Coulomb's Law: Electrostatics Practice Questions | PDF

Coulomb’s Law Coulomb’s law is a law of physics describing the electrostatic interaction between electrically charged particles. It was studied and first published in 1783 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. The...

Coulomb’s Law Coulomb’s law is a law of physics describing the electrostatic interaction between electrically charged particles. It was studied and first published in 1783 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. The scalar form of Coulomb’s law will only described the magnitude of the electrostatic force between two electric charges. If the direction is required then the vector form is required as well. The magnitude of the electrostatic force (F) on a charge (q 1) due to the presence of the second charge (q2), is given by F =ko | | q1 q2 r 2 where r is the distance between the two charges and ko a proportionality constant (Coulomb’s constant). The proportionality constant ko is related to define properties of space and can be computed exactly. 2 1 c μo k o= = =c2 ∙10−7 H /m 4 π ϵo 4 π 9 2 2 ko=9 x 10 N m /C By definition in SI units, the speed of light in vacuum, denoted by c is approximately equal to 3 x 10 8 m /s and the magnetic constant ( μ o ) is equal to 4 πx 10−7 H /m , leading to the value for the electric constant ( ϵ o ) as ϵ o=8.85 x 10 F /m. −12 This formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each object and inversely proportional to the square of the distance between them. Sample problem 1. Three point charges are placed on the x-axis as shown in the figure. Find the net force on the −5 μC charge due to the two other charges. q1 =3.0μC q2 =−5.0μC q3 =8.0μC _____ 20 cm ______ ____________30 cm _______  1 Sample problem 2. A plastic ball has a charge of 10−12 C. (a) Does it contain an excess of deficiency of electrons compared with its normal state of electrical neutrality? (b) How many such electrons are involved? Sample problem 3. What is the magnitude and direction of the force on the charge of −9 −8 4 x 10 C that is 5 cm from a charge of 5 x 10 C ? Sample problem 4. The two identical balls in a vacuum, each of mass 0.10 g are carrying identical charges and are suspended by two threads of equal length. At equilibrium they position themselves as identical (as shown in the figure). Find the charge on either ball.  60 ° 60 ° 40 cm 2 Exercises 1. How many electrons are contained in 1.0 C of charge? What is the mass of the electrons in 1.0 C of charge? 2. If two equal point charges, each of 1 C, were separated in air by a distance of 1 m, what would be the force between them? 3. Determine the force between two free electrons spaced 1.0 angstrom (10-10 m) apart in a vacuum. 4. What is the force of repulsion between two argon nuclei that are separated in vacuum by 1.0 nm (10-9 m)? The charge on an argon nucleus is +18 e. 5. Two equally charged small balls are 3 cm apart in air and repel each other with a force of 40 μN. Compute the charge on each ball. 6. Four equal point charges of +3.0μC are placed in air at the four corners of a square that is 40 cm on a side. Find the force on any one of the charges. 7. Four equal-magnitude point charges (3.0μC) are placed in air at the corners of a square that is 40 cm on a side. Two, diagonally opposite each other, are positive, and the other two are negative. Find the force on either negative charge. 8. Charges of +2.0 ,+3.0 and −8 μC are placed in air at the vertices of an equilateral triangle of side 10 cm. Calculate the magnitude of the force acting on the −8 μC charge due to the other two charges. q1 =2 μC q2 =3 μC q3 =−8 μC 9. One charge of q1 =+5 μC is placed in air at exactly x=0, and the second charge q2 =7 μC at x=100cm. Where can a third be placed so as to experience zero net force due to the other two? 10. Three point charges are placed at the following locations on the x-axis: q1 =+2 μC at x=0 , q2 =−3 μC at x=40 cm, and q3 =−5 μC at x=120cm. Find the force  (a) on q2 charge, (b) on q3 charge. 