Derive an expression for the equivalent resistance when two resistors are connected in parallel

Before we determine the value of the resistor in the circuit, it is necessary to understand in which manner resistors are connected with each other. Resistors can be connected in either in series or parallel combination. In this article, let us know about the resistors in series and parallel combinations.

When resistors are connected end-to-end with each other, they are said to be connected in a series combination. Consider two resistors, namely R1 and R2, connected in series combination. The total voltage across the resistors is given by V = R1+R2.

When resistors are connected across each other in a circuit, they are said to be connected in parallel combination. Consider two resistors, namely R1 and R2, connected in parallel combination. The total resistance across the voltage is given by 1/R = 1/R1+1/R2.

Part A Viva QuestionsPart B ExperimentPart B Viva Questions

Resistors In Series And Parallel

a) Determination of the equivalent resistance of two resistors when connected in parallel.

A circuit is said to be in a parallel connection when the resistors are connected in such a way that they branch out from one point. The potential difference in a parallel circuit is the same for each resistor and the current flow is not the same for all resistors. The total current flow through the circuit is calculated by summing up the current flow through each resistor. Parallel circuit connections find application in household electric distribution. The reason why a parallel circuit is preferred is to avoid short circuits and to monitor the flow of current for different devices. The other type of circuit is a series connection. Below is an experiment to determine the equivalent resistance when the resistors are connected in parallel.

Aim

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

Theory

If the resistors are connected in parallel along with a battery, then the total current I is calculated as a sum of the separate value of current through each branch. It is given as:

I = I1+I2+I3+….

Materials Required

1. A battery
2. A plug key
3. Connecting wires
4. An ammeter
5. A voltmeter
6. Rheostat
7. A piece of sandpaper
8. Two resistors of different values

Procedure

1. Make all the connections as shown in the experimental setup I by keeping the key off.
2. Insert the key when the circuit is connected appropriately.
3. For resistors R1 and R2, note three readings of ammeter and voltmeter.
4. Connect the circuit as shown in the experimental setup II.
5. Resistors and voltmeter both are connected in parallel.
6. Record three different readings of ammeter and voltmeter and also use a rheostat.
7. Remove the key.
8. With the help of the observation table, do the calculations.

Observation Table

Result

Precautions

1. The connecting wires that are used should be thick copper wire and using sandpaper the insulation at the end of the wires should be removed.
2. To avoid the introduction of external resistance, the connections should be tight.
3. To make connections, the circuit diagram should be referred to.
4. To make the current entry from the positive terminal and exit from the negative terminal, the ammeter should be connected in series.
5. Resistor and voltmeter should be connected in parallel.
6. The least count of ammeter and voltmeter should be calculated properly.
7. When there is no current flow, the ammeter and voltmeter should be at zero.
8. To avoid heating and change in the resistance value of the resistor, the current should flow while taking the readings.

Viva Questions

Q1. What happens to the total resistance when resistors are combined in parallel?

Ans: When the resistors are combined in parallel, the total resistance will be less.

Q2. When the resistors are connected in parallel, name the physical quantity that remains constant.

Ans: The physical quantity that remains constant when the resistors are connected in parallel is the voltage.

Q3. How to calculate equivalent resistance when three resistors are connected in parallel?

Ans: Following is the formula used to calculate the equivalent resistance when three resistors are connected in parallel:

Q4. What will be the net resistance if two resistors with 3 Ω and 6 Ω resistance are connected in parallel?

Ans: The net resistance of the resistors connected in parallel is given as:

\(\begin{array}{l}\frac{1}{R_{p}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\end{array} \)

\(\begin{array}{l}\frac{1}{R_{p}}=\frac{1}{3}+\frac{1}{6}\end{array} \)

\(\begin{array}{l}=\frac{3}{6}\end{array} \)

\(\begin{array}{l}\Rightarrow R_{p}=\frac{6}{3}=2\Omega\end{array} \)

Q5. What would be the total resistance when the resistors are connected in parallel with resistors value 3Ω?

