Home › Physics › Electricity and Magnetism › Laws of Combination of Resistances

Electricity and Magnetism

Laws of Combination of Resistances

Combination of resistances involves joining multiple resistors in series or parallel configurations to achieve a desired equivalent resistance in an electrical circuit. The equivalent resistance depends strictly on the topological arrangement and obeys Ohm's Law.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Calculation of equivalent resistance for simple series, parallel, and symmetrical bridge networks; basic application of Ohm's Law and Kirchhoff's Rules.

PGT extension

Infinite ladder networks, delta-wstar (triangle-star) transformations, complex symmetry in cubical or planar resistor networks, and temperature dependence of resistance combinations.

Prerequisites

Ohm's Law and resistivity
Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL)
Conservation of charge and energy in circuits

Series Combination of Resistors

Resistors are connected end-to-end such that the same electric current flows through each resistor sequentially.
The total potential difference across the combination is equal to the sum of the potential drops across individual resistors: $V = V_{1} + V_{2} + $... + V_n.
The equivalent resistance R_s is always greater than the largest individual resistance in the combination.

Parallel Combination of Resistors

Resistors are connected across the same two common nodes, ensuring the same potential difference exists across each resistor.
The total current entering the parallel junction divides among the branches such that $I = I_{1} + I_{2} + $... + I_n.
The reciprocal of the equivalent resistance R_p equals the sum of the reciprocals of individual resistances, making R_p smaller than the smallest individual resistor.

Delta-Star (Triangle-Wye) Transformation

Used for analyzing bridge circuits that cannot be simplified purely by standard series or parallel combinations.
Converts a three-terminal delta (mesh) connection of resistances into an equivalent star (tee) connection.
Simplifies complex network analysis by eliminating bridge cross-connections.

Symmetry in Resistor Networks

Mirror symmetry and perpendicular axis symmetry allow identification of equipotential nodes where no current flows.
Unneeded branches connected between equipotential nodes can be removed to simplify complex configurations like a cube resistance network.
Reduces computational complexity in advanced PGT numerical problems.

Derivations

Equivalent Resistance in Series Combination

Conditions: n resistors connected sequentially carrying identical current I

Let n resistors R₁, R₂, ..., R_n be connected in series across a voltage source V.
By Ohm's Law, the potential drop across each resistor is V_i = I R_i.
The total potential difference is $V = V_{1} + V_{2} + $... + V_n = I(R₁ + R₂ + ... + R_n).
Defining equivalent resistance R_s as V / I yields the final expression.

Result: $R_{s} = \sum_{i=1}^{n} R_{i}$

Equivalent Resistance in Parallel Combination

Conditions: n resistors connected across common nodes with identical potential difference V

Let n resistors R₁, R₂, ..., R_n be connected in parallel across voltage V.
The current through each branch is given by I_i = V / R_i.
Total current entering the junction is $I = I_{1} + I_{2} + $... + I_n = V(1/R₁ + 1/R₂ + ... + 1/R_n).
Using equivalent resistance R_p where $I = V / R_{p} gives the reciprocal relation$.

Result: \frac{1}{R_{p}} = su$m_{i=1}^{n} \frac{1}{R_{i}}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Series Equivalent Resistance	$$R_{s} = R_{1} + R_{2} + R_{3} + ... + R_{n}$$	$R_s = Equivalent resistance$, $R_i = Individual resistances$ Ohm (Omega)	Same current through all resistors
Parallel Equivalent Resistance	$$\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + ... + \frac{1}{R_{n}}$$	$R_p = Equivalent resistance in parallel$, $R_i = Branch resistances$ Ohm (Omega)	Same potential difference across all branches
Two Resistors in Parallel	$$R_{p} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}$$	R_1, $R_2 = Two individual parallel resistors$ Ohm (Omega)	Two-branch parallel network
n Identical Resistors in Series	$$R_{s} = n R$$	n = Number of resistors, R = Resistance of each resistor Ohm (Omega)	Identical resistors in series
n Identical Resistors in Parallel	$$R_{p} = \frac{R}{n}$$	n = Number of resistors, R = Resistance of each resistor Ohm (Omega)	Identical resistors in parallel
Current Division Rule (Two Branches)	$$I_{1} = I \left(\frac{R_{2}}{R_{1} + R_{2}}\right)$$	I = Total current, $I_1 = Current through R_1$, R_1, $R_2 = Parallel resistances$ Ampere (A)	Two resistors in parallel
Voltage Division Rule (Two Resistors)	$$V_{1} = V \left(\frac{R_{1}}{R_{1} + R_{2}}\right)$$	V = Total voltage, $V_1 = Potential drop across R_1$, R_1, $R_2 = Series resistances$ Volt (V)	Two resistors in series
Delta to Star Transformation (Conversion for R_A)	$$R_{A} = \frac{R_{ab} R_{ac}}{R_{ab} + R_{bc} + R_{ca}}$$	$R_A = Star resistance$, R_ab, R_bc, $R_ca = \delta resistances$ Ohm (Omega)	Three terminal resistive network

