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. $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$. 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: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$. 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: — $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$. 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.