It is noticed that the electric current is inversely divided on the resistors such that the biggest part of the current passes in the smallest resistor .

Calculating the equivalent resistance ( R‾ ) :

From Ohm’s law :  V = IR 

∴   I1 = V / R1   ,  I2 = V / R2   ,  I3 = V / R

Itotal = I1 + I2 + I3

V / R‾ = V / R1 + V / R2 + V / R3

∴   1 / R‾ = 1 / R1 + 1 / R2 + 1 / R3

That means that the reciprocal of the equivalent resistance for a group of resistors connected in parallel equals the sum of reciprocals of the resistances .

In case of only 2 resistors connected in parallel :  R‾ = R1 R2 / ( R1 + R2 )

If the resistors connected in parallel are all equal and each of value R and their number N , then : 

1 / R‾ = N / R

∴ R‾ = R / N

In the electric circuit connected in parallel thick wires are used at the terminals of the battery , Because the current intensity becomes maximum at the terminals of the battery , thus thick wires are used due to their small resistance which does not affect the intensity of the electric current , but in the other parts of the circuit , the current divides into smaller currents where thinner wires are used .

Electric current intensity of the branch ab :

R‾ = R1 R2 / ( R1 + R2 )

V1 = V 2 = Vab

I1 R1 = I2 R= I R‾

In case that terminals of a resistor are connected together with a conducting wire , its value is ignored when calculating the equivalent resistance because there is no potential difference between its terminals , In case of presence of a connecting wire of no resistance , its ends can be considered one point , In case of equal potential at the terminals of a resistor , then the value of this resistor is neglected when calculating the equivalent resistance .

The electric energy ( W ) 

When the electric current passes in a conductor , part of the electric energy is consumed .

V = W / Q     ,    ∴  W = V Q 

Where : ( W ) is the work done and represents the electric energy .

The consumed electric energy is measured in Joule and it is equivalent to Volt.Coulomb .

W = V Q     

∴ W = V I t = I² R t = V² t / R

The electric power ( Pw )

The electric power ( Pw ) is the consumed electric energy within one second , The electric power is measured in Watt which is equivalent to Joule/sec .

Pw = W / t = V I = I² R = V² / R

To compare between the consumed power in two connected resistors :

In series : Electric current intensity in both of them is the same .

( Pw )1 / ( Pw ) 2 = ( I² R1 / I² R2 ) = R1 / R2

( Pw )1 / ( Pw ) 2 = R1 / R2

In parallel : Electric potential difference at both of them is the same .

( Pw )1 / ( Pw ) 2 = ( V² / R1 ) × ( R2 / V² ) = R2 / R1

( Pw )1 / ( Pw ) 2 = R2 / R1

Electrical current , Potential difference , Electric resistance and Ohm’s law

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