In our daily life we use electricity for various activities. The electric lamps and tubes light our houses, we listen music on a tape recorder or radio, see different programmes on television, enjoy cool breeze from electric fan or cooler, and use electric pump to irrigate fields. In fact, electricity is a unique gift of science to mankind. We can not imagine life without electricity in the modern world. At home you might have observed that as soon as you switch on an electrtc lamp, it starts glowing. Why does it happen? What is the function of a switch?
In the preceding lessons of this module, you have studied about static electric charges and forces between them. In this lesson, you will learn about electric charges in motion. You will also learn that the rate of flow of charge through a conductor depends on the potential difference across it. You will also study the distribution of current in circuits and Kirchhoff’s laws which govern it. Elementary idea of primary and secondary cells will also be discussed in this lesson.
Physics is an experimental science and the progress it has made to unfold laws of nature became possible due to our ability to verify theoretical predictions or reproduce experimental results. This has led to continuous improvement in equipment and techniques. In this lesson you will learn about potentiometer, which is a very versatile instrument. It can be used to measure resistance as well as electro-motive force using null method.
OBJECTIVES
After seen this video, you should be able to :
- state Ohm’s law and distinguish between ohmic and non-ohmic resistances;
- obtain equivalent resistance for a series and parallel combination of resistors;
- distinguish between primary and secondary cells;
- apply Kirchhoff’s rules to closed electrical circuits;
- apply Wheatstone bridge equation to determine an unknown resistance;
- and explain the principle of potentiometer and apply it to measure the e.m.f and internal resistance of a cell.
ELECTRIC CURRENT
You have studied in the previous lesson that when a potential difference is applied across a conductor, an electric field is set up within it. The free electrons move in a direction opposite to the field through the conductor. This constitutes an electric current. Conventionally, the direction of current is taken as the direction in which a positive charge moves. The electrons move in the opposite direction. To define current precisely, let us assume that the charges are moving perpendicular to a surface of area A, as shown in Fig. 17.1. The current is the rate of flow of charge through a surface area placed perpendicular to the direction of flow. If charge Δq flows in time Δt, the average current is defined as :
If the rate of flow of charge varies with time, the current also varies with time. The instantaneous current is expressed as :
The electric current through a conductor is the rate of transfer of charge across a surface placed normal to the direction of flow.
The SI unit of current is ampere. Its symbol is A :
OHM’S LAW
In 1828, Ohm studied the relation between current in a conductor and potential difference applied across it. He expressed this relation in the form of a law, known as Ohm’s law.
According to Ohm’s law, the electric current through a conductor is directly proportional to the potential difference across it, provided the physical conditions such as temperature and pressure remain unchanged.
Let V be the potential difference applied across a conductor and I be the current flowing through it. According to Ohm’s law
According to Ohm’s law, the electric current through a conductor is directly proportional to the potential difference across it, provided the physical conditions such as temperature and pressure remain unchanged.
V ∝ I
Or V = RI
⇒ V I = R
where constant of proportionality R signifies the electrical resistance offered by a conductor to the flow of electric current. Resistance is the property of a conductor by virtue of which it opposes the flow of current through it. The I–V graph for a metallic conductor is a straight line (Fig. 17.3(a)).
The SI unit of resistance is ohm. It is expressed by symbol Ω (read as omega)
1 ohm = 1 volt/1 ampere
Most of the metals obey Ohm’s law and the relation between voltage and current is linear. Such resistors are called ohmic. Resistors which do not obey Ohm’s law are called non-ohmic. Devices such as vacuum diode, semiconductor diode, transistors.
show non ohmic character. For semiconductor diode, Ohm’s law does not hold good even for low values of voltage. Fig. 17.3(b) shows a non-linear I–V graph for a semiconductor diode.
ACTIVITY 17.1
Aim : To study conduction of electricity through an electrolyte.
