Determination of the Equivalent Resistance of Two Resistors When Connected in Series and Parallel

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To determine the equivalent resistance of resistors when connected in series and parallel, first, it is necessary to understand that the resistors are connected in two ways. One way is the series and the other can be in parallel. It depends on the flow of the current.

When two or more resistors are connected one after the other such that the same current flows through all the resistors, then the combination is said to be in Series.
When one end of all the resistors is connected to one terminal of a battery and the other end of all the resistors is connected to another terminal of the same battery then the combination of resistors is said to be in Parallel.
The combination of resistors each of them having individual resistance values gives equivalent resistance.

Table of Content

Resistors in Series and Parallel
How to find out equivalent resistance connected in series and parallel resistance
Experiment to understand the resistance when connected in parallel
Experiment to understand the resistance when connected in series
Things to Remember
Sample questions

Key Terms: Resistance, Resistors, Current flow, Series and parallels, Voltage, Terminals, Kirchhoff’s current law, Kirchhoff’s voltage law

Resistors in Series and Parallel

[Click Here for Sample Questions]

When we say that two resistors are connected in series it means that in a series combination of resistors the current flowing is the same throughout.

When two or more resistors are connected in series, the overall combined resistance R is the sum of the overall individual resistances.
Let R₁, R₂, and R₃ be the three resistors connected in a Series, then the Equivalent resistance R of the resistors is given by

R = R₁ + R₂+ R₃

When we say that two resistors are connected in parallel it means one end of all the registers is connected to a common point and the other end of all the registers is connected to the other common point then that resistance is connected in parallel.

The reciprocal of the resultant resistance is equal to the sum of reciprocals of all the resistors connected in parallel.
Let R₁, R₂, and R₃ be the three resistors connected in Parallel, then the Equivalent resistance R of the resistors is given by

1/R= 1/R₁ + 1/R₂ + 1/R₃

Also Read:

Electric Current	Kirchhoff’s Rule
Meter Bridge	Potentiometer

How to find out equivalent resistance connected in series and parallel resistance combinations in any network

[Click Here for Sample Questions]

Let’s understand the procedure in the series from below Fig 1.

Here, three resistors have been added to the series R₁, R₂, and R₃. A B, C, and D are the terminals. The last terminal of every resistance will be attached to the first terminal of the last series.

Suppose the current which is flowing from the series is I and the corresponding R₁ voltage is V₁, R₂ voltage is V₂, and R₃ voltage is V₃.
Always remember that the voltage is always different and the current flow is always the same throughout the series because voltage changes with resistance.

Read More: Internal Resistance Formula

By Kirchhoff’s voltage law (KVL), we have

V= V₁+ V₂+ V₃

By Ohm’s Law, we have

V= IR

⇒ IR = IR₁ + IR₂ + R₃

⇒ R = R₁ + R₂+ R₃

Now let's understand the procedure in parallel through Fig. 2

Fig. 2: Parallel Resistance

Let R₁, R₂, and R₃ be resistors connected in parallel. Terminal points are A and B. Current passes from 3 ways are I₁, I₂, and I₃. And the voltage is the same throughout the parallels.

From Kirchhoff’s current law (KCL), If at any point of the parallels, the additions of the currents are the same. So,

I = I₁ + I₂ + I₃

By Ohm law, we have

V= IR ⇒ I = V/R

⇒ V/R = V/R₁ + V/R₂ + V/R₃

⇒ 1/R = 1/R₁ + 1/ R₂ + 1/ R₃

Also Check:

Conductor	Resistors in parallel
Wheatstone Bridge	Electrical Current

Experiment to understand the resistance when connected in parallel

[Click Here for Sample Questions]

Aim

To determine the equivalent resistance of two resistors when connected in parallel.

Requirements

Voltmeter and ammeter of suitable range, two unknown resistances, rheostat, plug key, battery eliminator or accumulator, connecting wire, and sandpaper

Process

Choose the voltmeter and the ammeter of a suitable range. Note the least count and zero error of the voltmeter and ammeter.
Make the electrical circuit In suitable combinations.
Then, connect the ammeter across resistor R1. Put the plugin in the key and adjust the Rheostat to obtain the desired current in the ammeter. Obtain four sets of such readings of different currents.
Then, connect the ammeter across resistor R2 and again obtain four sets of readings as above.
Lastly, connect the voltmeter across both registers. Here also opt-in for four sets of readings as above.

