After carefully studying this chapter, you should be able to do the following:
➢Explain the concept of voltage, current, and resistance.
➢State and explain Ohm’s law.
➢Distinguish between electrical power and energy.
➢Distinguish between series and parallel connections of resistances.
➢Calculate branch currents and potential drops in parallel and series-parallel circuits.
➢Reduce a ladder network into a single equivalent resistance.
➢Explain the basic function of circuit elements like resistance, inductance, and capacitance.
➢Distinguish between self-inductance and mutual inductance and establish their relationship.
➢Determine the factors on which resistance, inductance, and capacitance depend.
➢Explain how an inductor and a capacitor can work as energy-storing devices.
➢Explain the role of magnetic core material in an inductor.
➢Explain the role of dielectric material in a capacitor.
➢Distinguish between series and parallel connections of resistors and capacitors.
➢Be conversant with the network terminologies.
➢Distinguish between a voltage source and a current source.
➢Convert a voltage source into a current source and vice-versa.
In this chapter, we shall review the basic concepts related to electrical networks. The students must have studied these concepts in their earlier classes. However, a brief review will help them to understand the contents of chapters that would follow. We will review the basic concepts of voltage, current, resistance, Ohm’s law, electrical power and electrical energy, series and parallel connections of resistors and division of currents and voltages, and circuit parameters such as resistor, inductor and capacitor.
When two oppositely charged bodies are connected by a wire, flow of electrons occurs from one body to the other. This movement of electrons constitutes a flow of electric current. The wire that allows the flow of electrons is called a conductor.
Electromotive force is an electric pressure that causes current to flow through a conducting wire. A conducting wire of large cross-sectional area allows more current to flow than a conductor with small cross-sectional area. Therefore, a conductor with a large cross-sectional area offers less resistance to flow of current than a conductor with a small cross-sectional area. Therefore, we can say that resistance is the opposition to the flow of current through a conductor.
Voltage or Electro motive force (EMF) or potential difference is an electric pressure that causes the flow of current, that is, the movement of electrons through a conductor. The opposition offered by the conductor to the flow of current is called its resistance.
The direction of electron flow is from the negatively charged body to the positively charged body. However, current flow is considered from the positive to the negative terminal.
The current that flows in one direction is called direct current (DC). In alternating current (AC), the flow of charge reverses directions alternately every fraction of a second (every half-cycle). The cycle repeats over and over again.
In an electric circuit, current, voltage and resistance are related by an important law, called the Ohm’s law. Ohm’s law is expressed as follows:
where I is in amperes, V is in volts and R is in ohms.
Ohm’s law states that the current flowing through a resistance is directly proportional to the potential difference between its ends and inversely proportional to the value of the resistance, provided the temperature remains constant.
By applying Ohm’s law, the current flowing through a resistor can be calculated by measuring the voltage drop across it, provided the value of the resistance is known.
Conductance (G) is the inverse of resistance (R). In terms of conductance, Ohm’s law can be represented as follows:
The unit of G is siemens and that of R is ohms.
When electric current flows through a conducting material, work is done in moving the electrons. Further, heat is dissipated as a result of the work done.
Work is also done when electricity is converted into light, sound and heat as in an electric lamp, a speaker and an electric heater, respectively.
Power (P) is defined as the time rate of doing work (W). The power supplied to any electrical equipment is as follows:
Energy consumed is expressed as watt-hour or kilowatt-hour (kWh). Energy consumed or work done is as follows:
Energy = Power × Time
W = P × t Wh or kWh;
The unit of work is joule. If unit of power is watt and unit of time is second,
Calorie is the unit of energy
1 calorie = 4.2 joules
We can convert electrical energy in kWh into calories as
Thus, we have explained the relationship between work, power, and energy.
A number of resistors need to be connected in series, parallel, or series-parallel in practical circuits. It is often required to determine their equivalent resistance and also calculate current and voltage distribution in these resistances. The steps are explained in the following sections.
When a number of resistors are connected in series, there is only one path through which the current can flow. The magnitude of current is the same in all parts or components of a series circuit.
When a number of resistances are connected in series, their equivalent resistance is given as follows:
Series-connected resistors can be used as a voltage divider as shown in Figure 1.1.