Material Required: Ammeter, Voltmeter, a jar containing copper sulphate solution, two copper plates, a battery, plug key, connecting wires and a rheostat.
How to Proceed :
17.2.1 Resistance and Resistivity
Let us now study the factors which affect the resistance of a conductor. You can perform two simple experiments. To do so, set up a circuit as shown in Fig. 17.6.
ACTIVITY 17.2
Take a long conducting wire of uniform cross section. Cut out pieces of different lengths, say l1 , l2 , l3 , etc from it. This makes sure that wires have same area of cross-section. Connect l 1 between A and B and note down the current through this wire. Let this current be I. Perform the same experiment with wires of lengths l2 and l3 , one by one. Let the currents in the wires be I2 and I3 respectively. Plot a graph between l -1 and I. You will find that the graph is a straight line and longer wires allow smaller currents to flow. That is, longer wires offer greater resistance [Fig.17.7(a)]. Mathematically, we express this fact as
R ∝ l (17.6)
ACTIVITY 17.3
Take wires of the same length of a given material but having different areas of cross section, say A1 , A2 , A 3etc. Connect the wires between A and B one by one and note down the currents I1 , I2 , I3 etc. in each case. A plot of I and A will give
a straight line. Wires of greater cross sectional area allow greater currents to flow. You may say that wires of larger area of cross-section offer smaller resistance [Fig. 17.7 (b)]. Mathematically, we can write
R ∝ 1 /A
On combining Eqns.(17.6) and (17.7), we can write.
R ∝ l/A
or R = ρ l/A
where ρ is a constant for the material at constant temperature. It is called the specific resistance or resistivity of the material. By rearranging terms, we can write
ρ = RA/ l
Unit of conductivity is Ohm-1 metre-1 or mho-metre-1 or Sm-1.
Resistivity depends on the nature of the material rather than its dimensions, whereas the resistance of a conductor depends on its dimensions as well as on the nature of its material.
You should now study the following examples carefully.
Example 17.1 : In our homes, the electricity is supplied at 220V. Calculate the resistance of the bulb if the current drawn by it is 0.2A.
Solution :
Example 17.2 : A total of 6.0 × 1016 electrons pass through any cross section of a conducting wire per second. Determine the value of current in the wire.
Solution : Total charge passing through the cross-section in one second is
Example 17.3 : Two copper wires A and B have the same length. The diameter of A is twice that of B. Compare their resistances.
Solution : From Eqn. (17.8) we know that
Since diameter of A = 2 × diameter of B , we have rA = 2rB. Hence
Resistance of B will be four times the resistance of A.
Example 17.4 : The length of a conducting wire is 60.0 m and its radius is 0.5cm. A potential difference of 5.0 V produces a current of 2.5 A in the wire. Calculate the resistivity of the material of the wire.
INTEXT QUESTIONS 17.1
1. (a) A current I is established in a copper wire of length l. If the length of the wire is doubled, calculate the current due to the same cell.
(b) What happens to current in an identical copper wire if the area of cross section is decreased to half of the original value?
2. The resistivity of a wire of length l and area of cross section A is 2 × 10–8Ωm. What will be the resistivity of the same metallic wire of length 2l and area of cross section 2A?
3. A potential difference of 8 V is applied across the ends of a conducting wire of length 3m and area of cross section 2cm2 . The resulting current in the wire is 0.15A. Calculate the resistance and the resistivity of the wire.
4. Do all conductors obey Ohm’s law? Give examples to support your answer.
5. 5 × 1017 electrons pass through a cross-section of a conducting wire per second from left to right. Determine the value and direction of current.
17.3 GROUPING OF RESISTORS
An electrical circuit consists of several components and devices connected together. Some of these are batteries, resistors, capacitors, inductors, diodes, transistors etc. (They are known as circuit elements.) These are classified as resistive and reactive. The most common resistive components are resistors, keys, rheostats, resistance coils, resistance boxes and connecting wires. The reactive components include capacitors, inductors and transformers. In addition to many other functions performed by these elements individually or collectively, they control the current in the circuit. In the preceding lesson you learnt how grouping of capacitors can be used for controlling charge and voltage. Let us now discuss the role of combination of resistors in controlling current and voltage.