3 Electric Field An electric field is a region of space in which a charge would be acted upon by an electric force. An electric field may be produced by one or more charges, and it may be uniform or it may vary in magnitude and/or direction from place to place. If charge q at a certain point is acted on by the force F , the electric field E at that point is defined as the ratio between F and q: F E= q From the Lorentz Force Law that the magnitude of the electric field ( E ) created by single point charge ( q ) at a certain distance ( r ) is given by 1 q E= ∙ 2 4 π ϵo r For a positive charge, the direction of the electric field points along lines directed radially away from the location of the point charge, while the direction is the opposite for a negative charge. The SI units of electric field are volts per meter ( V /m ) or Newtons per coulomb ( N /C ). Electric Field Lines Field lines are a means of describing a force field, such as an electric field, using imaginary lines to indicate the direction and magnitude of the field. The direction of an electric field line at any point is the direction in which a positive charge would move if place there. The field lines are drawn close together where the field is strong and far apart if the field is weak. weak field Strong field Sample problem 1 What is the electric field that acts on the electron in a hydrogen atom, which is 5.3 x 10-11 m from the proton that is the atom’s nucleus? 4 Sample problem 2 The electric field in a certain neon sign is 5000 V/m. (a) What force does this field exert on a neon ion of mass 3.3 x 10 -26 kg and charge +e? (b) What is the acceleration of the ion? Sample problem 3 How strong an electric field is required to exert a force on a proton equal to its weight at sea level? Sample problem 4 Compute: (a) the electric field E in air at a distance of 30 cm from a point charge −9 −10 q1 =5.0 x 10 C , (b) the force on a charge q2 =4.0 x 10 C , placed 30 cm from q1, and (c) the force on a charge q3 =−4.0 x 10−10 C , placed 30 cm from q1(in absence of q2). Exercises 1. A point charge of −3.0 x 10−5 C is placed at the origin of coordinates in vacuum. Find the electric field at the point x=5.0m on the x-axis. 2. Four equal-magnitude (4.0 μC ) charges in vacuum are placed at the four corners of a square that is 20 cm on each side. Find the electric field at the center of the square (a) if the charges are all positive, (b) if the charges  alternate in sign around the perimeter of the square, (c) if the charges have the following sequence around the square: plus, plus, minus, minus. 3. A 0.20-g ball in air hangs from a thread in a uniform vertical electric field of 3.0 kN/C directed upward. What is the charge on the ball if the tension in the thread is (a) zero and (b) 4.0 mN? 4. Determine the acceleration of the proton (q=+e ,m=1.67 x 10−27 kg) immersed in an electric field of strength 0.50 kN/C in vacuum. How many times is this acceleration greater than that due to gravity? −31 5. An electron (q=−e ,me=9.1x 10 kg) is projected out along the +x-axis in vacuum with an initial speed of 3.0 x 106 m/ s. It goes 45 cm and stops due to a uniform electric field in the region. Find the magnitude and direction of the field. 5 6. How much force is exerted on a charge of 10−6 C by an electric field 50 V/m? 7. An electron is present in an electric field of 10 4 V /m. (a) Find the force on the electron. (b) Find the electron’s acceleration. 8. A potential difference of 100 V is applied by a battery across a pair of metal plates 5 cm apart. (a) What is the electric field between the plates? (b) How much force does a charge of 10−8 C experience in this field? (c) How much kinetic energy does this charge acquire when it goes from the positive plate to the negative plate? 9. A charge of −2 x 10−9 C in an electric field between metal plates 4 cm apart is acted on by a force of 10−4 N. (a) What is the strength of the field? (b) What is the potential difference between the plates? 10.Three charges are placed on three corners of a square, as shown below. Each side of the square is 30 cm. (a) Compute Ē (magnitude and direction) at the fourth corner. (b) What would be the force on a 6.0 μC charge placed at the vacant corner? q1 =8 μC q2 =−5.0μC E2 E3 q3 =−4 μC E1 Electric Potential Difference The Potential Difference between points A and B is the work done against electrical forces in carrying a unit positive test charge from A and B. We represent the potential difference between A and B by V b−V a or just by V , when there is no ambiguity. Its units are those of work per charge and are designated as volts. 1 V = 1 J/C Because work is a scalar quantity, so too is a potential difference. Like work, potential difference may be positive or negative. The work W done in transporting a charge q from one point A to second point B is W =q ( V b−V a )=qV where the appropriate sign ¿ must be given to the charge. If both V b−V a and q are positive (or negative) the work done is positive. If V b−V a and q have opposite sign the work done is negative. 