As in Figure 1.1,
By applying Ohm’s law, we get the following:
Voltage Divider Rule in Series Circuits
From the above, the voltage divider rule is expressed as follows:
Resistors are said to be connected in parallel when the same voltage is applied across each resistor. However, the current through each parallel branch will be different.
When a number of resistances are connected in parallel, their equivalent resistance is derived as the following form:
when two resistances, R1 and R2 are connected in series, Req = R1 + R2 and when connected in parallel, .
Current Divider Rule in Parallel Circuit
Figure 1.2 shows two resistors, R1 and R2, which are connected in parallel across a voltage source V.
By applying Ohm’s law, we can write the following:
Further, by assuming that current I is divided into currents I1 and I2, we obtain the following:
Therefore, the current divider rule in a parallel circuit as in Figure 1.2, is as follows:
A series-parallel circuit is shown in Figure 1.3. The current and voltage drops in the circuit are calculated as
The voltage across terminals A and B is equated to the voltage drops in the two parallel paths. Therefore,
Figure 1.4 shows a ladder network. This is a special case of series-parallel network. By replacing series-connected resistors into their equivalents, such circuits can be reduced into a simple circuit.
To reduce the circuit into a simple circuit, we proceed in steps as shown in Figure 1.5. Across terminals C and D, R5 and R6 are in series and connected in parallel with R4 to form Rx. Then across terminals A and B, R3 and Rx are in series and connected in parallel with R2 to form Ry. The equivalent resistance is Ry.
Resistance of the whole network across the voltage V is therefore, equals to R1 + Ry.
Example 1.1 Calculate the current supplied by the 12V battery in the network shown in Figure 1.6.
Solution: Points A and C can be brought together. Similarly, points B and D can be brought together. First, we will bring A and C together; then, we combine B and D together as shown in Figure 1.7. The process is completed by taking equivalents of parallel resistances.
Example 1.2 Calculate the resistance between the terminals P and Q in the circuit shown in Figure 1.8.
Solution: The circuit is redrawn as shown in Figure 1.9. The resistance between terminals P and Q is calculated by taking the equivalent of series resistances.
So, the equivalent resistence between terminals P and Q is 8 Ω.
Example 1.3 What will be the equivalent resistance between terminals P and Q of the ladder network shown in Figure 1.10.
Solution: By considering series and parallel connection of resistors, the equivalent resistance across terminals P and Q is determined as shown in Figure 1.11.
So, the equivalent resistance, Req between terminals P and Q is
Example 1.4 Four resistances of equal value, R are connected as shown in Figure 1.12. Calculate the equivalent resistance across terminals P and Q.
Solution: We can redraw the circuit as shown in Figure 1.13 and find the equivalent resistance as shown using successive steps.
Thus, the equivalent resistance is R.
Example 1.5 In the circuit shown in Figure 1.14, calculate values of branch currents I1 and I2.
Solution: In the circuit, total current I is divided into I1 and I2. We will first calculate the total current I and then calculate the branch currents using current divided rule.
The basic circuit elements are resistors, inductors, and capacitors. These are described in brief in the following sections.
The resistors offer resistance to current flow. The common types of resistors are wire wound type and carbon composition type. Resistors can be of fixed value or variable or adjustable value. However, standard values of fixed resistors range from 2.7Ω to 22 MΩ with tolerances varying from ± 20% to ± 1%.
High-power resistors are generally wire wound and are mounted on ceramic tubes. Further, carbon composition resistors are normally of low power rating that ranges from fractions of a watt to a few watts.
When there is sudden growth (sudden increase) or decay (sudden decrease) in current flowing through a coil, a changing flux will be produced. According to Faraday’s law, an EMF is induced in a coil due to changing current flowing through it. The amount of induced EMF in the coil depends upon the self-inductance of the coil. The EMF will also be induced in another neighbouring coil due to the changing current in the coil which causes mutual inductance between the coils. Figure 1.15 shows that EMF (e) is induced in a coil when a changing magnetic flux, which is produced due to changing current flow, links the coil. EMF is induced only when the flux is changing. If the flux is constant, no EMF will be induced.
A changing current produces an EMF of self-induction e in a coil that varies directly with the time rate of change of current. The induced emf can be expressed as follows:
where L is called the self inductance or simply inductance of the coil
The inductance L of a coil is one henry, when an EMF of 1V is induced by the changing current of 1 A/s flowing through the coil.