Two types of groupings of resistors are in common use. These are : series grouping and parallel grouping. We define equivalent resistance of the combination as a single resistance which allows the same current to flow as the given combination when the same potential difference is applied across it.
17.3.1 Series Combination
That is, the equivalent resistance of a series combination of resistors is equal to the sum of individual resistances. If we wish to apply a voltage across a resistor (say electric lamp) less than that provided by the constant voltage supply source, we should connect another resistor (lamp) in series with it.
17.3.2 Parallel Combination
You may connect the resistors in parallel by joining their one end at one point and the other ends at another point. In parallel combination, same potential difference exists across all resistors. Fig. 17.9 shows a parallel combination of two resistors R1and R2 . Let the combination be connected to a battery of voltage V and draw a current I from the source
Note that the equivalent resistance of parallel combination is smaller than the smallest individual resistance. You may easily see this fact by a simple electrical circuit having a resistor of 2 Ω connected across a 2V battery. It will draw a current of one ampere. When another resistor of 2 Ω is connected in parallel, it will also draw the same current. That is, total current drawn from the battery is 2A. Hence, resistance of the circuit is halved. As we increase the number of resistors in parallel, the resistance of the circuit decreases and the current drawn from the battery goes on increasing.
In our homes, electrical appliances such as lamps, fans, heaters etc. are connected in parallel and each has a separate switch. Potential difference across each remains the same and their working is not influenced by others. As we switch on bulbs and fans, the resistance of the electrical circuit of the house decreases and the current drawn from the mains goes on increasing (Fig.17.10).
Solution :
(i) Three resistors (5Ω, 10Ω and 30Ω) are connected in parallel. Therefore, equivalent resistance is given by
INTEXT QUESTIONS 17.2
1. There are two bulbs and a fan in your bed room. Are these connected in series or in parallel? Why?
2. The electric supply in a town is usually at 220 V. Sometimes the voltage shoots upto 300 V and may harm your T V set and other gadgets. What simple precaution can be taken to save your appliances?
3. Calculate the equivalent resistance between points A and B for the following circuit :
TYPES OF RESISTORS
We use resistors in all electrical and electronic circuits to control the magnitude of current. Resistors usually are of two types :
- carbon resistors
- wire wound resistors
In a wire wound resistor, a resistance wire (of manganin, constantan or nichrome) of definite length, which depends on the required value of resistance, is wound two-fold over an insulating cylinder to make it non-inductive. In carbon resistors, carbon with a suitable binding agent is molded into a cylinder. Wire leads are attached to the cylinder for making connections to electrical circuits. Resistors are colour coded to give their values :
R = AB × 10C Ω, D
where A, B and C are coloured stripes. The values of different colours are given in Table 17.1. As may be noted,
– first two colours indicate the first two digits of the resistance value;
– third colour gives the power of ten for the multiplier of the value of the resistance; and
– fourth colour (the last one) gives the tolerance of the resistance, which is 5% for golden colour, 10% for silver colour and 20% for body colour
Suppose that four colours on a resistor are Blue, Grey, Green and Silver. Then
The first digit will be 6 (blue)
The second digit will be 8 (Grey)
The third colour signifies multiplier 105 (Green)
The fourth colour defines tolerance = 10% (Silver)
Hence value of the resistance is
17.5 TEMPERATURE DEPENDENCE OF RESISTANCE
The resistivity of a conductor depends on temperature. For most metals, the resistivity increases with temperature and the change is linear over a limited range of temperature :
Eqn. (17.14) can be rearranged to obtain an expression for temperature coefficient of resistivity :
17.6 ELECTROMOTIVE FORCE (EMF) AND POTENTIAL DIFFERENCE
EMF is the short form of electromotive force. EMF of a cell or battery equals the potential difference between its terminals when these are not connected (open circuit) externally. You may easily understand the difference between e.m.f. and potential difference of a cell by performing the following activity.
