Resistances in Series and Parallel — Revision Notes

⚡ 30-Second Revision

Series: — $I$ is same, $V$ divides. $R_{eq} = R_1 + R_2 + ...$. $V_i = V_{total} \frac{R_i}{R_{eq}}$.
Parallel: — $V$ is same, $I$ divides. $rac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$. For 2 resistors: $R_{eq} = \frac{R_1R_2}{R_1+R_2}$. $I_i = I_{total} \frac{R_{eq}}{R_i}$.
Identical R: — Series $R_{eq} = nR$. Parallel $R_{eq} = R/n$.
Power: — Series $P propto R$ (for constant $I$). Parallel $P propto 1/R$ (for constant $V$).

2-Minute Revision

Resistors combine in two primary ways: series and parallel. In a series combination, resistors are connected end-to-end, creating a single path for current. The key is that the *current is the same* through all resistors, while the *total voltage divides* among them. The equivalent resistance ($R_{eq}$) is simply the sum of individual resistances: $R_{eq} = R_1 + R_2 + ...$. This increases the total resistance. The voltage across any resistor $R_i$ is $V_i = V_{total} \frac{R_i}{R_{eq}}$.

In a parallel combination, resistors are connected across the same two points, offering multiple paths for current. Here, the *voltage is the same* across all resistors, while the *total current divides* among the branches.

The reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances: $rac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$. For two resistors, $R_{eq} = \frac{R_1R_2}{R_1+R_2}$.

This configuration decreases the total resistance. The current through any resistor $R_i$ is $I_i = I_{total} \frac{R_{eq}}{R_i}$. Remember that power dissipation in series is proportional to $R$ ($P propto R$), but in parallel it's inversely proportional to $R$ ($P propto 1/R$).

5-Minute Revision

A thorough understanding of series and parallel resistance combinations is non-negotiable for NEET. Let's consolidate the key aspects.

Series Combination:

Connection: — End-to-end, single current path.
Current: — $I_{total} = I_1 = I_2 = ...$ (Same through all).
Voltage: — $V_{total} = V_1 + V_2 + ...$ (Divides proportionally to resistance).
Equivalent Resistance: — $R_{eq} = R_1 + R_2 + ... + R_n$. Always greater than the largest individual resistance.
Voltage Division Rule: — $V_i = V_{total} \frac{R_i}{R_{eq}}$.
Power Dissipation: — $P = I^2R$. Since $I$ is constant, $P propto R$. Higher resistance dissipates more power.
Example: — Two $5,Omega$ resistors in series. $R_{eq} = 5+5 = 10,Omega$. If $1,\text{A}$ flows, $V_1 = 1 \times 5 = 5,\text{V}$, $V_2 = 1 \times 5 = 5,\text{V}$. Total $V = 10,\text{V}$.

Parallel Combination:

Connection: — Across same two points, multiple current paths.
Voltage: — $V_{total} = V_1 = V_2 = ...$ (Same across all).
Current: — $I_{total} = I_1 + I_2 + ...$ (Divides inversely proportionally to resistance).
Equivalent Resistance: — $rac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$. Always less than the smallest individual resistance. For two resistors: $R_{eq} = \frac{R_1R_2}{R_1+R_2}$.
Current Division Rule: — $I_i = I_{total} \frac{R_{eq}}{R_i}$. For two resistors: $I_1 = I_{total} \frac{R_2}{R_1+R_2}$.
Power Dissipation: — $P = \frac{V^2}{R}$. Since $V$ is constant, $P propto \frac{1}{R}$. Lower resistance dissipates more power.
Example: — Two $10,Omega$ resistors in parallel. $R_{eq} = \frac{10 \times 10}{10+10} = \frac{100}{20} = 5,Omega$. If $10,\text{V}$ is applied, $I_1 = 10/10 = 1,\text{A}$, $I_2 = 10/10 = 1,\text{A}$. Total $I = 2,\text{A}$.

Mixed Circuits: Systematically simplify by identifying the innermost series or parallel combinations, calculating their equivalent resistance, and redrawing the circuit. Repeat until a single equivalent resistance is found. Always be mindful of the specific question: are you finding total resistance, current through a specific branch, or voltage across a component? Use the appropriate formula and rule for each step.

Prelims Revision Notes

For NEET, quick recall of series and parallel resistance properties and formulas is vital.

Series Combination:

Definition: — Resistors connected end-to-end, forming a single path.
Current: — $I_{total} = I_1 = I_2 = ...$ (Current is *same* through all).
Voltage: — $V_{total} = V_1 + V_2 + ...$ (Voltage *divides*).
Equivalent Resistance: — $R_{eq} = R_1 + R_2 + ... + R_n$. This means adding resistors in series *increases* the total resistance.
Voltage Division: — For any resistor $R_i$ in series, V_i = V_{total} left( \frac{R_i}{R_{eq}} \right).
Power Dissipation: — $P = I^2R$. Since $I$ is constant, $P \propto R$. The resistor with higher resistance dissipates more power.
Special Case: — For $n$ identical resistors $R$ in series, $R_{eq} = nR$.

Parallel Combination:

Definition: — Resistors connected across the same two points, providing multiple paths.
Voltage: — $V_{total} = V_1 = V_2 = ...$ (Voltage is *same* across all).
Current: — $I_{total} = I_1 + I_2 + ...$ (Current *divides*).
Equivalent Resistance: — $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$. This means adding resistors in parallel *decreases* the total resistance.
Shortcut for two resistors: — $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$.
Current Division: — For any resistor $R_i$ in parallel, $I_i = I_{total} \left( \frac{R_{eq}}{R_i} \right)$. For two resistors $R_1, R_2$: $I_1 = I_{total} \left( \frac{R_2}{R_1+R_2} \right)$.
Power Dissipation: — $P = \frac{V^2}{R}$. Since $V$ is constant, $P \propto \frac{1}{R}$. The resistor with lower resistance dissipates more power.
Special Case: — For $n$ identical resistors $R$ in parallel, $R_{eq} = \frac{R}{n}$.

Key Strategy for Mixed Circuits: Always simplify step-by-step. Identify the simplest series or parallel groups, calculate their equivalent resistance, and redraw the circuit. Repeat until the entire circuit is reduced to a single equivalent resistance. Be careful with units and calculations, especially fractions for parallel combinations.

Vyyuha Quick Recall

Same In Series, Voltage Divides. Parallel Voltage Same, Inverse Reciprocal Equivalent.