If N is the number of turns, in terms of rate of change of flux linkage (Nϕ) with respect to i, inductance L of the coil is expressed as the following:
where, μ = μ0 μr is the permeability of the flux path; l is the length of the flux path; A is the area through which flux passes; B is the flux density.
It is to be noted that inductance of a coil gets increased many times if the permeability of the flux path is increased by placing a magnetic material as the core material for the coil.
When changing flux of one coil links another coil, that is, if the two coils are magnetically coupled, an EMF is also induced in the second coil also. Two coils have a mutual inductance of 1 H when an EMF of 1V is induced in one coil by the changing current at the other coil at the rate of 1 A/second.
The amount of flux linkage from one coil to the other is indicated by a factor called coefficient of coupling. Figure 1.16 shows two coupled coils having self-inductance of L1 and L2, respectively. According to equation (1.13) stated earlier,
On multiplying L1 and L2, we obtain the following expression:
The mutual inductance between the two coils is due to the EMF induced in the second coil for the change in current in the first coil.
Let e2 be the EMF induced in the second coil, as shown in Figure 1.16.
where M is the mutual inductance and e2 is the EMF induced in the second coil; di1/dt is the rate of change of current in the first coil.
According to Faraday’s law, the equation of induced EMF in the second coil is given as follows:
where K is the coefficient of coupling between the two coils.
We know, flux ϕ = BA, B is the flux density, and A is the area of the core of the coil.
Types of Inductors
Power Supply Inductors. The wave shape of power supplies and of a fluctuating DC can be improved by using an inductor. An inductor opposes any change in the current level; hence, the fluctuations are reduced due to the connection of an inductor in series.
Laboratory Inductors. Two coupled coils can be used to make a variable inductor. The coils may be connected either in series or in parallel. The total inductance value is changed by adjusting the position of one coil with respect to the other.
Moulded Inductors. Small moulded inductors of range 1-10 μH are made with a maximum current carrying capacity of 70 mA. Their values are colour-coded as is done for resistors. These inductors are of minute size and are used in electronic circuits.
Energy Stored in an Inductor
Energy is stored in an inductor in the form of an electromagnetic field when a current (I amperes) flows through the coil of inductance (L henries). The amount of energy stored (W) is given by the following expression:
Example 1.6 Two coils having number of turns 1000 and 2000, respectively, are placed side by side. The self-inductance of coil with 1000 turns is 800 mH. Only 80% of the flux produced by coil with 1000 turns links the second coil. Calculate the emf induced in coil 2 when current changes at the rate of 10 A/s in the coil with 1000 turns. In addition, calculate the mutual inductance of the two coils.
Solution: The two coils having magnetic coupling is shown in Figure 1.17.
Emf induced in coil 2, e2 is given as
Example 1.7 Two coils are placed side by side. A current of 4A amperes flowing through one coil of 400 turns produces a flux of 2 mWbs. When the current in the coil is suddenly reversed in 1 ms, an emf of 20 volts is induced in the second coil. The coefficient of coupling between the two coils is 50%. Calculate the self and mutual inductance of the two coils.
Solution: Let induced emf in the second coil be e2.
Current in the first coil has been reversed from 4A to −4A. So, total change of current is 8A and has taken place in 1 ms.
A capacitor is made using two thin metal plates separated by a layer of insulating material. The layer of insulating material between the capacitor plates is known as the dielectric. Typical dielectric materials are mica, paper, rubber, air, etc. Figure 1.18 shows a capacitor that has the ability to store electric charge.
A voltage V is applied across the capacitor by closing the switch S. The plates of the capacitor will aquire positive and negative charges as shown in Figure 1.18 (a). Free electrons from the positive plate will be attracted by the positive terminal of the battery thereby leaving positive charge on it and electrons will be supplied from the negative terminal of the battery to the other plate of the capacitor. These charges will be retained by the capacitor plates even after the switch S is opened. This is because the dielectric material placed between the plates will oppose any flow of charge through it. In this way, a capacitor stores charge.
It is interesting to note that both a voltage cell and a capacitor store charge. However, a capacitor cannot be used to supply a load current. This is because when load current is drawn from a voltage cell, the chemical action within the cell continuously creates free electrons to supply the current. In a capacitor, such action does not take place, and therefore, a capacitor cannot be considered as a voltage source.
