Chapter 8.3 - Resistivity of a Material

In the previous section we saw the relation between voltage and current. We saw that resistance is the ratio between voltage and current. In this section, we will see more details about resistors. 

Let us do an activity. The steps are written below:
1. Make a circuit with the following components:
An ammeter, switch, cells and a bulb. 
2. One end of the circuit must be free and flexible
This free end should have a pointer J which should be able to touch the ends of the following conductors:
(i) An iron conductor PA
(ii) An aluminium conductor PB (having same length as PA)  
(iii) A nichrome conductor PC (having same length as PA and PB)
(iv) A nichrome conductor PD (having same length as PA, PB and PC, but twice the thickness)  
(v) A nichrome conductor PE (having same thickness as PA, PB and PC, but twice the length)
• The circuit diagram is shown in fig.8.19 below:

Fig.8.19

3. Turn on the switch
Complete the circuit by touching the pointer at A
• Note down the ammeter reading 
• Note down the intensity of light from the bulb  
• Turn off the switch. This completes one trial.
4. Turn on the switch
Complete the circuit by touching the pointer at B
• Note down the ammeter reading 
• Note down the intensity of light from the bulb  
• Turn off the switch. This completes the second trial.
5. Turn on the switch
Complete the circuit by touching the pointer at C
• Note down the ammeter reading 
• Note down the intensity of light from the bulb  
• Turn off the switch. This completes the third trial.
- - - 
- - - 
Continue for the remaining points E and D

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The trials are complete. The observations are tabulated below:

Table.8.3

From the table we can note the following three points:
point 1:
1. Consider the first 3 trials:
• We have, same voltage, same length and same thickness
• But currents are different. What is the reason?
Ans: We have: I = ^V⁄_R
• V is the same. So if there is a change in I, it has to be due to the change in R
• That is., the resistance offered by PA, PB and PC are different
2. All three have same length and same thickness. Only difference is in 'material'. So we can write: 
■ Iron , aluminium and nichrome will offer different resistances because they are different materials
• Aluminium offers the least resistance among the three. So highest current is in trial 2
Point 2:
1. Consider the trials 3 and 4:
• We have, same voltage, same length and same material
• But currents are different. What is the reason?
Ans: We have: I = ^V⁄_R .
• V is the same. So if there is a change in I, it has to be due to the change in R
• That is., the resistance offered by PC, and PD are different
2. Both have same length and same material. Only difference is in 'thickness'. So we can write: 
■ Even if length and materials are the same, two resistors having different thicknesses will offer different resistances
• In our present case, PD has greater thickness. It has greater current. So we can write:
■ When the thickness increases, resistance decreases
A similar situation is shown in fig.8.20 below:

Fig.8.20

• People want to pass through a tunnel. 
    ♦ In fig.8.20(a), the tunnel is narrow. People will find it difficult to pass freely
    ♦ In fig.8.20(b), the tunnel is broad. People will find it easy to pass freely
• Current will also find it easy to pass through a thick wire
Point 3:
1. Consider the trials 3 and 5:
• We have, same voltage, same thickness and same material
• But currents are different. What is the reason?
Ans: We have: I = ^V⁄_R
• V is the same. So if there is a change in I, it has to be due to the change in R
• That is., the resistance offered by PC, and PE are different
2. Both have same thickness and same material. Only difference is in 'length'. So we can write: 
■ Even if thickness and materials are the same, two resistors with different lengths will offer different resistances
• In our present case, PE has greater length. It has lesser current. So we can write:
■ When the length increases, resistance increases
We can confirm this by doing an extra trial:
(i) In the trial 5, we touched the end of PE. Do it again. Note the intensity of light. 
(ii) Slowly slide the pointer from E to P
• We can see that, the intensity of the light increases as the pointer moves from E to P
(iii) The reason is that, the length through which current has to travel through nichrome goes on decreasing. 
• So resistance goes on decreasing. 
• Thus current goes on increasing. 
• Thus the intensity goes on increasing

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Another activity:
This activity is performed to find the relation between temperature and resistance. We will write the steps:
1. Make a simple circuit with the following components:
A 6 V bulb, a 6 V cell and a switch

2. Measure the resistance of the bulb when the switch is turned off. 
• A multimeter can be used for measuring the resistance even when the switch is turned off
3. Remove the multimeter. Turn on the switch. Let the bulb glow for some time.
4. Turn off the switch. Immediately measure the resistance again. 
• We can see that, this second reading is higher
■ What is the reason for this higher reading?
Ans: When the bulb glowed for some time, it's temperature increased.
• So in the second reading, we were measuring the resistance of a 'hot resistor'.
• So we can write: When temperature increases, resistance increases

