JEE Physics/ Lakshmikanta Satapathy/ Direct current Theory part 4/ Equivalent Resistance of a series and parallel combination of three resistances

1. Physics Helpline
L K Satapathy
Direct Current Theory - 4
Resistances in Series
1 2 3sR R R R  
Resistances in Parallel
1 2 3
1 1 1 1
PR R R R
  

2. Physics Helpline
L K Satapathy
Cells in Parallel
Direct Current Theory - 4
Resistances in Series :
The figure shows the combination of three resistors R1 , R2 and R3 in series
E
I
R1 R2 R3
V1 V2 V3
Two or more resistances are said to be connected in
series, if the same current passes through each
resistance, when some potential difference is applied
across the combination.
Let a DC source of emf E is connected across the combination
Due to this the same current I flows through each of them as shown
Let V1, V2 and V3 be the PD across resistors R1 , R2 and R3 respectively
Then according to Ohm’s law , we have
1 1 2 2 3 3, . . . (1)V IR V IR and V IR  

3. Physics Helpline
L K Satapathy
Cells in Parallel
Direct Current Theory - 4
The PD across the combination is given by
1 2 3 1 2 3E V V V IR IR IR     
1 2 3( ) . . . (2)E I R R R   
Let RS be the resistance of the series combination and current through it is I
. . . (3)sE IR 
1 2 3(2)&(3) ( )sIR I R R R    
1 2 3sR R R R   
 Equivalent resistance of a series combination > the individual resistances
 Current is same through all the resistors in series
 Total resistance of the circuit is equal to the sum of the individual resistances
 Voltage across any part  resistance of that part

4. Physics Helpline
L K Satapathy
Cells in Parallel
Direct Current Theory - 4
Resistances in Parallel :
E
I
R1
R2
R3
I1
I2
I3
Two or more resistors are said to be connected in parallel, if
PD across each of them is equal to the applied PD. Consider
three resistors R1 , R2 and R3 connected in parallel as shown in
figure. A cell of emf E is connected across the combination. If
I1 , I2 and I3 be the currents through the resistors R1 , R2 and
R3 respectively, then the total current in the circuit is given by
Applying Ohm’s Law , we get
1 2 3 . . . (4)I I I I  
1 1 2 2 3 3E I R I R I R  
1 2 3
1 2 3
, , . . . (5)
E E E
I I I
R R R
   
1 2 3
1 2 3 1 2 3
1 1 1
(4) . . . (6)
E E E
I I I I E
R R R R R R
 
           
 

5. Physics Helpline
L K Satapathy
Cells in Parallel
Direct Current Theory - 4
If RP be the resistance of the combination , then
. . . (7)P
P
E
E IR I
R
  
1 2 3 1 2 3
1 1 1 1 1 1 1
(6)&(7)
P P
E
E
R R R R R R R R
 
         
 
 Resistance of a parallel combination < the individual resistances
 Total current = sum of individual currents
 PD across each resistance = PD across the combination
 Current through each branch is inversely proportional to the resistance of
that branch

6. Physics Helpline
L K Satapathy
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