When a voltage is applied across the two plates of a capacitor, the dielectric material gets subjected to electric pressure pressing for the flow of current through it. If the voltage applied is too high, the stress becomes more and at a certain high voltage, the material may break down. The dielectric strength of a given material is the voltage per unit thickness at which the dielectric may breakdown and hence cannot prevent the current flow. Dielectric strength of air is 3 kV/mm, whereas for mica, it is 200 kV/mm. So, mica is a better dielectric material than air.
The capacitance of a capacitor is expressed as follows:
where A is the area of the plates, d is the distance between the plates and ε is equal to ε0εr where, ε0 is the permittivity of free space and εr is the relative permittivity and also called dielectric constant. The value of ε0 is 8.85 × 10-12 F/m approximately. The value of dielectric constant for vacuum or air is 1 and for mica, it varies from 3 to 7. The value of dielectric constant for paper varies from 4 to 6, and for ceramic, the value may be as high as 1000. If the space between the two capacitor plates is filled by paper or polythene, or mica or other dielectric, the value of capacitance gets increased.
Types of Capacitors
Air Capacitors.Air capacitors are made using two sets of metal plates; one set fixed and the other set movable. Movable set of plates is adjusted by a rotatable shaft so that the area of the plates opposite to each other is changed.
Paper Capacitors.Paper capacitors are made by placing a layer of paper between two layers of metal foils. The metal foils and paper are rolled up and external connections are brought out from the metal foils.
Plastic Film Capacitors.These are made the same way as paper capacitors except that the paper is replaced by very thin film of plastic material like Polystyrene or Mylar.
Mica Capacitors.Alternate layers of metal foil and mica are used to make mica capacitors. Connections are taken out from metal foils. The whole of the assembly is encapsulated (placed inside) in moulded plastic jacket.
Ceramic Capacitors.A ceramic disc is used as the dielectric. Thin films of metal are deposited on each side of the ceramic disc. Connections are taken out from the metallic sides. The whole assembly is encapsulated in a container made of a plastic material.
Electrolytic Capacitors.Electrolytic capacitors are made the same way as paper capacitors. Thin sheets of aluminium foils separated by an electrolyte is rolled up and enclosed in a small aluminium cylinder. Terminals are brought out from the two metallic foils. A direct voltage is applied to the capacitor that causes a thin layer of aluminium oxide to form on the positive plate towards the side of the electrolyte. The aluminium oxide formed becomes the dielectric. The electrolyte and the foil on one side makes one plate while the positive sheet of foil on the other side forms the second plate. Electrolytic capacitors have one terminal identified for positive connection. An electrolytic capacitor should be connected with correct polarity; otherwise, the capacitor may explode causing serious injuries to persons handling the device.
Series and Parallel Connections of Capacitors
Series Connection.Series connection of capacitors results in increasing the total thickness of the dielectric between the two outermost plates. For example, when three parallel-plate capacitors are connected in series, the total capacitance for three series-connected capacitors as shown in Figure 1.19 is expressed as the following:
Parallel Connection.Parallel connection of capacitors is equivalent to increasing the total plate area. When three capacitors are connected in parallel, as has been shown in Figure 1.20, their total capacitance Cp is given as follows:
The capacitance of 1 farad is the capacitance of the capacitor that contains a charge of 1 coulomb when the potential difference between the plates is 1 volt.
From the definition of farad, charge Q can be expressed as follows:
where C is the capacitance in farad, V is the voltage between the plates in volts and Q is the charge in coulombs. When a capacitor is charged, energy is supplied to it which the capacitor stores as stored energy. Current is the rate of flow of charge, so that charge, Q = I × t.
Energy supplied in charging the capacitor is,
When a circuit containing a resistance and an inductance in series is switched on to a DC voltage source, maximum level of counter EMF is induced in the inductor, thereby making the initial current to be zero. However, the counter EMF gradually falls to zero and hence the current grows gradually to its maximum value.
When a circuit containing a resistance and a capacitance in series is switched on to DC voltage, the charging current is maximum initially and then the current falls to zero.
Before discussing various laws and theorems, certain terminologies related to DC networks are described first. These include the definition of terms used, voltage and current sources, series and parallel circuits and voltage and current source transformations.
While discussing network theorems, laws, and electrical and electronic circuits, one often comes across the following terms.