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From the above two activities, we come to know about four 'factors which affect the resistance'.
Factor 1. Resistance depends on the nature of the material
Factor 2. Resistance depends on the thickness 
• Thickness is same as 'area of cross section'
• We saw this:
When area of cross section (A) increases, resistance (R) decreases
• So when one quantity increases, the other quantity decreases
• This is an inverse relation. It can be represented mathematically as: R ∝ ¹⁄_A
• That is., R is proportional to ¹⁄_A
Factor 3. Resistance depends on the length
• We saw this:
When length (l) decreases , resistance (R) decreases
• So when one quantity decreases, the other quantity also decreases
• This is a direct relation. It can be represented mathematically as: R ∝ l
• That is., R is proportional to l.
Factor 4. Resistance depends on the temperature
• We saw this:
When temperature increases, resistance also increases
• So when one quantity increases, the other quantity also increases
• This is a direct relation. We do not need to represent this mathematically at present. We will see it in higher classes when we do advanced problems.

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But for our present problems, we need to combine the mathematical representations in (2) and (3):
• Note that 'l' is in the numerator and 'A' is in the denominator.
• So, when we combine them, we get: R ∝ ^l ⁄_A
• We can avoid the '∝' symbol by introducing a 'constant of proportionality'. See details here. 
That is., R = (a constant) × ^l ⁄_A
• This constant is given a special name: 'Resistivity of the material which is used to make that conductor'
    ♦ Or simply 'Resistivity'
• It's symbol is 'ρ'. It is the Greek letter 'rho'
• So we can write: R = ρ × ^l ⁄_A
• Rearranging this, we get: ρ = ^(RA)⁄_l

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Now consider a simple problem:
■ The resistance of a resistor is R Ω. It's length is 1 m. It's area of cross section is 1 m². Calculate the resistivity of the material used for making that resistor
Solution:
• We have: ρ = ^(RA)⁄_l 
• Substituting the known values, we get: ρ =  ^(R×1)⁄₁ = R×1 = R

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From the above problem, we can formulate a definition for resistivity. We will write it in steps:
1. Consider a piece of any material. 
• Let it be made up of any material like iron, copper, nichrome etc.,
• Let it's length be 1 m
• Let it's cross sectional area be 1 m² 
2. We want to know the resistance which this resistor is able to apply (to a current that flows through it)
• For that, we can use the relation that we derived previously: R = ρ × ^l ⁄_A
• Substituting the known values, we get: R = ρ × ¹⁄₁ = (ρ × 1) = ρ
• So we get R = ρ.
3. We started off with a piece of material whose length is 1 m and area of cross section is 1 m².
• We obtained the result: 
Resistance of that piece = resistivity of the material with which that piece is made
• This result helps us to give a definition for resistivity. The official definition is:
■ Resistivity of a substance is the resistance of the conductor of unit length and unit area of cross section.
• The resistivity of a substance is a constant at fixed temperature. 
• Resistivity will be different for different materials

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Now we will try to establish a unit for resistivity. The steps are shown below:

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Types of resistors:
• We use different types of resistors in electric circuits. Some images can be seen here.
• If we want to purchase a particular resistor, we must know it's 'resistance in ohms'
    ♦ So the 'resistance in ohms' must be marked on every resistor.
• It is convenient to 'use colour code' rather than to 'write the exact numerical value' on the resistor
■ Let us see how colour coding works. We will use an example:
Consider fig.8.21 below. It is taken from wikimedia commons:
Source 1.
Source 2. 

Fig.8.21

1. There are 4 colour bands. First we will consider the first three bands
2. The first band is red. 
    ♦ It indicates the digit '2'. This is taken from the chart shown on right side
• The second band is violet 
    ♦ It indicates the digit '7'
So in this step we get '27'
3. The third band indicates the number with which '27' is to be multiplied
• Our third band is green. So we have to multiply by 100K
• 'K' indicates 1000. So 100K is 100 × 1000 = 100000
4. So we have to multiply '27' by 100000. We get: 2700000
• This is the value of the resistance. We can write:
• The resistance of the given resistor is 2700000 Ω.
• This can be shortened as: 2700 kΩ.
5. Now we come to the fourth band. It is silver in colour.
• So the tolerance is ±10% 
• The '±' sign indicates 'above or below'. It can be explained as follows:
    ♦ 10% of 2700 = (2700 × ¹⁰⁄₁₀₀) = (2700 × 0.1) = 270
    ♦ 2700 + 270 = 2970
    ♦ 2700 - 270 = 2430
• So the actual resistance can be any value from 2430 kΩ to 2970 kΩ.
• If the fourth band is absent, tolerance should be taken as ±20%

