How to Calculate Equivalent Resistance

Key Concepts

• Net resistance in series connection
• Net resistance in parallel connection

Introduction: 

Overall current is less as the total resistance increases in series connection and overall current is more as the total resistance decreases in parallel connection. In this section, we are going to derive the formula to calculate the net resistance in a series connection and a parallel connection. We are also going to calculate the voltage and the current in each circuit.  

Explanation: 

Series connection of resistors: 

When the resistors are connected such that one end of a resistor is connected to an end of another and so on, then the circuit formed is a series connection of resistors.  

In this circuit, the same amount of current flows through different resistors and the voltage across each is different. If any of the resistors stop working, then the whole circuit breaks.  

In the above circuit, the total resistance is given as  

R = R₁ + R₂ +……….+ R_n 

Now since the same current flows through each resistor, we will multiply the whole equation with current, I.  

I × R = (I × R₁) + (I × R₂) + …….. + (I × R_n) 

Þ V = V₁ + V₂ + …….. + V_n 

Hence, in a series connection of resistors,  

R = R₁ + R₂ +……….+ R_n 

I =

12𝟏𝟐

V = V₁ + V₂ + …….. + V_n  

Parallel connection of resistors: 

When the resistors are connected such that one end of all resistors is connected to a single point and similarly, the other end of all resistors is connected to a single point, then the circuit formed is a parallel connection of resistors.  

In this circuit, the voltage is the same across all resistors and the current branches out when it enters one point of connection and recombines when it leaves the other end. If any of the resistor stops working, the whole circuit is not affected by it.  

In the above circuit, the total resistance is given as  

1R=1R1+1R2+ . . .  . . + 1Rn𝟏𝑹=𝟏𝑹𝟏+𝟏𝑹𝟐+ . . .  . . + 𝟏𝑹𝒏

Now since the voltage is the same across each resistor, we will multiply the whole equation with voltage, V. 

(V×1R)=VR1+VR2+ . . .  . . + VRn𝑽×𝟏𝑹=𝑽𝑹𝟏+𝑽𝑹𝟐+ . . .  . . + 𝑽𝑹𝒏

Þ I = I₁ + I₂ + …….. + I_n 

Hence, in a parallel connection of resistors,  

1R=1R1+1R2+ . . .  . . + 1Rn𝟏𝑹=𝟏𝑹𝟏+𝟏𝑹𝟐+ . . .  . . + 𝟏𝑹𝒏

V = same 

I = I₁ + I₂ + …….. + I_n 

Questions and answers 

Question 1: Find out the net resistance, current and voltage across each resistor. 

Answer: 

Question 2: Find out the net resistance, current and voltage across each resistor. 

 Figure 4: Question 2 figure 

Answer: 

Answer:  

Summary

Series connection:

1. Current is same across each resistor.
2. Net resistance is the sum of each resistance in the electric circuit.
3. Sum of voltage across each resistance is equal to net voltage.
4. Net resistance is more than each resistance.

Parallel Connection:

1. Voltage is same across each resistor
2. Net resistance is the reciprocal of the sum of reciprocal of each resistance.
3. Sum of current across each resistance is equal to net current flowing through the electric circuit.
4. Net resistance is less than each resistance.