6 The Absolute Potential The absolute potential at a point is the work done against electric forces in carrying a unit positive test-charge from infinity to the point. Hence, the absolute potential at point B is the difference in potential from A at ∞ to B. Consider a point charge q, in vacuum and in a point P at a distance r from the point charge. The absolute potential at point P due to the charge q is qo V =ko r 9 2 2 where ko=9 x 10 Nm /C is the coulomb’s constant for vacuum. The absolute potential at infinity ( at r=∞ ) is zero. Electrical Potential Energy ( PE E ) To carry a charge q from infinity to a point where the absolute potential is V, work in the amount qV must be done on the charge. This work appears as electrical potential energy ( PE E ). Similarly, where a charge q is carried through a potential difference V, work in the amount qV must be done on the charge. This work results in a charge qV in the PE E of the charge. For potential rise, V will be positive and PE E will increase if q is positive. But for potential drop, V will be negative and the PE E of the charge will decrease if q is positive. V Related to E Suppose that in a certain region the electric field is uniform and is in the x-direction. Call its magnitude E x. Because E x is the force on a unit positive test-charge, the work done is moving the test-charge through a distance x is ( ¿ W =F x x ) V =E x x The field between two large, parallel, oppositely charged, closely space metal plates is uniform. We can therefore use the equation to relate the electric field E between the plates to the plate separation d and their potential difference V. For parallel plates, V =Ed Sample problem 1 In the figure below, the potential difference between the metal plates in air is 40 V. (a) Which plate is at the higher potential? (b) How much work must be done to carry a +3.0 C charge from B to A? From A to B? (c) How do we know that the electric field is in the direction indicated? (d) If the plate separation is 5.0 mm, what is the magnitude of Ē? A B + - Ē + - + - + - + - d 7 Sample problem 2 How much work is required to carry an electron from the positive terminal of 12-V battery to the negative terminal? Sample problem 3 How much electrical potential energy does a proton lose as it falls through a potential drop of 5 KV? Sample problem 4 An electron starts from rest and falls through a potential rise of 80 V. What is its final speed? Sample problem 5 (a) What is the absolute potential at each of the following distances from a charge of 2.0μC in air: r = 10 cm and r = 50 cm? (b) How much work is required to carry a 0.05−μC charge from the point at r = 50 cm to r = 10 cm? 8 Sample problem 6 The nucleus of a tin atom in vacuum has a charge of +50 e. (a) Find the absolute potential V at a radial distance of 1.0 x 10−12 m from the nucleus. (b) If a proton is released from this point, how fast will it be moving when it is 1.0 m from the nucleus? Sample problem 7 In the figure below, charge at A is +200pC , while the charge at B is −100pC. (a) Find the absolute potentials at points C and D. (b) How much work must be done to transfer a charge of +500pC from point C to point D? +200pC D C −100pC A B  20 cm 60 cm 20 cm Exercises 1. Two metal plates are attached to the two terminals of a 1.5-V battery. How much work is required to carry a +5.0μC charge across the gap (a) from the negative to the positive plate, (b) from the positive to the negative plate? 2. The plates describe in problem 1 are in vacuum. An electron −31 (q=−e ,me=9.1x 10 kg) is released at the negative plate and falls freely to the positive plate. How fast is it going just before it strikes the plates? −27 3. A proton (q=e ,mp =1.67 x 10 kg) is accelerated from rest through a potential difference of 1.0 MV. What is its final speed? 4. An electron gun shoots electron at a metal plate that is 4.0 mm away in vacuum. The plate is 5.0 V lower in potential than the gun. How fast must the electrons be moving as they leave the gun if they are to reach the plate? 5. The potential difference between the two large metal plates is 120 V. The plate separation is 3.0 mm. Find the electric field between the plates. 9 6. An electron is shot with speed 5.0 x 106 m/ s parallel to a uniform electric field of strength 3.0 kV/m. How far the electron go before it stops? 7. A potential difference of 24 kV maintains a downward-directed electric field between two horizontal parallel plates separated by 1.8 cm in vacuum. Find the charge on an oil droplet of mass 2.2 x 10−13 kg that remains stationary in the field between the plates. 8. An electron has a speed of 6.0 x 105 m/ s as it passes point-A on its way to point-B. Its speed at B is 12.0 x 10 5 m /s. What is the potential difference between A and B, and which is the higher potential? 9. A point charge of 2.0μC is placed at the origin of coordinates. A second, of −3.0μC , is placed on the x-axis at x = 100 cm. At what point (or points) on the x-axis will the absolute potential be zero? 10.In problem 9, what is the difference in potential between the following two points on the x-axis: point-A at x = 0.1 m and point-B at x = 0.9 m? Which point is at the higher potential? 