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Variable resistance
1. In the above discussion, we derived the relation: R ∝ l
• That is., Resistance is proportional to l.
• If we increase the length of a conductor, the resistance that it applies (against the flow of current through it) will increase
• If we decrease the length of a conductor, the resistance that it applies (against the flow of current through it) will decrease

2. So if we can make a 'resistor device' in which, the length can be increased or decreased, we will get different 'resistance values' from a single resistor

■ Rheostat is a device designed on the basis of this principle
• A rheostat can be seen here. It is obtained from wikimedia commons. 

• The conductor is wound over an insulator. A sliding contact moves on the horizontal bar at the top.
• As the slider is moved towards left or right, the contact point with the conductor changes
• So the 'length of the conductor coming inside the circuit' changes. 
• Thus the desired resistance can be obtained

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• If we can change the resistance value, we can change the current.
■ Rheostat is a device used to regulate the current in a circuit by changing the resistance  
• The symbol of a variable resistor is:

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In the next section, we will see how more than one resistors are connected in a circuit.

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Chapter 8.2 - Relation between Voltage and current

In the previous section we saw how to connect voltmeter and ammeter to circuits. In this section, we will see the relation between voltage and current. 

Let us do an activity. The steps are written below:
1. Make a circuit with the following components:
An ammeter, switch, a cell and a bulb. 
• The circuit diagram is shown in fig.8.16 below:

Fig.8.16

2. Turn on the switch
• Note down the ammeter reading 
• Note down the intensity of light from the bulb  
• Turn off the switch. This completes one trial.
3. Add one more cell in the circuit. The two cells should be connected in series
4. Turn on the switch
• Note down the ammeter reading 
• Note down the intensity of light from the bulb  
• Turn off the switch. This completes the second trial.
5. Add one more cell in the circuit. The three cells should be connected in series
6. Turn on the switch
• Note down the ammeter reading 
• Note down the intensity of light from the bulb  
• Turn off the switch. This completes the third and final trial.

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The trials are complete. The observations are tabulated below:

Table.8.1

From the table we can make the following two conclusions:
1. When the number of cells (connected in series) increase, ammeter reading increases
• The increase in ammeter reading indicates increase in current
■ So we can write:
When the number of cells (connected in series) increase, the current in the circuit increases
2. When the number of cells (connected in series) increase, intensity of light increases
• That means, when current increases, intensity of light also increases
■ What is the reason?
We will write the answer in steps:
(i) We know that, light is produced as a result of the heating of the filament of the bulb
(ii) More current passing through the filament means that more electrons passing through it per second
(iii) So the filament will glow with more intensity
• We can write this in another way also:
When current increase, the heat produced also increases

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Another activity:
This activity is performed to find the relation between current and potential difference. We will write the steps:
1. Make a circuit with the following components:
An ammeter, a voltmeter, switch, a 1.5 V cell and a 30 cm long nichrome wire. 
• The circuit diagram is shown in fig.8.17 below:

Fig.8.17

2. Turn on the switch
• Note down the voltmeter reading. This reading should be entered in the table in the 'V' column
• Note down the ammeter reading. This reading should be entered in the table in the 'I' column 
• Turn off the switch. This completes one trial.
3. Add one more 1.5 V cell in the circuit. The two cells should be connected in series
4. Turn on the switch
• Note down the voltmeter reading. This reading should be entered in the table in the 'V' column
• Note down the ammeter reading. This reading should be entered in the table in the 'I' column 
• Turn off the switch. This completes the second trial.
5. Add one more 1.5 V cell in the circuit. The three cells should be connected in series
6. Turn on the switch
• Note down the voltmeter reading. This reading should be entered in the table in the 'V' column
• Note down the ammeter reading. This reading should be entered in the table in the 'I' column 
• Turn off the switch. This completes the third and final trial.