ACTIVITY 17.4
introduces a resistance r, called internal resistance of the cell. Let current I be flowing in the circuit. Potential drop Ir across internal resistance r due to current flow acts opposite to the e.m.f. of the cell. Hence, the voltmeter reading will be
E – Ir = V
or E = V + Ir
Thus while drawing current from a cell, e.m.f. of the cell is always greater than the potential difference across external resistance, unless internal resistance is zero.
E.M.F. of a cell depends on :
– the electrolyte used in the cell;
– the material of the electrodes; and
– the temperature of the cell.
Note that the e.m.f. of a cell does not depend on the size of the cell, i.e. on the area of plates and distance between them. This means that if you have two cells of different sizes, one big and one small, the e.m.f.s can be the same if the material of electrods and electrolyte are the same. However, cells of larger size will offer higher resistance to the passage of current through it but can be used for a longer time.
Example 17.8 :When the current drawn from a battery is 0.5A, potential difference at the terminals is 20V. And when current drawn from it is 2.0A, its voltage reduces to 16V. Calculate the e.m.f. and internal resistance of the battery.
Solution : Let E and r be the e.m.f. and internal resistance of battery. When current I is drawn from it, the potential drop across internal resistance of the cell is Ir. Then we can write
17.6.1 Elementary Idea of Primary and Secondary cells
Magnetism We have seen that to pass electric current through a conductor continuously we have to maintain a potential difference between its ends. For the purpose, generally, we use a device called chemical cell.
Chemical calls are of two types :
- Primary Cells : In these cells, the chemical energy is directly converted into electrical energy. The material of a primary cell is consumed as we use the cell and, therefore, it cannot be recharged and reused. Dry cell, Daniel Cell, Voltaic Cell etc are examples of primary cells.
- Secondary Cells : These are chemical cells in which electrical energy is stored as a reversible chemical reaction. When current is drawn from the cells the chemical reaction runs in the reverse direction and the original substances are obtained. These cells, therefore, can be charged again and again. Acid-accumulator, the type of battery we use in our inverter or car, is a set of secondary cells.
17.7 KIRCHHOFF’S RULES
You now know that Ohm’s law gives current–voltage relation for resistive circuits. But when the circuit is complicated, it is difficult to know current distribution by Ohm’s law. In 1842, Kirchhoff formulated two rules which enable us to know the distribution of current in complicated electrical circuits or electrical networks.
(i) Kirchhoff’s First Rule (Junction Rule) : It states that the sum of all currents directed towards a junction (point) in an electrical network is equal to the sum of all the currents directed away from the junction
Refer to Fig. 17.16.If we take currents approaching point A as positive and those leaving it as negative, then we can write
I = I1 + I2 + I3
or I – (I1 + I2 + I3 ) = 0
In other words, the algebraic sum of all currents at a junction is zero.
Kirchhoff’s first rule tells us that there is no accumulation of charge at any point if steady current flows in it. The net charge coming towards a point should be equal to that going away from it in the same time. In a way, it is an extension of continuity theorem in electrical circuits.
(ii) Kirchhoff’s Second Rule (Loop Rule) : This rule is an application of law of conservation of energy for electrical circuits. It tells us that the algebraic sum of the products of the currents and resistances in any closed loop of an electrical network is equal to the algebraic sum of electromotive forces acting in the loop.
While using this rule, we start from a point on the loop and go along the loop either clockwise or anticlockwise to reach the same point again. The product of current and resistance is taken as positive when we traverse in the direction of current. The e.m.f is taken positive when we traverse from negative to positive electrode through the cell. Mathematically, we can write.