Graphs and experiments

V-I Characteristics for Series and Parallel Combinations

Draw: A graph of Current (I) on the x-axis versus Potential Difference (V) on the y-axis, showing straight lines with different slopes for series and parallel circuits.

Interpret: The slope of the V-I graph represents the equivalent resistance ($R = V/I)$. A steeper slope indicates a higher equivalent resistance (series combination), whereas a shallower slope indicates a lower equivalent resistance (parallel combination).

Exam use: Frequently tested in conceptual multiple-choice questions comparing slopes of series and parallel configurations.

Metre Bridge Experiment for Resistance Combination Verification

Draw: Circuit diagram of a Metre Bridge connected to a resistance box and unknown resistors in series/parallel configurations with a galvanometer and jockey.

Interpret: At the null point, the ratio of lengths (l/(100-l)) equals the ratio of resistances, allowing experimental verification of R_{s_{and R_{p_formulas.}}}

Exam use: Tested in practical physics applications and numerical problems based on null-deflection balancing length.

Infinite Resistance Ladder Network

Draw: An infinite repeating ladder network of alternating series and parallel resistors connected across input terminals.

Interpret: Due to infinite repetition, removing one repeating unit leaves the input equivalent resistance unchanged, setting up a quadratic equation for R_{eq_.}

Exam use: Advanced PGT examination problem type assessing asymptotic network analysis.

Solved numericals

TGT core

Three resistors of resistances 2 Ω, 3 Ω, and 6 Ω are connected in parallel. Calculate the equivalent resistance of the combination.

Recall the formula for parallel combination: 1/$R_p = 1/R_1 + 1/R_2 + 1/R_3$.
Substitute the given values: 1/$R_p = 1/2 + 1/3 + 1/6$.
Take the common denominator (6): 1/$R_p = (3 + 2 + 1) / 6 = 6 / 6 = 1 Ω^-1$.
Invert to find R_p: $R_p = 1 Ω$.

Answer: 1 Ω

TGT core

A wire of resistance R is cut into 5 equal pieces. If these pieces are connected in parallel, find the equivalent resistance of the combination in terms of R.

When a wire is cut into 5 equal parts, the resistance of each individual piece becomes R' = R / 5.
When n identical resistors of value R' are connected in parallel, the equivalent resistance is $R_p = R' / n$.
Substitute R' = R / 5 and $n = 5: R_p = (R / 5) / 5 = R / 25$.

Answer: R / 25

PGT extension

Calculate the equivalent resistance between terminals A and B of a cube skeleton where each edge has a resistance r.

Apply symmetry principles to the cubical network connected between diagonally opposite corners (A and B).
Current entering node A divides symmetrically into 3 edges, each carrying current I/3. The potential at these three nodes is equal.
At the next set of 3 symmetric nodes, current recombines symmetrically before exiting at node B.
Sum the potential drops along a path from A to B: $V_AB = (1/3)r + (1/6)r + (1/3)r = (5/6)r$.

Answer: (5/6) r

Assumptions and limitations

Resistors are assumed to be ohmic, meaning their resistance remains constant irrespective of the current or applied voltage.
Connecting wires and terminal contacts are assumed to have zero electrical resistance unless explicitly stated otherwise.
Temperature variations during current flow are neglected in basic combination laws (Temperature coefficient of resistance alpha is taken as zero).

