How are resistances connected in series? What are their characteristics? Derive the formula for equivalent resistance.

When resistors are connected end-to-end in a single path for the flow of current, they are said to be in series. In this type of connection, the current remains the same through all the resistors, but the voltage divides among them.

Characteristics of Series Resistance Connection

1. Same Current: The same current flows through all the resistors.
2. Voltage Division: The total voltage across the series combination is equal to the sum of the voltage drops across individual resistors.
3. Increased Resistance: The total (equivalent) resistance is the sum of the individual resistances, making the overall resistance larger.
4. Failure Effect: If one resistor fails (i.e., breaks the circuit), the entire circuit stops functioning.

Derivation of Equivalent Resistance Formula

Consider three resistors R1,R2,R3R_1, R_2, R_3R1​,R2​,R3​ connected in series with a voltage source $V$. The same current $I$ flows through each resistor.

Step 1: Apply Ohm’s Law to Each Resistor

From Ohm’s Law:

V=IR

For individual resistors:

V1=IR1,V2=IR2,V3=IR3​

Step 2: Total Voltage

The total voltage across the series combination is the sum of the individual voltage drops:

V=V1+V2+V3​

Substituting values:

V=IR1+IR2+IR3​

Step 3: Define Equivalent Resistance

If Req​ is the equivalent resistance of the series combination, we write:

V=IReq​

Since the total voltage is the sum of individual voltages:

IReq=IR1+IR2+IR3​

Step 4: Cancel Current (Since $I$ is common)

Req=R1+R2+R3​

General Formula

For $n$ resistors connected in series:

Req=R1+R2+R3+⋯+Rn​

This shows that the equivalent resistance of resistors in series is simply the sum of their individual resistances.