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Derive an expression for the current in a circuit with external resistance R when

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Derive an expression for the current in a circuit with external resistance R when (a) n identical cells of emf E and internal resistance r are connected in series (b) m identical cells are connected in parallel
asked Jul 25, 2014 in Physics by sohandash (240 points)

1 Answer

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Please refer to 

http://www.electrical4u.com/series-parallel-battery-cells/

(A) Series combination

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

  • Figure below shows the two cells of emf's E1 and E2 and internal resistance r1 and r2 respectively connected in series combination through external resistance

    image 
     
  • Points A and B in the circuit acts as two terminals of the combination
  • Applying kirchoff's loop rule to above closed circuit
    -Ir2-Ir1-IR+E1+E2=0
    or
    I=E1+E2/R+(r1+r2)
    Where I is the current flowing through the external resistance R
  • Let total internal resistance of the combination by r=r1+r2 and also let E=E1+E2 is the total EMF of the two cells
  • Thus this combination of two cells acts as a cell of emf E=E1+E2 having total internal resistance r=r1+r2 as shown above in the figure

    image

(B) Parallel combinations of cells

  • Figure below shows the two cells of emf E1 and E2 and internal resistance r1 and r2 respectively connected in parallel combination through external resistance 
    image 
     
  • Applying kirchoff's loop rule in loop containing E1 ,r1 and R,we find
    E1-IR-I1r1=0 ------------------------(1)
    Similarly applying kirchoff's loop rule in loop containing E2 ,r2 and R,we find
    E2-IR-(I-I1)r2=0 ------------------------(2)
  • Now we have to solve equation 1 and 2 for the value of I,So multiplying 1 by r2 and 2 by r1 and then adding these equations results in following equation
    IR(r1+r2)+r2r1I-E1r2-E2r1=0
    which gives

    image 
    We can rewrite this as 
    image 
    E is the resulting EMF due to parallel combination of cells and r is resulting internal resistance. 
answered Jul 28, 2014 by admin (4,450 points)
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