Common misconceptions

Common error: Adding resistors in parallel increases the total equivalent resistance of the circuit.

Correct: Adding resistors in parallel decreases the equivalent resistance.

Parallel connection provides additional alternate conducting paths for current, lowering the overall obstruction to flow.

Common error: The current is divided equally among all resistors in a parallel combination regardless of their resistance values.

Correct: Current divides inversely proportional to the resistance of each branch.

Branches with smaller resistance carry larger currents according to Ohm's Law ($I = V/R)$.

Common error: Connecting identical resistors in series increases the current drawn from a fixed voltage source.

Correct: Connecting resistors in series increases total resistance, thereby decreasing the total current drawn from the source.

From Ohm's Law, current is inversely proportional to total series resistance ($I = V / nR)$.

Quick revision

Series combination maximizes equivalent resistance ($R_s = Σ R_i)$, whereas parallel combination minimizes it (1/$R_p = Σ 1/R_i)$.
Series Equivalent Resistance: $R_{s} = R_{1} + R_{2} + R_{3} + $... + R_{n}
Parallel Equivalent Resistance: \frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + ... + \frac{1}{R_{n}}

Must remember

In a series combination of resistors, the equivalent resistance $R_s = R_1 + R_2 + ... + R_n$ is always greater than the largest individual resistance.
In a parallel combination, the reciprocal of equivalent resistance $1/R_p = 1/R_1 + 1/R_2 + ... + 1/R_n$ is always smaller than the smallest individual resistance.
The current divides inversely proportional to the resistance values among parallel branches according to Ohm's Law.
Delta-Star transformation is utilized for analyzing bridge circuits that cannot be simplified purely by standard series or parallel combinations.
Mirror symmetry and perpendicular axis symmetry in resistance networks allow the identification of equipotential nodes where no current flows.

Glossary

Series CombinationAn arrangement where resistors are connected end-to-end such that the same electric current flows through each resistor sequentially.

Parallel CombinationAn arrangement where resistors are connected across the same two common nodes, ensuring the same potential difference exists across each resistor.

Delta-Star TransformationA mathematical technique to convert a three-terminal delta (mesh) connection of resistances into an equivalent star (tee) connection.

Equipotential NodePoints in a circuit network that share the exact same electrical potential, resulting in zero current flow between them if connected.

Series vs Parallel Resistor Combinations

Parameter	Series Combination	Parallel Combination
Current through each resistor	Same	Different (divides inversely with resistance)
Potential difference across each	Different (divides directly with resistance)	Same
Equivalent Resistance ($R_{eq}$)	$R_s = \sum R_i$ (Greater than largest)	$1/R_p = \sum (1/R_i)$ (Smaller than smallest)
Total circuit current	Decreases as more resistors are added	Increases as more branches are added

Practice Quiz

Choose an option to reveal the explanation.

1. What is the equivalent resistance of $n$ identical resistors each of resistance $R$ connected in series?

Explanation: When $n$ identical resistors of resistance $R$ are connected in series, the equivalent resistance is $R_s = R + R + ... \text{ (n \times)} = n R$.

2. What is the equivalent resistance of $n$ identical resistors each of resistance $R$ connected in parallel?

Explanation: For $n$ identical resistors in parallel, $1/R_p = 1/R + 1/R + ... \text{ (n \times)} = n/R$, which gives $R_p = R/n$.

3. How does the total equivalent resistance change when additional resistors are connected in parallel?

Explanation: Connecting resistors in parallel provides additional conducting paths for the electric current, thereby reducing the overall equivalent resistance of the network.

4. If two resistors of $3\,\Omega$ and $6\,\Omega$ are connected in parallel, what is their equivalent resistance?

Explanation: Using the formula for two resistors in parallel: $R_p = (R_1 R_2)/(R_1 + R_2) = (3 \times 6)/(3 + 6) = 18 / 9 = 2\,\Omega$.

5. Which transformation is specifically used to simplify bridge circuits that cannot be solved by simple series or parallel reductions?