Ans: The total resistance of the resistors connected in parallel is given as:

\(\begin{array}{l}\frac{1}{R_{p}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\end{array} \)

\(\begin{array}{l}\frac{1}{R_{p}}=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=\frac{3}{3}=1\end{array} \)

\(\begin{array}{l}\Rightarrow R_{p}=1\Omega\end{array} \)

b) Determination of the equivalent resistance of two resistors when connected in series.

In a series connection, the current flows through one resistor and then the other. The current flowing through each resistor is the same as the current flows in one direction. The total voltage is the sum of all the voltage drops across the resistors. There are few drawbacks of a series circuit, and they are:

• Either because of the overflow of current or loss of current, short-circuiting may happen.
• The electrical appliance may be damaged due to a high flow of current.
• Only one device cannot be switched off, without turning off both the devices.

Aim

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

Theory

Depending on the combination and connections in a circuit, the resistance can be increased or decreased. The difference between the series and parallel circuit is based on the arrangement of the resistors. Resistors are said to be connected in series if their ends are joined. The potential difference across each resistor would be different but the current would be the same.

If two resistors are connected in series then;

Resistance, R = R1+R2

Current, I = constant

Potential difference, V = V1+V2

On applying Ohm’s law, we get,

V1 = IR1

V2 +IR2

V = V1+V2

V = I(R1+R2)

∴ R = R1+R2

Materials Required

• Two resistors of different values
• A battery of 6 volts
• Ammeter
• Plug key
• Connecting wires
• A piece of sandpaper
• Voltmeter
• Rheostat

Circuit Diagram

Procedure

1. With the help of circuit diagram, make the connections.
2. Do not switch on the key.
3. The ammeter should be connected in series, voltmeter in parallel, and the rheostat in series.
4. Make the connections as shown in the experimental setup and check of +ve and -ve terminals of the battery.
5. By inserting the key, record the ammeter and voltmeter readings.
6. Note three readings by adjusting the rheostat.
7. Note down the readings of a voltmeter by connecting it to each resistor.
8. Measure the potential difference, V1 across the first resistor by plugging in the key.
9. Measure the potential difference, V2 across the second resistor by plugging in the key.
10. Calculate the relationship between V, V1, and V2.

Observation Table

Result

Hence, it is verified that Rs=R1+R2.

Precautions

1. Voltmeter and resistor should always be in parallel.
2. The least count of voltmeter and ammeter should be calculated properly.
3. Connections should be as per the experimental setup.
4. When no current flows through the ammeter and voltmeter, the pointers should be at zero mark.
5. The connecting wires that are used should be thick copper wire and using sandpaper the insulation at the end of the wires should be removed.
6. To avoid the introduction of external resistance, the connections should be tight.
7. To avoid heating and change in the resistance value of the resistor, the current should flow while taking the readings.

Viva Questions

Q1. What will be the total resistance of two resistors connected in series with value 2 Ω and 3 Ω?

Ans: The total resistance is given as:

R = R1 + R2

∴ R = 2 Ω + 3 Ω = 5 Ω

Q2. Determine the voltmeter reading for a series circuit if a current of 5A is flowing through 2 Ω and 3 Ω resistors.

Ans: The voltmeter reading across the 2 Ω resistor is given as:

V =IR

V1= 5 × 2 = 10V

The voltmeter reading across the 3 Ω resistor is given as:

V =IR

V2= 5 × 3 = 15V

∴ V = V1+V2

V = 10 + 15 = 25V

Q3. How should the resistor be connected so that the resultant resistance is increased?

Ans: To get the resultant resistance maximum, the resistors should be connected in series.

Q4. What will be the total resistance if two equal resistors are connected in series?

Ans: If two equal resistors are connected in series, then the total resistance is given as:

R + R = 2R.

Q5. Where should the voltmeter be connected when the resistors are in series?

Ans: The voltmeter should be connected parallel when the resistors are in series.

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