11.A proton is accelerated by a potential difference of 15,000 V. What is its kinetic energy? The Current A Current ( I ) of electricity exists in a region when a net electric charge is transported from one point to another in that region. Suppose the charge is moving through a wire. If a charge q is transported in a given cross section of the wire in a time t, then the current through the wire is q I= t Here, q is in coulombs, t is in seconds, and I is in amperes (1 A = 1 C/s). by custom the direction of the current is taken to be in the direction of flow of positive charge. thus, a flow of electrons to the right corresponds to a current to the left. A battery is a source of electrical energy. If no internal energy losses occur in the battery, then the potential difference between its terminals is called the electromotive force (emf) of the battery. Unless otherwise stated, it will be assumed that the terminal potential difference of a battery is equal to its emf. The unit for emf is the same for the potential difference, the volt. The Resistance and Ohm’s Law The electrical resistance of a circuit component or device is defined as the ratio of the voltage applied to the electric current which flows through it. V R= I If the resistance is constant over a considerable range of voltage, Ohm’s Law, I =V /R , can be used to predict the behavior of a material. Although the definition above involves DC current and voltage, the same definition holds for AC application of resistors. Whether or not a material obeys Ohm’s Law, its resistance can be described in terms of its bulk resistivity. The resistivity, and thus the resistance, is temperature dependent. Over sizable ranges of temperature, this temperature dependence can be predicted from a temperature coefficient of resistance. 10 The Resistivity The electrical resistance of the wire would be expected to be greater for a longer wire, less for a wire of larger cross-sectional area, and would be expected to depend upon the material out of which the wire is made. Experimentally, the dependence upon these properties is straightforward one for a wide range of conditions, and the resistance of the wire can be expressed as ρl R= A The factor in the resistance which takes into account the nature of the material is the resistivity. Although it is temperature dependent, it can be used at a given temperature to calculate the resistance of the wire of given geometry. The inverse of resistivity is called conductivity. There are contexts where the use of conductivity is more convenient.  1 Electrical conductivity =σ= ρ Resistance Varies With Temperature If a conductor has a resistance Ro at a temperature T o, then the resistance R at a temperature T is R=Ro +α Ro (T −T o) where α is the temperature coefficient of the material of a conductor. Usually α varies with temperature, and so this relation is applicable only over a small temperature range. Circular Mil In engineering practice, the unit of area of a round conductor is often the 1 circular mil∨cmil. The mil is a unit of length equal to 0.001 in., which is in. 1000 A circular mil is a unit of area to the area of a circle whose diameter is 1 mil. The 2 area A cm in circular mils of a circle whose diameter in mils is Dm is equal to Dm: 2 Acm=D mil The advantage of using the circular mil as a unit of area is that it avoids multiplication and division by π. When the length of the conductor is given by feet and its area in circular mils, the unit of resistivity is the ohm−cmil per foot. D = 1 mil A =1 cmil 11 Sample problem 1 A wire carries a current of 1 A. How many electrons pass any point in the wire each second? A 120-V toaster has a resistance of 12 𝝮. What must be the minimum rating of Sample problem 2 the fuse in the electric circuit to which the toaster is connected? Sample problem 3 A 120-V electric heater draws a current of 25 A. What is its resistance? A 20-m length of a certain wire has a resistance of 15 𝝮. What length of this Sample problem 4 wire would have a resistance of 8 𝝮? Sample problem 5 What is the resistance of a copper wire 0.5 mm in diameter and 20 m long? The resistivity of copper is 1.7 x 10−8 Ω m. 12 A platinum wire 80 cm long is to have a resistance of 0.1 𝝮. What should its Sample problem 6 diameter be? The resistivity of platinum is 1.1 x 10−7 Ω m. Sample problem 7  A copper wire has a resistance of 10.0 𝝮 at 20℃. (a) What will its resistance be at 80 ℃ ? (b) At 0 ℃ ? The temperature coefficient of resistance of copper is 0.004/℃. Sample problem 8 A resistance thermometer makes use of the variation of the resistance of a platinum