Explanation: Delta-Star (Triangle-Wye) transformation is used for analyzing and simplifying bridge circuits by converting three-terminal delta connections into equivalent star connections.

6. In a series combination of three unequal resistors, which resistor experiences the highest potential drop?

Explanation: In series, current $I$ is constant. By Ohm's Law $V = I R$, the potential drop is directly proportional to resistance, so the largest resistance has the highest voltage drop.

7. What is the primary purpose of identifying equipotential nodes through network symmetry?

Explanation: Symmetry (mirror or perpendicular axis) helps identify equipotential nodes where no current flows, allowing unneeded branches to be removed to simplify complex configurations like a cube network.

8. Two resistors $R_1$ and $R_2$ ($R_1 < R_2$) are connected in parallel across a voltage source $V$. Which resistor carries more current?

Explanation: Current divides inversely with resistance ($I = V/R$). Since $R_1 < R_2$, branch $R_1$ carries a larger current than branch $R_2$.

9. What is the equivalent resistance of a series combination containing $R_1 = 2\,\Omega$, $R_2 = 4\,\Omega$, and $R_3 = 6\,\Omega$?

Explanation: For series combination, $R_s = R_1 + R_2 + R_3 = 2 + 4 + 6 = 12\,\Omega$.

10. What can be said about the equivalent series resistance $R_s$ compared to the individual resistances?

Explanation: By definition, the series equivalent resistance is the sum of all individual resistances, making it always greater than the largest individual resistance in the combination.

11. According to the Current Division Rule for two parallel resistors $R_1$ and $R_2$ with total current $I$, what is the current $I_1$ through $R_1$?

Explanation: The current through branch $R_1$ is given by the formula $I_1 = I \left(\frac{R_2}{R_1 + R_2}\right)$, showing inverse proportionality to its own resistance.

12. According to the Voltage Division Rule for two series resistors $R_1$ and $R_2$ across total voltage $V$, what is the voltage drop $V_1$ across $R_1$?

Explanation: The potential drop across $R_1$ in a series circuit is $V_1 = V \left(\frac{R_1}{R_1 + R_2}\right)$, following direct proportionality to its resistance.

13. Which PGT-level network configuration typically requires advanced symmetry techniques or infinite ladder analysis to find equivalent resistance?

Explanation: A cubical resistance network exhibits complex 3D symmetry, requiring nodal analysis, mirror symmetry, or delta-star transformations to evaluate equivalent resistance between different terminal pairs.

14. Why does adding resistors in series decrease the total current drawn from a fixed voltage source?

Explanation: Adding resistors in series increases the total equivalent resistance ($R_s = \sum R_i$), which, for a constant voltage source, decreases total current as dictated by Ohm's Law ($I = V/R$).

15. What is the star resistance $R_A$ corresponding to delta resistances $R_{ab}$, $R_{bc}$, and $R_{ca}$?

Explanation: In delta to star conversion, any star resistance connected to a terminal is equal to the product of the two adjacent delta resistances connected to that same terminal divided by the sum of all three delta resistances.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. Three identical resistors are connected in parallel, drawing a total current $I$ from a source. If one resistor is removed, what happens to the total current drawn from the source (assuming constant voltage)?

Explanation: Initially, $R_p = R/3$ and $I = 3V/R$. With one resistor removed, two remain in parallel: $R_p' = R/2$ and $I' = 2V/R$. Thus, $I' = (2/3)I$, meaning the current decreases to $2/3$ of its original value.

2. A wire of resistance $R$ is stretched uniformly to double its original length. If this stretched wire is then cut into two equal halves and connected in parallel, what is the equivalent resistance of the combination?

Explanation: Stretching length to $2$ times makes resistance $R' = n^2 R = 4R$ (since volume/mass remains constant, cross-sectional area halves). Cutting into two equal halves gives two resistors of $2R$ each. Connecting them in parallel yields $R_{eq} = (2R \times 2R)/(2R + 2R) = 4R^2 / 4R = R$.

3. In a cubical network consisting of 12 identical resistors each of resistance $r$ along each edge, what is the equivalent resistance between two diagonally opposite corners of the cube?

