Electrical Circuits: System of Resistors in Series and Parallel Configurations

• System of Resistors: Ohm’s law can be applied to combinations of resistors. There are two methods of joining the resistors together.

System of Resistors in Series Circuits

• An electric circuit in which resistors having resistances R1, R2 and R3, respectively , are joined end to end is said to be a system of resistors in series.  
• In a system of resistors in series combination of resistors the current is the same in every part of the circuit or the same current through each resistor. 
• The total potential difference across a system of resistors in series is equal to the sum of potential difference across the individual resistors.

V = V1 + V2 + V3 

Let I be the current through the circuit. The current through each resistor is also I.

By applying the Ohm’s law to the entire circuit, we have

V = I R 

On applying Ohm’s law to the three resistors separately, we further have

V1 = I R1 

V2 = I R2 and

V3 = I R3 

Since, V = V1 + V2 + V3 

So, I R = I R1 + I R2 + I R3 

Or Rs = R1 + R2 + R3 

• Thus, when several resistors are joined in a system of resistors  in series, the resistance of the combination Rs equals the sum of their individual resistances, R1 , R2 , R3 and is thus greater than any individual resistance. 
• Disadvantage of a series circuit: In a series circuit the current is constant throughout the electric circuit. 
  • Thus it is impracticable to connect an electric bulb and an electric heater in series, because they need currents of widely different values to operate properly. 
  • Another major disadvantage of a series circuit is that when one component fails the circuit is broken and none of the components works.

System of Resistors in Parallel: Exploring Current Division and Reduced Resistance in Electrical Circuits

• When the arrangement of three resistors  in a system of resistors joined in parallel with a battery, it is observed that the total current I, is equal to the sum of the separate currents through each branch of the combination.

              I = I1 + I2 + I3 

• Let Rp be the equivalent resistance of the parallel combination of resistors.
• By applying Ohm’s law to the parallel combination of resistors, we have

               I =  V/Rp

On applying Ohm’s law to each resistor, we have

I1 =   V/R1

I2 =  V/R2  and

I3 =  V/R3

Since I = I1 + I2 + I3 

So, V/RP  = V/R1 + V/R2  +  V/R3

Or,  1/RP = 1/R1 + 1/R2  + 1/R3  

• Thus, the reciprocal of the equivalent resistance of a group of resistances joined in parallel is equal to the sum of the reciprocals of the individual resistances. 
• Advantage: A system of resistors in parallel circuits divides the current through the electrical gadgets. The total resistance in a parallel circuit is decreased. 
  • This is helpful particularly when each gadget  in a system of resistors has different resistance and requires different current to operate properly.

System of Resistors and Electric Power: Understanding Wattage, Kilowatts, and Energy Consumption in Electrical Circuits

• The rate of doing work is power. This is also the rate of consumption of energy. 
• The power P is given by

        P = VI

       Or P = I2R = V2/R  

• The SI unit of electric power is watt (W).
• One Watt is the power consumed by a device that carries 1 A of current when operated at a potential difference of 1 V. Thus,

              1 W = 1 volt × 1 ampere = 1 V A  

• The unit ‘watt’ is very small to use. Therefore, we use a much larger unit called ‘kilowatt’ (= 1000 watts).
• Since electrical energy is the product of power and time, the unit of electric energy is, therefore, watt hour (W h).
• One watt hour is the energy consumed when 1 watt of power is used for 1 hour. 
• The commercial unit of electric energy is kilowatt hour (kW h), commonly known as ‘unit’.

               1 kW h = 1000 watt × 3600 second

                            = 3.6 × 106 watt second

                            = 3.6 × 106 joule (J) 

Conclusion
Studying Ohm’s Law and electric circuits with a system of resistors offers insights into efficient current distribution. Whether in series or parallel, this systematic arrangement optimizes device performance. Understanding power consumption and energy efficiency becomes key for effective design and utilization.