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The trials are complete. The observations are tabulated below:

Table.8.2

• Note that, the last column is filled up by calculating ^V⁄_I for each trial
• From the table we can make the following conclusions:
Conclusion 1: When cells are connected in series, the 'available potential difference' increases.
■ Since the potential difference is measured in volts, we say:
When cells are connected in series, voltage increases.
• When a single 1.5 V cell is connected in series, the voltmeter reading = 1.5 V
• When two 1.5 V cells are connected in series, the voltmeter reading = (1.5 × 2) = 3 V
• When three 1.5 V cells are connected in series, the voltmeter reading = (1.5 × 3) = 4.5 V
Conclusion 2: 
We will arrive at this conclusion by writing the required steps:
(i) From the third and fourth columns we get:
• When voltage increases, current also increases. 
(ii) This can be written mathematically as: V ∝ I
• That is., V is proportional to I 
(iii) We can avoid the '∝' symbol by introducing a 'constant of proportionality'. See details here. 
That is., V = (a constant) × I 
⟹ ^V⁄_I = a constant
(iv) That means:
• We can do any number of trials we like. 
• In each of those trials, we can calculate ^V⁄_I using the V and I obtained in that trial.
• Then we can compare those ^V⁄_I values. All those values will be the same.
• This is indeed true as can be seen from the last column. All values are '10'
■ So the conclusion is:
^V⁄_I is a constant

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• This property was first discovered by the German scientist George Simon Ohm
■ He formulated the Ohm's law. It states that:
When temperature remains constant, the current through a conductor is directly proportional to the potential difference between it's ends.
• Note that, the temperature should remain constant.
• This is important because, if we increase or decrease the temperature, the internal molecular and ionic properties of a conductor will change. So the resistance given (against the flow of current) by the conductor will change. We cannot do the calculations if the temperature changes.

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• So we have: ^V⁄_I = a constant
• This constant is given a special name: 'Resistance of the conductor' or simply 'Resistance'
• It is denoted by the letter 'R'
• So we can write: ^V⁄_I = R

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How can we apply this law to a practical situation?
We will write the answer in steps:
1. Consider the activity that we saw just above.
• We have: Current = 0.15 A (when 1.5 V cell is connected)
2. What if we want a higher current with the same 1.5 V?
• We have: ^V⁄_I = R = 10 
• That is: R = ^1.5⁄_I = 10
3. The current 'I' is in the denominator. So if we decrease 10, I will increase.
4. How can we decrease 10?
• We can decrease it by decreasing the length of the nichrome wire.
• That is., if we decrease the length of the nichrome wire, the 'resistance to the flow of current' will decrease. 
• So the current will increase even without any increase in the number of cells.
■ The reverse is also applicable. That is., if we want to reduce the current without decreasing the voltage, we can increase the length of the nichrome wire.

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■ So 'resistance' is a very convenient way to increase or decrease current. The convenience is that:
Using resistance, we can change the current without changing the voltage  
• Nichrome wire is often used to provide resistance in circuits. 
    ♦ A longer nichrome wire will give a higher resistance
    ♦ A shorter nichrome wire will give a lower resistance. 
• Some images of nichrome wire can be seen here.
• The components in a circuit whose function is to 'provide a resistance to the flow of current' are called resistors
■ The official definition is:
Resistors are conductors used to include a particular resistance in a circuit
• Some resistors available in the market can be seen here.
• In circuit diagrams, they are shown using the symbol:

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Unit of resistance

1. In the definition, note the words: 'particular resistance'  
• It means that, we must know 'how much resistance' is to be provided in a circuit. 
• Then only we can purchase a resistor
2. So we must be able to 'measure resistance'.
• For 'measuring resistance', we must need an appropriate unit.
3. Let us try to establish a unit:
• We know that R = ^V⁄_I. 
    ♦ Unit of voltage 'V' is volt
    ♦ Unit of current 'I' is ampere
• So unit of R is ^volt⁄_ampere
• This '^volt⁄_ampere' is given a special name: Ohm 
• It's symbol is 'Ω'. It is the Greek letter 'omega'

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■ So what can we say about '1 Ω'?
• That is., we want to know the peculiarity about a resistor, whose resistance is '1 Ω'
We will write the steps:
1. Consider the circuit shown in fig.8.18 below:

Fig.8.18

• A voltmeter is connected to know the 'potential difference across the two ends of a resistor'
• Recall that 'potential difference across the two ends of a resistor' is same as 'voltage across the two ends of a resistor'
• An ammeter is connected to know the current flowing through the circuit
• Let the voltmeter reading be 1 V
    ♦ Then we can say: The voltage between the ends of the resistor is 1 volt.
• Let the ammeter reading be 1 A
    ♦ Then we can say: A current of 1 ampere is flowing through the circuit. 
    ♦ That is., a current of 1 ampere is flowing through the resistor
• So the resistance 'R' of the resistor shown in that circuit is: ^{1 V}⁄_{1 A}. = 1 Ω
■ The official definition is:
If a conductor connected to a voltage of one volt, passes a current 1 A, then the conductor will have '1 Ω' resistance