Read More:

Unit of Specific Resistance

Uses of Resistor

Observation

The Least count of the voltmeter = 0 V
The least count of the ammeter = 0 A
Zero error of the voltmeter = 0 V
Zero error of the ammeter = 0 A

Check Important Difference Between Ammeter and Voltmeter

Let's put the above observation in the table:

Steps	Set of Observations	Ammeter Reading (I)		Voltmeter Reading (V)		V/I (ohms)	Mean (ohms)
Steps	Set of Observations	Observed	Corrected	Observed	Corrected	V/I (ohms)	Mean (ohms)
For R₁	1.	0.15	0.15	0.10	0.10	0.66	0.66 [(0.66+0.66+0.66)/3]
	2	0.30	0.30	0.20	0.20	0.66
	3	0.45	0.45	0.30	0.30	0.66
For R₂	1	0.60	0.60	0.40	0.40	0.66	0.66 [(0.66+0.66+0.66)/3]
	2	0.75	0.75	0.50	0.50	0.66
	3	0.90	0.90	0.60	0.60	0.66
For R_p When R₁ and R₂ are connected together in parallel	1	0.75	0.75	0.24	0.24	0.33	0.33
	2	1.05	1.05	0.34	0.34	0.32
	3	1.35	1.35	0.45	0.45	0.33

Read Further: Current Density Formula

Therefore, Total Resistance

1/R_p= 1/R₁ +1/R₂

⇒ 1/R_p= 1/0.66 + 1/0.66

⇒ 1/R_p= 3.03

⇒ R_p= 1/0.303 = 0.33 ohms

Mean equivalent resistance R_p when R₁ and R₂ are connected together in parallel = 0.33 ohms.

Result

The equivalent resistance of two resistors in parallel = 0.33 ohms.
The relation of R_p, R₁, and R₂ is verified.

Check More:

Difference between Resistance and Resistivity	Derivation of Drift Velocity
Resistivity	Resistivity Temperature Dependence

Experiment to understand the resistance when connected in series

[Click Here for Sample Questions]

Aim

To determine the equivalent resistance of two resistors connected in series

Requirements

Two resistors, a rheostat, a voltmeter, an ammeter, a plug key, dry cells, connecting wires, and a sandpaper

Process

Choose the voltmeter and the ammeter of a suitable range. Note the least count and zero error of the voltmeter and ammeter.
Connect the circuits as shown in the diagram.
Then, connect the voltmeter across resistor R1. Put the plugin in the key and adjust the Rheostat to obtain the desired current in the ammeter. Obtain four sets of such readings of different currents.
Then, connect the voltmeter across resistor R2 and again obtain four sets of readings as above.
Lastly, connect the voltmeter across both resistors. Here also opt-in for four sets of readings as above.
Remove the plug and disconnect the circuit

Observation

The Least count of the voltmeter = 0 V
The least count of the ammeter = 0 A
Zero error of the voltmeter = 0 V
Zero error of the ammeter = 0 A

Check Further: Types of Current

Let's put the above observation in the table:

Steps	Set of Observations	Ammeter Reading (I)		Voltmeter Reading (V)		V/I (ohms)	Mean (ohms)
Steps	Set of Observations	Observed	Corrected	Observed	Corrected	V/I (ohms)	Mean (ohms)
R₁	1	0.15	0.15	0.45	0.45	3.0	3
	2	0.20	0.20	0.60	0.60	3.0
	3	0.26	0.26	0.80	0.80	3.02
R₂	1	0.30	0.30	0.90	0.90	3.0	3.03
	2	0.35	0.35	1.10	1.10	3.1
	3	0.40	0.40	1.20	1.20	3.0
For R_s When R₁ and R₂ connected together in series	1	0.28	0.28	1.68	1.68	6	6.03
	2	0.26	0.26	1.58	1.58	6.07
	3	0.27	0.27	1.62	1.62	6.00

Therefore, Total Resistance

R_s= R₁ +R₂

⇒ R_s= 6.02 ohms

Mean equivalent resistance R_s when R₁ and R₂ are connected together in series = 6.02 ohms.

Result

The equivalent resistance of two resistors in series = 6.02 ohms.
The relation of R_s, R₁, and R₂ is verified.