1.Circuit: A conducting path through which an electric current either flows or is intended to flow is called a circuit.
2.Electric network: A combination of various circuit elements, connected in any manner, is called an electric network.
3.Linear circuit: The circuit whose parameters are constant, that is, they do not change with the application of voltage or current is called a linear circuit.
4.Non-linear circuit: The circuit whose parameters change with the application of voltage or current is called a non-linear circuit.
5.Circuit parameter: The various elements of an electric circuit are called its parameters, such as resistance, inductance, and capacitance.
6.Bilateral circuit: A bilateral circuit is one whose properties or characteristics are the same in either direction, for example, transmission line.
7.Unilateral circuit: A unilateral circuit is one whose properties or characteristics change with the direction of its operations, for example, diode rectifier.
8.Active network: An active network is one that contains one or more sources of EMF.
9.Passive network: A passive network is one that does not contain any source of EMF.
10.Node: A node is a junction in a circuit where two or more circuit elements are connected together. For convenience, the nodes are labelled by letters.
11.Branch: The part of a network that lies between two junctions is called a branch.
12.Loop: A loop is a closed path in a network formed by a number of connected branches.
13.Mesh: Any path that contains no other paths within it is called a mesh. Thus, a loop contains meshes, but a mesh does not contain a loop.
For example, let us consider an electric circuit as shown in Figure 1.21.
2.Number of branches (B) = 5 (that is, AB, BC, BD, CD, AD)
3.Independent meshes (M ) = B − N + 1
= 5 − 4 + 1 = 2 (that is, ABDA, BCDB)
4.Number of loops = 3 (that is, ABDA, BCDB, and ABCDA). It is seen that a loop ABCDA encloses two meshes, that is, mesh I and mesh II.
A source is a device that converts mechanical, thermal, chemical, or some other form of energy into electrical energy. There are two types of sources: voltage source and current source.
Voltage sources are further categorised as ideal voltage source and practical voltage source. Examples of voltage sources are batteries, dynamos, alternators, etc. An ideal voltage source is defined as the energy source that gives constant voltage across its terminals irrespective of current drawn through its terminals. The symbol of ideal voltage source is shown in Figure 1.22(a). In an ideal voltage source, the terminal voltage is independent of the value of the load resistance (RL) connected. Whatever is the voltage of the source, the same voltage is available across the load terminals of RL, that is, VL = VS under loading condition as shown in Figure 1.22(b). There is no drop of voltage in the source supplying current to the load. The internal resistance of the source is therefore zero.
In a practical voltage source, voltage across the load will be less than the source voltage due to voltage drop in the resistance of the source itself when a load is connected, as shown in Figure 1.22(c).
In certain applications, a constant current flow through the circuit is required. When the load resistance is connected between the output terminals, a constant current (IL) will flow through the load.
Some examples of current source are photoelectric cells, collector current in transistors, etc. The symbol of current source is shown in Figure 1.23.
Practical Voltage and Current Sources
A practical voltage source like a battery has the drooping load characteristics due to some internal resistance. A voltage source has small internal resistance in series, whereas a current source has some high internal resistance in parallel.
1.For ideal voltage source, Rse = 0.
2.For ideal voltage source, Rsh = ∞.
A practical voltage source is shown as an ideal voltage source in series with a resistance. This resistance is called the internal resistance of the source, as shown in Figure 1.24(a). A practical current source is shown as an ideal current source in parallel with its internal resistance, as shown in Figure 1.24(b).
From Figure 1.24(a), we can write the following expression:
VL (open circuit), that is, VL (OC) = VS
That is, when the load RL is removed, the circuit becomes an open circuit and the voltage across the source becomes the same as the voltage across the load terminals.
When the load is short-circuited, the short-circuit current, IL(SC) = Vs/Rse
In the same way, from Figure 1.24(b), we can obtain the following expression:
Independent and Dependent Sources
The magnitude of an independent source does not depend upon the current in the circuit or voltage across any other element in the circuit. The magnitude of a dependent source gets changed due to some other current or voltage in the circuit. An independent source is represented by a circle, while a dependent source is represented by a diamond-shaped symbol. Dependent voltage sources are also called controlled sources.
There are four kinds of dependent sources:
1.voltage-controlled voltage source (vcvs)
2.current-controlled current source (cccs)
3.voltage-controlled current source (vccs)
4.current-controlled voltage source (ccvs)
The dependent voltage sources find applications in electronic circuits and devices.