In more general form, Kirchhoff’s second rule is stated as : The algebraic sum of all the potential differences along a closed loop in a circuit is zero.
Example 17.9 : Consider the network shown in Fig. 17.18. Current is supplied to the network by two batteries. Calculate the values of currents I 1 , I2 and I3 . The directions of the currents are as indicated by the arrows.
Solution : Applying Kirchhoff’s first rule to junction C, we get
17.7.1 Wheatstone Bridge
You have learnt that a resistance can be measured by Ohm’s law using a voltmeter and an ammeter in an electrical circuit. But this measurement may not be accurate for low resistances. To overcome this difficulty, we use a wheatstone bridge. It is an arrangement of four resistances which can be used to measure one of them in terms of the other three.
Consider the circuit shown in Fig. 17.19 where
- P and Q are two adjustable resistances connected in arms AB and BC.
- R is an adjustable known resistance.
- S is an unknown resistance to be measured.
- A sensitive galvanometer G along with a key K2 is connected in the arm BD.
- A battery E along with a key K1 is connected in the arm AC. On closing the keys, in general, some current will flow through the galvanometer and you will see a deflection in the galvanometer. It indicates that there is some potential difference between points B and D. We now consider the following three possibilities:
- Point B is at a higher potential than point D : Current will flow from B towards D and the galvanometer will show a deflection in one direction, say right
- Point B is at a lower potential than point D : Current will flow from point D towards B and the galvanometer will show a deflection in the opposite direction.
- Both points B and D are at the same potential: In this case, no current Magnetism will flow through the galvanometer and it will show no deflection, i.e. the galvanometer is in null condition. In this condition, the Wheatstone bridge is said to be in the state of balance.
The points B and D will be at the same potential only when the potential drop across P is equal to that across R. Thus
- The balance condition given by Eqn. (17.27) at null position is independent of the applied voltage V. In other words, even if you change the e.m.f of the cell, the balance condition will not change.
- The measurement of resistance does not depend on the accuracy of calibration of the galvanometer. Galvanometer is used only as a null indicator (current detector).
The main factor affecting the accuracy of measurement by Wheatstone bridge is its sensitivity with which the changes in the null condition can be detected. It has been found that the bridge has the greatest sensitivity when the resistances in all the arms are nearly equal.
Example 17.9: Calculate the value of R shown in Fig.17.20. when there is no current in 50Ω resistor
Solution: This is Wheatstone bridge where galvanometer has been replaced by 50Ω resistor. The bridge is balanced because there is no current in 50Ω resistor. Hence,
INTEXT QUESTIONS 17.3
1. Refere to figure below. Calculate the value of currents in the arms AB, AD and BD.
17.8 POTENTIOMETER
You now know how to measure e.m.f. of a source or potential difference across a circuit element using a voltmeter. (An ideal voltmeter should have infinite resistance so that it does not draw any current when connected across a source of e.m.f.) Practically it is not possible to manufacture a voltmeter which will not draw any current. To overcome this difficulty, we use a potentiometer, which draws no current from it. It employs a null method. The potentiometer can also be used for measurement of internal resistance of a cell, the current flowing in a circuit and comparison of resistances.
17.8.1 Description of a Potentiometer
A potentiometer consists of a wooden board on which a number of resistance wires (usually ten) of uniform cross-sectional area are stretched parallel to each other. The wire is of maganin or nichrome. These wires are joined in series by thick copper strips. In this way, these wires together act as a single wire of length equal to the sum of the lengths of all the wires. The end terminals of the wires are provided with connecting screws.
A metre scale is fixed on the wooden board parallel to wires. A jockey (a sliding contact maker) is provided with the arrangement. It makes a knife edge contact at any desired point on a wire. Jockey has a pointer which moves over the scale. It determines the position of the knife edge contact. In Fig. 17.21 a ten wire potentiometer is shown. A and B are ends of the wire. K is a jockey and S is a scale. Jockey slides over a rod CD.