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Using basic algebra, the equation R = ^V⁄_I can be written in two other forms also:
• I = ^V⁄_R 
• V = IR
• We can use any one of the three equations. The choice depends on the requirements in the problem
• We can use the following triangle to remember the equations:

■ This is called the VIR triangle
• If we want to calculate V, then we put a finger over V. 
    ♦ That leaves I and R 
    ♦ I and R are on the same level
    ♦ So we get V = IR
• If we want to calculate I, then we put a finger over I
    ♦ That leaves V and R
    ♦ V is at top and R is at bottom
    ♦ So we get I = ^V⁄_R
• If we want to calculate R, then we put a finger over R
    ♦ That leaves V and I
    ♦ V is at top and I is at bottom
    ♦ So we get R = ^V⁄_I

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Solved example 8.1
(a) In a circuit, the voltmeter connected across a 4 Ω resistor showed a reading of 12 V. What was the current flowing through that resistor at that time?
(b) In a circuit, a voltmeter is connected to a 3 Ω resistor. The ammeter reading shows that 2 A current is flowing through it. What is the potential difference across the resistor?
(c) In a circuit, a voltmeter is connected to a resistor. The voltmeter reading shows that, the potential difference across the resistor is 6 V. The ammeter reading shows that 3 A current is flowing through it. What is the resistance of the resistor?
Solution:
Part (a):
1. The given values are: V = 12 V, R = 4 Ω
We have to calculate I
2. We have: I = ^V⁄_R 
Substituting the known values, we get: I = ¹²⁄₄ = 3 A
Part (b):
1. The given values are: I = 2 A, R = 3 Ω
We have to calculate V
2. We have: V = IR
Substituting the known values, we get: V = 2 × 3 = 6 V
Part (c):
1. The given values are: V = 6 V, I = 3 A
We have to calculate R
2. We have: R = ^V⁄_I 
Substituting the known values, we get: R = ⁶⁄₃ = 2 Ω

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• We have seen the relation between voltage and current. 
• Let us draw a graph connecting the two
• We can use the values recorded in table 8.2 above. The resulting graph is shown below:

Let us see the features of the graph:
• The bold yellow line is our required graph. 
• We can see that, it is a straight line. Why is it straight?
Let us analyse:
1. We have the relation: V= IR
• In this relation, R is a constant. Both V and I are variables. That is:
    ♦ V can take different values like 1.5, 3, 4.5 etc., 
    ♦ I can take different values like 0.15, 0.3, 0.45 etc., 
    ♦ But R will always remain a constant
2. So this is similar to the equation y = kx
• Where x and y are variables and k is a constant
3. The graph of 'y = kx' will always be a straight line
• Further more, this graph will always pass through the origin of the graph
• We can see that this is indeed true for our case also. 
    ♦ If we extend our graph down wards, it will pass through the origin. 
    ♦ This is indicated by the dashed yellow line.
4. Also note that, 'k' is multiplied with 'x'
• So the constant is multiplied with the 'variable which is plotted along the x axis'
• In our case, the constant R is multiplied with the variable I. So I is plotted along the x axis
• We cannot plot I along y axis

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In the next section, we will see more details about resistors.

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Chapter 8.1 - Combination of Cells to form Battery

In the previous section we saw how a cell provides the required electromotive force. In this section, we will see combination of cells. Later in this section, we will see the methods of connecting voltmeter and ammeter.
First we will see the method for measuring current. We will write it in steps:
1. We know that electric current is the flow of charges. 
• So if more charges flow, can we say that there is more current?
• To answer this question, we must introduce 'time' also into the calculations. 
2. Consider two different circuits A and B. Each has a conductor connected to a cell. This is shown in fig.8.9 below:

Fig.8.9

• Let in a time of 5 seconds, Q_A charge flow through the conductor in the circuit A
• Let in a time of 5 seconds, Q_B charge flow through the conductor in the circuit B
3. Note that time is the same 5 seconds. So if Q_A > Q_B, we can say that there is more current in circuit A
• So current can be defined as the charge flowing in one second through a conductor.
4. The unit for charge is coulomb. It's symbol is C
• The unit of time is seconds. It's symbol is s
• If in a time of 5 s, 10 C charge flows through a conductor, then:
The charge which flows in one second is ¹⁰⁄₅ = 2
5. So '2' is the charge which flows in one second and so it is the current.
• In other words:
Current in that conductor = ^charge⁄_time = ¹⁰⁄₅ = 2 ^coulombs⁄_second ⟹ 2 ^C⁄_s
■ Current is the quantity of charge that flows through a conductor in a circuit in one second 
6. This '^C⁄_s' is given a special name: Ampere. It's symbol is A
• So we say: The current in that conductor is 2 amperes or 2 A
7. The device used to measure current is called ammeter. 
• Some images of can be seen here. The position of the needle gives the current in the circuit.

