Class 10 > Unit # 15: Current Electricity > Combination of Resistors

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Equivalent Resistance: Series and Parallel Combinations

Q.5 Derive expressions for the equivalent resistance when resistors are connected (i) In Series (ii) In Parallel. Also give advantages and disadvantages of each combination.

(i) Resistances in Series

"Resistors are said to be connected in series if they are joined end-to-end such that there is only one single conducting path for the electric current to flow through all of them."

Consider three resistors having resistances $R_1$, $R_2$, and $R_3$ connected in series across a battery providing a total potential difference $V$. If $I$ is the current flowing from the battery, the same current passes through each resistor.

According to Ohm's Law, the potential drops across individual resistors are:

• Potential drop across $R_1$: $V_1 = I R_1$
• Potential drop across $R_2$: $V_2 = I R_2$
• Potential drop across $R_3$: $V_3 = I R_3$

The total potential difference $V$ across the combination is equal to the sum of individual potential drops:

$V = V_1 + V_2 + V_3 \quad \text{--- (i)}$

If $R_e$ represents the equivalent resistance of the circuit connected across the same battery, then:

$V = I R_e$

Substituting $V_1$, $V_2$, $V_3$, and $V$ into equation (i):

$I R_e = I R_1 + I R_2 + I R_3$

$I R_e = I (R_1 + R_2 + R_3)$

$R_e = R_1 + R_2 + R_3$

Conclusion: In a series combination, the equivalent resistance is equal to the algebraic sum of all individual resistances and is greater than any individual resistance.

Advantages and Disadvantages of Series Combination

Advantages	Disadvantages
Useful when multiple lights or components need to be operated simultaneously. Limits overall circuit current, making it safer in certain low-power setups. Simple wiring makes it easy to control multiple devices with a single master switch.	All appliances share a single switch; individual devices cannot be turned off separately. If one component fails (burns out or opens), the entire circuit breaks and all devices stop working. Voltage is distributed among components, so appliances do not receive full supply voltage.

(ii) Resistances in Parallel

"Resistors are said to be connected in parallel if one end of all resistors is connected to a common point and the other end is connected to another common point, ensuring that the potential difference across each resistor is the same."

Consider three resistors $R_1$, $R_2$, and $R_3$ connected in parallel across a voltage source $V$. The total current $I$ from the battery divides into three branch currents $I_1$, $I_2$, and $I_3$ at junction point $a$.

Applying Ohm's Law to each branch:

$I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_3 = \frac{V}{R_3}$

According to Kirchhoff's Current Law, the total current is equal to the sum of branch currents:

$I = I_1 + I_2 + I_3 \quad \text{--- (ii)}$

If $R_e$ represents the equivalent resistance of the parallel combination, then total current is:

$I = \frac{V}{R_e}$

Substituting $I_1$, $I_2$, $I_3$, and $I$ into equation (ii):

$\frac{V}{R_e} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$

$\frac{V}{R_e} = V \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right)$

$\frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Conclusion: The reciprocal of the equivalent resistance in a parallel combination is equal to the sum of the reciprocals of individual resistances.

Advantages and Disadvantages of Parallel Combination

Advantages	Disadvantages
Each electrical appliance can be turned on or off independently using its own switch. Every appliance receives the full rated voltage of the power supply. If one appliance fails or breaks down, the other appliances continue to function normally.	Because parallel circuits draw higher total current as more devices are added, they can be less safe without proper fusing. Wiring is more complex and requires more copper conductor material.

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