Explanation: Using symmetry and potential distribution across a cubical network between opposite body diagonal corners, the equivalent resistance is calculated as $(5/6)r$.

4. Four resistors of values $1\,\Omega$, $2\,\Omega$, $3\,\Omega$, and $4\,\Omega$ are connected in series. If a potential difference of $20\text{ V}$ is applied across the combination, what is the potential drop across the $3\,\Omega$ resistor?

Explanation: Total resistance $R_s = 1 + 2 + 3 + 4 = 10\,\Omega$. Current $I = V / R_s = 20 / 10 = 2\text{ A}$. The voltage drop across the $3\,\Omega$ resistor is $V_3 = I \times R_3 = 2 \times 3 = 6\text{ V}$.

5. An infinite ladder network is constructed with repeating units of series resistance $r$ and parallel resistance $2r$ connected across a voltage source. What is the condition for equivalent resistance of an infinite ladder?

Explanation: For an infinite ladder, adding one more repeating unit does not change the total equivalent resistance $R_{eq}$. This leads to an algebraic equation yielding a finite limiting value for $R_{eq}$.

6. Two resistors $R_1 = 6\,\Omega$ and $R_2 = 3\,\Omega$ are connected in parallel. If the total current entering the combination is $9\text{ A}$, what is the current flowing through $R_1$?

Explanation: Using the Current Division Rule: $I_1 = I \left(\frac{R_2}{R_1 + R_2}\right) = 9 \left(\frac{3}{6 + 3}\right) = 9 \left(\frac{3}{9}\right) = 3\text{ A}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the equivalent resistance between terminals A and B if two resistors of $10\,\Omega$ each are connected in series?

Explanation: For series combination, $R_s = R_1 + R_2 = 10 + 10 = 20\,\Omega$.

2. If four identical resistors of resistance $R$ are connected in parallel, the equivalent resistance is:

Explanation: For $n$ identical resistors in parallel, $R_p = R / n$. Here $n = 4$, so $R_p = R / 4$.

3. Which combination of four $2\,\Omega$ resistors can yield an equivalent resistance of $2\,\Omega$?

Explanation: Two branches each having two $2\,\Omega$ resistors in series gives $4\,\Omega$ per branch. Connecting two such $4\,\Omega$ branches in parallel gives $(4 \times 4)/(4 + 4) = 16/8 = 2\,\Omega$.

4. In a parallel network of resistors, the potential difference across each individual resistor is:

Explanation: Resistors connected in parallel share the exact same two common nodes, meaning the potential difference across each branch is identical and equals the total potential across the combination.

5. What is the equivalent resistance of three resistors $2\,\Omega$, $3\,\Omega$, and $6\,\Omega$ connected in parallel?

Explanation: $1/R_p = 1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = 6/6 = 1\,\Omega^{-1}$. Thus, $R_p = 1\,\Omega$.

6. In a series circuit, if one resistor gets short-circuited (resistance becomes $0$), what happens to the current through the remaining active resistors?

Explanation: Short-circuiting one resistor reduces the total series resistance of the circuit. With a fixed voltage source, a lower total resistance results in a higher total current through the remaining active resistors according to Ohm's Law.

7. What is the primary role of symmetry analysis in complex resistor networks?

Explanation: Symmetry analysis (mirror or perpendicular axis) identifies nodes at identical potentials (equipotential nodes), allowing no-current branches to be removed or shorted to simplify complex networks.

8. If $n$ identical resistors of resistance $R$ give an equivalent resistance $R_s$ in series and $R_p$ in parallel, what is the ratio $R_s / R_p$?

Explanation: $R_s = n R$ and $R_p = R / n$. Therefore, $R_s / R_p = (n R) / (R / n) = n^2$.

9. What happens to the equivalent resistance of a parallel circuit when one of the parallel branches is opened (disconnected)?

Explanation: Removing a parallel branch removes a conducting path, which decreases total conductance and therefore increases the equivalent resistance of the remaining parallel combination.

10. Which of the following statements is correct regarding the Delta-Star transformation?

Explanation: Delta-Star transformation specifically converts a three-terminal delta (mesh) connection of resistances into an equivalent star (tee) connection to simplify complex bridge circuits.