A voltage source can be represented by an equivalent current source. Similarly, a current source can be represented by an equivalent voltage source. This source transformation often provides simplified solutions of circuit problems.
Transformation of Voltage Source into Current Source and Current Source into Voltage Source
A voltage source is equivalent to a current source and vice-versa if they produce equal values of IL and VL when connected to the load RL. They should also provide the same open-circuit voltage and short-circuit current.
If voltage source is converted into current source as in Figure 1.25, we consider the short-circuit current equivalence. The short-circuit currents in the two equivalent circuits are, respectively, Vs/Rse and Is. Therefore, the equivalent current source is represented as .
If the current source is converted into voltage source, as in Figure 1.26, we will consider the open-circuit voltage equivalence. The open circuit voltage in the two circuits are, repectively, Ish Rsh and Vs. Therefore, the equivalent voltage source is Vs = Ish Rsh.
The two circuits are equivalent when their open circuit voltages and short circuit currents are the same.
A few examples will further clarify this concept.
Example 1.8 Convert a voltage source of 20V with internal resistance of 5Ω into an equivalent current source.
Solution: Given Vs = 20V Rse = 5Ω
The short-circuit current, . The voltage source is represented by a current source of 4A.
The internal resistance will be the same as Rse in both the cases. So, Rsh is shown as Rse.
The condition for equivalence is checked from the following Voc should be same and Isc should also be same.
Isc in Figure 1.27(a) is 4 A. This is because Isc = 20/5 = 4 A; Isc in Figure 1.27(b) is 4 A. This is because when terminals A and B are short-circuited, the whole of 4 A will flow through the short-circuit path.
These two circuits are equivalent because the open-circuit voltage and short-circuit current are same in both the circuits.
Example 1.9 Convert a current source of 100A with internal resistance of 10Ω into an equivalent voltage source.
For an equivalent voltage source, the following can be derived:
The open-circuit voltage and short-circuit current are same in the two equivalent circuits as shown in Figure 1.28(a) and 1.28(b), respectively.
Short Answer Type
1.Define resistance, current and potential difference.
2.Distinguish between electrical power and electrical energy.
3.State Ohm’s law, and mention the conditions.
4.Distinguish between self-inductance and mutual inductance.
5.State the factor on which resistance and inductance of a coil depends.
6.Establish the relationship between self-inductance and mutual inductance.
7.Explain why the inductance of a coil increases when we place an iron bar inside it.
8.What do you mean by coefficient of coupling between two coils and what is its significance?
9.What are the various types of inductors and capacitors generally in use?
10.A resistor connected in series gives a higher value of equivalent resistance but a capacitor connected in series gives a lower value of equivalent capacitance. Explain.
11.What do you mean by dielectric constant and what is its significance?
12.What is an electrolytic capacitor?
13.When capacitors are connected in parallel, the total capacitance increases. Explain how.
14.How do you calculate the energy stored in an inductor and in a capacitor?
15.Explain why a charged capacitor cannot be considered as a voltage source.
16.How do you convert a voltage source into an equivalent current source and vice-versa?
1.For the resistive circuit shown in Figure 1.29, calculate the current flowing through the 10 Ω resistor and voltage drop across the 5 Ω resistor.
[Ans. 1A, 10 V]
2.Calculate the resistance between the terminals A and B in the network shown in Figure 1.30. What will be the resistance across the same terminals when terminals D and B are short-circuited?
[Ans. 2.5 Ω, 1 Ω]
3.Find the value of V1 in the circuit shown in Figure 1.31.
[Ans. 2.25 V]
4.Calculate the resistance across terminals AB of the network shown in Figure 1.32.
[Ans. 5 Ω]
5.Calculate the value of resistance across terminals PQ in the network shown in Figure 1.33.
[Ans. 6 Ω]
6.A voltage source of 20V has an internal series resistance of 2Ω. What will be its equivalent current source?
[Ans. Is = 10 A, Rsh = 2Ω]
7.Find the equivalent resistance between the terminals A and B of the network shown in Figure 1.34.
[Ans. 1 Ω]
8.Find the equivalent resistance between the terminals P and Q of the network shown in Figure 1.35.
[Ans. 5.33 Ω]