Equivalent Resistance — Revision Notes

⚡ 30-Second Revision

Series: — $R_{eq} = R_1 + R_2 + ... + R_n$ (Current same, Voltage divides)
Parallel: — $rac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$ (Voltage same, Current divides)
Two Parallel Resistors: — $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$
n Identical Resistors (R) in Series: — $R_{eq} = nR$
n Identical Resistors (R) in Parallel: — $R_{eq} = \frac{R}{n}$
Wheatstone Bridge Balanced: — $rac{R_1}{R_2} = \frac{R_3}{R_4}$ (Middle arm ignored)

2-Minute Revision

Equivalent resistance ($R_{eq}$) simplifies complex resistor networks into a single resistance. For resistors in series, current is constant, voltage divides, and $R_{eq}$ is the sum of individual resistances ($R_{eq} = R_1 + R_2 + ...

$). This increases the total resistance. For resistors in parallel, voltage is constant, current divides, and the reciprocal of$R_{eq}$is the sum of reciprocals ($rac{1}{R_{eq}} = rac{1}{R_1} + rac{1}{R_2} + ...

$). This decreases the total resistance, making$R_{eq}$smaller than the smallest individual resistance. For two parallel resistors, use the product-over-sum rule:$R_{eq} = rac{R_1 R_2}{R_1 + R_2}$.

Mixed circuits are solved by step-by-step reduction, simplifying innermost series/parallel combinations first. Recognize Wheatstone bridges; if balanced, the central resistor can be removed for calculation.

Always trace current paths and identify common potential points to correctly classify connections.

5-Minute Revision

Equivalent resistance is a crucial concept for simplifying circuits. When resistors are connected in series, they form a single path for current. The current ($I$) through each resistor is the same, but the total voltage ($V_{total}$) is divided among them ($V_{total} = V_1 + V_2 + ...

$). The equivalent resistance is the direct sum:$R_{eq} = R_1 + R_2 + ... + R_n$. This means adding resistors in series always increases the total resistance. For example, if$2,Omega$and$3,Omega$are in series,$R_{eq} = 5,Omega$.

When resistors are connected in parallel, they share the same two connection points, meaning the voltage ($V$) across each resistor is identical. The total current ($I_{total}$) splits among the branches ($I_{total} = I_1 + I_2 + ...

$). The reciprocal of the equivalent resistance is the sum of the reciprocals:$rac{1}{R_{eq}} = rac{1}{R_1} + rac{1}{R_2} + ... + rac{1}{R_n}$. For two resistors, the shortcut is$R_{eq} = rac{R_1 R_2}{R_1 + R_2}$.

Adding resistors in parallel always decreases the total resistance, making $R_{eq}$ smaller than the smallest individual resistance. For example, if $6,Omega$ and $3,Omega$ are in parallel, $R_{eq} = \frac{6 \times 3}{6+3} = \frac{18}{9} = 2,Omega$.

For mixed combinations, simplify the circuit step-by-step. Start with the simplest series or parallel groups, calculate their $R_{eq}$, and redraw the circuit. Repeat until a single equivalent resistance remains.

Pay attention to Wheatstone bridges; if balanced ($rac{R_1}{R_2} = \frac{R_3}{R_4}$), the resistor in the middle arm carries no current and can be ignored. Also, look for symmetry to identify equipotential points.

Remember that a short circuit (zero resistance wire) in parallel with a resistor effectively removes that resistor from the circuit, while an open circuit (break) means infinite resistance in that path.

Always ensure your final $R_{eq}$ is physically logical (e.g., parallel $R_{eq}$ must be less than the smallest component resistance).

Prelims Revision Notes

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Definition: — Equivalent resistance ($R_{eq}$) is the single resistance that can replace a network of resistors, drawing the same total current from a source for a given voltage. It simplifies circuit analysis.
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Resistors in Series:

* Connection: End-to-end, single path for current. * Current: Same through all resistors ($I_{total} = I_1 = I_2 = ...$). * Voltage: Divides across resistors ($V_{total} = V_1 + V_2 + ...$). * Formula: $R_{eq} = R_1 + R_2 + ... + R_n$. * Effect: $R_{eq}$ is always greater than the largest individual resistance. Increases total resistance.

1
Resistors in Parallel:

* Connection: Terminals connected to same two points, multiple paths for current. * Voltage: Same across all resistors ($V_{total} = V_1 = V_2 = ...$). * Current: Divides among branches ($I_{total} = I_1 + I_2 + ...$). * Formula: $rac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$. * Shortcut (Two Resistors): $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$. * Effect: $R_{eq}$ is always smaller than the smallest individual resistance. Decreases total resistance.

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Identical Resistors:

* 'n' resistors of value 'R' in series: $R_{eq} = nR$. * 'n' resistors of value 'R' in parallel: $R_{eq} = \frac{R}{n}$.

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Solving Mixed Circuits:

* Simplify step-by-step: Identify innermost series/parallel combinations. * Calculate their $R_{eq}$ and redraw the circuit. * Repeat until a single $R_{eq}$ is found.

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Wheatstone Bridge:

* Configuration: Four resistors in a diamond shape with a fifth resistor/galvanometer across the middle. * Balanced Condition: $rac{R_1}{R_2} = \frac{R_3}{R_4}$. If balanced, no current flows through the middle arm, and it can be removed for $R_{eq}$ calculation (then it becomes two parallel branches, each with two series resistors).

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Special Cases:

* Short Circuit: A zero-resistance wire in parallel with a resistor effectively bypasses the resistor, making its contribution to $R_{eq}$ zero. * Open Circuit: A break in a path means infinite resistance, and no current flows through that branch.

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Tips: — Trace current paths, identify common potential points, and always check if your answer is physically reasonable.

Vyyuha Quick Recall

Same Current, Add Resistance (Series) Parallel Voltage, Reciprocal Resistance (Parallel)