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Now we can discuss about combination of cells
■ A cell is a single unit which produces electrical energy. But a battery is a combination of two or more cells.
The connection between the cells can be done in two ways:
I. Series connection
• In this method, the cells are connected one after the other
• The positive of one cell is connected to the negative of another cell
• This is shown in fig.8.10(a) below. It shows 3 cells connected in series.

Fig.8.10

• Fig.8.10(b) shows the symbolic representation of three cells connected in series
■ The fig.8.11 below explains the 'symbolic representation of cells':

Fig.8.11

The explanation can be written in steps:
1. In fig.8.11(a) we see two vertical lines. 
• The longer vertical line indicates the positive terminal of the cell
• The shorter vertical line indicates the negative terminal of the cell
2. Also in fig.8.11(a), the horizontal lines indicate the leads taken from the terminals of the cell
3. When the cells are connected in series, the symbolic representation will be as shown in fig.8.11(b)
• The longer vertical lines and shorter vertical lines are drawn alternately
    ♦ This indicates the positive of one cell being connected to the negative of another cell

II. Parallel connection
In this method, similar poles are connected together. This is shown in fig.8.12(a) below:

Fig.8.12

• Fig.8.12(b) shows the symbolic representation

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• In the discussion so far in this chapter, we have seen two devices:
    ♦ Voltmeter to measure the potential difference between two points in a circuit
    ♦ Ammeter to measure the current flowing through a circuit
• Now we will see how each of these devices are connected to a circuit
• First we will see the voltmeter. We will write the connection details in steps:
1. Consider the circuit in fig.8.13 below:

Fig.8.13

• The switch is turned on. So current will flow through the circuit. This current is indicated by the letter 'I'
• When 'I' passes through the bulb, it will glow. 
2. We know that, for the current I to pass through the bulb, there must be a potential difference between the two points X and Y. 
• We want to know this potential difference. 
• For that, we will use the voltmeter. 
3. But how do we connect the voltmeter into the circuit?
• Consider fig.8.14(a) below:

Fig.8.14

• The voltmeter is connected in series with the bulb. 
• In this connection, the voltmeter will surely give us a reading. 
• But the reading is useless to us. Let us see the reason:
4. The current I flows from the positive terminal of the battery to it's negative terminal. 
• This direction is indicated by the arrow marks. 
5. During this flow, some energy will be lost in the form of heat energy. 
• So the actual energy supplied by the battery will not be available at every part of the circuit. 
• As the distance from the battery increases, energy loss will be more
6. Now consider the voltmeter in the fig.a. The reading in it will correspond to the potential difference between it's own terminals. 
• We want the potential difference between X and Y
7. The current leaves the voltmeter meter and has to flow some more distance to reach X and Y.
• During this flow, there will be a further 'drop in potential'. 
• So we will not get the required value between X and Y. 
• Thus the reading is useless to us
8. Now consider fig.8.14(b)
• The voltmeter is connected in parallel with the bulb. 
• The leads from the voltmeter are connected to X and Y
• In this connection, we will get the potential difference between X and Y
• So this is the correct method. A voltmeter should always be connected in parallel. 
■ Note that in fig.b, the current I splits into I₁ and I₂ at the point X. But they combine together to form I at Y

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Now we will see how the ammeter is connected. We will write the steps:
1. Consider the same fig.8.13 that we saw previously. This time, we want to know the current flowing through the bulb. 
2. Consider fig.8.15(a) below:

Fig.8.15

• We can see that the current I in the circuit is flowing through the ammeter also. 
• So it will give the correct reading
3. But in fig.8.15(b), only the portion I₁ is flowing through the ammeter. 
• This is because, the current I splits into I₁ and I₂ at X. 
• So this connection will not give the correct reading. 
• An ammeter should always be connected in series

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In the next section, we will see the relation between Voltage and Current.

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