17.8.2 Measurements with a Potentiometer
If r is the resistance per unit length of the wire, and k is the potential drop across unit length of the wire, then
The measurements with potentiometer have following advantages :
- When the potentiometer is balanced, no current is drawn from the circuit on which the measurement is being made.
- It produces no change in conditions in a circuit to which it is connected.
- It makes use of null method for the measurement and the galvanometer used need not be calibrated.
17.8.3 Comparison of E.M.Fs of two Cells
You have learnt to measure the e.m.f. of a cell using a potentiometer. We shall now extend the same technique for comparison of e.m.fs of two cells. Let us take, for example, a Daniel cell and a Leclanche cell and let E1 and E2 be their respective e.m.fs.
Refer to circuit diagram shown in Fig.17.23. The cell of e.m.f. E1 is connected in the circuit through terminals 1 and 3 of key K1 . The balance point is obtained by moving the jockey on the potentiometer wire as explained earlier. Note that e.m.f of cell E should be greater than the emfs of E1 and E2 seperately. (Otherwise, balance point will not be obtained.) Let the balance point on potentiometer be at point Y1 and length AY1 = l1 . The cell of e.m.f. E2 is connected in the circuit through terminals 2 and 3 of the key K2 . Suppose balance is obtained at point Y2and length AY2 = l2 .
17.9 DRIFT VELOCITY OF ELECTRONS
Let us now understand the microscopic picture of electrical conduction in a metal. The model presented here is simple but its strength lies in the fact that it conforms to Ohm’s law.
We assume that a metallic solid consists of atoms arranged in a regular fashion. Each atom usually contributes free electrons, also called conduction electrons. These electrons are free to move in the metal in a random manner, almost the same way as atoms or molecules of a gas move about freely in the a container. It is for this reason that sometimes conduction electrons are referred to as electron gas. The average speed of conduction electrons is about 106 ms-1.
We know that no current flows through a conductor in the absence of an electric field, because the average velocity of free electrons is zero. On an average, the number of electrons moving in +x direction is same as number of electrons moving in –x direction. There is no net flow of charge in any direction.
The conduction electrons frequently collide with the atoms in the solid. The free electrons drift slowly in a direction opposite to the direction of the applied electric field. The average drift velocity is of the order of 10– 4ms–1. This is very small compared to the average speed of free electrons between two successive collisions (106ms-1). On applying an electric field, the conduction electrons get accelerated.
The excess energy gained by the electrons is lost during collisions with the atoms. The atoms gain energy and vibrate more vigorously. The conductor gets heated up. Fig. 17.25 shows how the motion of electrons is modified when an electric field is applied is applied.
Let us now obtain an expression for the drift velocity of conduction electrons. Let e and m be the charge and mass respectively of an electron. If E is the electric field, the force on the electron is eE. Hence acceleration experienced by the electron is given by
17.10 POWER CONSUMED IN AN ELECTRICAL CIRCUIT
Let us examine the circuit in Fig. 17.26 where a battery is connected to an external resistor R . The positive charges (so to say) flow in the direction of the current in the resistor and from negative to positive terminal inside the battery. The potential.
difference between two points gives kinetic energy to the charges. These moving Magnetism charges collide with the atoms (ions) in the resistor and thus lose a part of their kinetic energy. This energy increases with the temperature of the resistor. The loss of energy by moving charges is made up at the expense of chemical energy of the battery.
The electrical power lost in a conductor as heat is called joule heat. The heat produced is proportional to : (i) square of current (I), (ii) resistance of conductor (R), and (iii) time for which current is passed (t).
The statement Q = I 2 Rt, is called Joule’s law for heating effect of current.
Example: 17.11 : A 60W lamp is connected to 220V electricity supply in your home. Calculate the power consumed by it, the resistance of its filament and the current through it.
