Question

With the help of a circuit diagram derive the formula for the resultant resistance of three resistances connected:

1. in series and
2. in parallel

Answer 1

• Resistors in Series:
  A series circuit connects the components one after the other to form a ‘single-loop. A series circuit has only one loop through which current can pass. If the circuit is interrupted at any point in the loop, no current can pass through the circuit, and hence no electric appliances connected in the circuit will work. Series circuits are commonly used in devices such as flashlights. Thus, if

1. Resistors are connected end to end so that the same current passes through each of them, then they are said to be connected in series.
   

2. Let, three resistances R₁, R_2, and R₃ be connected in series.

3. Let the current flowing through them be I.

4. According to Ohm’s Law, the potential differences V₁, V_2, and V₃ across R₁, R2, and R₃ respectively, are given by:
   V₁ = IR₁ …. (1)
   V₂ = IR₂ …. (2)
   V₃ = IR₃ …. (3)
   The sum of the potential differences across the ends of each resistor is given by:
   V = V₁ + V₂ + V₃
   using equations (1), (2), and (3), we get
   V = IR₁ + IR₂ + IR₃ …. (4)

5. The effective resistor is a single resistor, which can replace the resistors effectively, so as to allow the same current through the electric circuit.

6. Let, the effective resistance of the series-combination of the resistors, be R_S. Then,
   V = IR_S ….(5)
   Combining equations (4) and (5)
   IR_S = IR₁ + IR₂ + IR₃
   R_S = R₁ + R₂ + R₃
   Thus, we can understand that

7. When a number of resistors are connected in series, their equivalent resistance or effective resistance is equal to the sum of the individual resistances.

8. When V resistors of equal resistance R are connected in series, the equivalent resistance is ‘nR’. i.e., R_S = nR

9. The equivalent resistance in a series combination is greater than the highest of the individual resistances.

• Resistors in Parallel:
  A parallel circuit has two or more loops through which current can pass. If the circuit is disconnected in one of the loops, the current can still pass through the other loop(s). The wiring in a house consists of parallel circuits.
  

1. Consider that three resistors R₁, R_2, and R₃ are connected across two common points A and B.

2. The potential difference across each resistance is the same and equal to the potential difference between A and B. This is measured using the voltmeter.

3. The current I arriving at A divides into three branches I₁, I_2, and I₃ passing through R₁, R_2, and R₃ respectively.
   According to Ohm’s law, we have,
   `"I"_1 = "V"/"R"_1`    ....(1)
   `"I"_2 = "V"/"R"_2`    ....(2)
   `"I"_3 = "V"/"R"_3`    ....(3)
   The total current through the circuit is given by
   I = I₁ + I₂ + I₃
   Using equations (1), (2) and (3), we get
   I = `"V"/"R"_1 + "V"/"R"_2 + "V"/"R"_3`     ...(4)
   Let the effective resistance of the parallel combination of resistors be R_P Then,
   I = `"V"/"R"_"P"`   ...(5)
   Combining equations (4) and (5), we have
   `"V"/"R"_"P" = "V"/"R"_1 + "V"/"R"_2 + "V"/"R"_3`
   `1/"R"_"P" = 1/"R"_1 + 1/"R"_2 + 1/"R"_3`

4. When a number of resistors are connected in parallel, the sum of the reciprocals of the individual resistances is equal to the reciprocal of the effective or equivalent resistance.

5. When ‘n’ resistors of equal resistances R are connected in parallel, the equivalent resistance is `"R"/"n"`  

   `1/"R"_"P" = 1/"R" + 1/"R" + 1/"R" ... + 1/"R" = "n"/"R"`

   Hence, `"R"_"P" = "R"/"n"`

6. The equivalent resistance in a parallel combination is less than the lowest of the individual resistances.