Physics Related Links:

Ohm's Law	Resistors in Series
Difference Between AC and DC	Important Formula in electricity

Things to Remember

The wires used for connecting the ammeter and voltmeter and the batteries should be of thick copper wire and must use sandpaper as the insulators.
If the ammeter is connected in series, then the current will enter from the positive terminal and exit from the negative terminal.
On the other hand, remember that the voltmeter and the resistors are connected in parallel.
The value of the resistance of the register changes, so with the continuous flow of current note the readings properly.
While performing the experiment, must note that the connections are properly tied. If not then it can attract external resistance.
The current must flow between the ammeter and the voltmeter otherwise the values will be Nil.

Sample Questions

Ques. What are the necessary precautions to be taken when resistors are connected in series? (3 Marks)

Ans: The necessary precautions to be taken when resistors are connected are as follows,

Do not switch the key.
Check the positive and the negative terminals of the battery and then make the setup properly.
While the current is flowing take the readings properly during that time as after that you cannot do the same.
Adjust the rheostat while taking the readings.
Measure the differences of V1 and we do properly and determine the relationships between V, V₁, and V₂.

Ques: Where to connect the voltmeter when the resistors are connected in series? (3 Marks)

Ans: The voltmeter should be parallel when the resistors are connected in series. It must make sure while making the experiment that it is properly connected otherwise the current will not flow in a correct manner and you can get wrong readings. Again, when it is in series it must be noted that the current is flowing in a single direction and the voltages differ with the different resistance.

Ques. Why are the resistors connected in parallel? (3 Marks)

Ans: If the positive result is expected to be less that is the total resistance is less, then the resistors are connected in parallel. If the resistors are connected in parallel, the quantity, that is the voltage, does not change and remains constant throughout. Again, the current flow changes as per the resistance. Hence, if the resistors are connected in parallel, the RP is less which is highly considered.

Ques. What are materials required when the resistors connected in parallels are experimented with? (2 Marks)

Ans: Materials required are as follows:-

Voltmeter and ammeter of suitable range.
Two unknown resistance
Rheostat
Plug Key
Battery eliminator or accumulator
Connecting wire
Sandpaper

Ques. What are the necessary drawbacks of the series circuit? (3 Marks)

Ans: The necessary drawbacks of the series resistance for which generally it is avoided are:-

The chances of short circuits are quite high. Since the current flows in a single direction and there is no break in it the overflow of current can take place or vice versa which can result in too short a circuit.
If the current flow increases to the desired level, then the appliances used can be destroyed.
The experiment or the function depends on both the devices whether an ammeter or voltmeter. Hence, if the experiment needs to be revised or any step needs to be revised, then both devices are required to turn off.

Ques. Determine the total resistance when connected in parallel. (2 Marks)

Ans: When we say that two resistors are connected in parallel it means one end of all the registers is connected to a common point and the other end of all the registers is connected to the other common point then that resistance is connected in parallel. Now the formula to calculate the resultant resistance when the number of registers is connected in parallel is

Reciprocals of Resultant resistant= Sum of reciprocals of all the resistors connected in parallel.

1/R= 1/R₁ + 1/R₂ + 1/R₃

Ques. Determine the total resistance when connected in series. (2 Marks)

Ans: When we say that two resistors are connected in series it means that in a series combination of resistors the current flowing is the same throughout. When two or more resistors are connected in series, the overall combined resistance R is the sum of the overall individual resistances.

R= R₁ +R₂+ R₃

Ques. What insulator to use and why? (2 Marks)

Ans: The copper wire is used as an insulator. The copper wire is regarded as the safest medium for current flow. One does not want any external resistance during the current flow otherwise readings cannot be taken properly. Again, the chances of short circuits are also less in copper wire. So, to avoid certain difficulties, and for safety purposes, copper wire is used as an insulator.

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Determination of the Equivalent Resistance of Two Resistors When Connected in Series and Parallel

Resistors in Series and Parallel

How to find out equivalent resistance connected in series and parallel resistance combinations in any network

Experiment to understand the resistance when connected in parallel

Experiment to understand the resistance when connected in series

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

4.
The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

5.
Write any two features of nuclear forces.

6.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

Comments

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Determination of the Equivalent Resistance of Two Resistors When Connected in Series and Parallel

Resistors in Series and Parallel

How to find out equivalent resistance connected in series and parallel resistance combinations in any network

Experiment to understand the resistance when connected in parallel

Experiment to understand the resistance when connected in series

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

4.The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

5. Write any two features of nuclear forces.

6.Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

4.
The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

5.
Write any two features of nuclear forces.

6.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]