element is 5 𝝮 at 20 ℃ and 9 𝝮 when it is inserted in a furnace, find the conductor with temperature. If the resistance of such a thermometer with a temperature of the furnace. The value of α for a platinum is 0.0036/℃. The specific resistance of the copper used in electric wire is 10.4 𝝮 cmil/ft. Sample problem 9 Find the resistance of 1500 ft of copper wire whose diameter is 0.080 in. 13 The specific resistance of Nichrome is 600 𝝮 cmil/ft. How long should a Sample problem 10 Nichrome wire 20 mils in diameter be for it to have a resistance of 5 𝝮? Electric Power The rate at which work is done to maintain an electric current is given by the product of the current I and the potential difference V.  P=IV When I is in amperes and V is in volts, P will be in watts. If the conductor or a device through which the current passes obeys Ohm’s Law, the power consumed may be expressed in the alternative forms 2 2 V P=IV =I R= R Also the electrical power delivered by an energy source as it carries a charge through the potential rise in a time is Work Power finished= time W P= , t Since W =qV , therefore qV P= t Conversions: 1 W = 1 J/s = 0.738 ft lb/s 1 hp = 33 000 ft lb/min 1 hp = 746 W = 42.4 Btu/min Sample problems 11 The current through a 50-𝝮 resistance is 2 A. How much power is dissipated as heat? 14 Sample problems 12 A 2-kW heater is to be connected to a 240-V power line whose circuit breaker is rated at 10 A. Will the breaker open when the heater is switch on? Sample problem 13  A generator driven by a diesel engine that develops 12 hp delivers 30 A at 240 V. What is the efficiency of a generator? Sample problem 14 A 12-V storage battery is charged by a current of 20 A for 1 h. (a) How much power is required to charge the battery at this rate? (b) How much energy has been provided during the process? Exercises 1. How many electrons pass through the filament of 75-W, 120-V light bulb per second? 2. Find the current in a 200-𝝮 resistor when the potential difference across it is 40 V. mm2 is found to be 3.6 𝝮. What is the resistivity of the iron? 3. The resistance of a 15-m length of iron wire whose cross-sectional area is 0.5 4. How long should a copper wire 0.4 mm in diameter be for it to have a resistance of 10.0-𝝮? The resistivity of copper is 1.7 x 10−8 Ω m. 5. Find the resistance of 8 m of aluminum wire 0.1 mm in diameter. The 6. The specific resistance of iron is 72 𝝮 cmil/ft. Find the resistance of 250 ft of resistivity of aluminum is 2.6 x 10−8 Ω m. 1 iron whose diameter is in. 4.62 Ω 16 resistance of a carbon resistor is 1000 𝝮 at 0℃ , find its resistance at 120 ℃. 7. The temperature coefficient of resistance of carbon is -0.0005/ ℃. If the 8. An iron has a resistance of 0.20 𝝮 at 20 ℃ and a resistance of 0.30 𝝮 at 110 ℃. Find the temperature coefficient of resistance of the iron used in the wire. 15 9. The maximum allowable voltage drop for a 70-m length of a certain cable that carries a current of 40 A is 5 V. What is the minimum cross-sectional area of the copper wire of the cable in square millimeters? The resistivity of copper is 1.7 x 10−8 Ω m. 9.52 mm2 10.What is the resistance of a 750-W, 120-V electric iron? 11.How much power is developed by an electric motor that draws a current of 4 A when it is operated at 240 V?  12.A 32-V storage battery has a capacity of 10 6 J. How long can it supply a current of 5 A? 13.Compute the work and the average power required to transfer 96 kC of 14.An electric iron of resistance 20 𝝮 takes a current of 5.0 A. Calculate the charge in one hour through a potential rise of 50 V. thermal energy in joules, developed in 30 s. 15.A coil develops 800 cal/s when 20 volts is supplied across it ends. Compute its resistance. Resistors in Series The Characteristics of a series connection of resistor is that the current is constant, meaning that the magnitude of current that passes through each resistor is equal all throughout the circuit. The total resistance is equal to the algebraic sum of all resistances connected in series. The terminal voltage is equal to the algebraic sum of all the voltage drops across each load. R1 VT R2 R3 I Characteristics: 1. I T =I 1=I 2 =I 3 =...= I n 2. RT =R1 + R 2 + R3 +...+ R n 3. V T =V 1 +V 2 +V 3 +...+V n 16 Sample problem 1 It is desired to limit the current in a 50-𝝮 resistor to 10 A when it is connected to a 600-V power source. (a) How should an auxiliary resistor be connected in the circuit and what should its resistance be? (b) What is the voltage drop across each resistor? Sample problem 2 (a) What is the equivalent resistance of three 5-𝝮 resistors connected in series? (b) If a potential difference of 60 V is applied across the combination, what is the current in each resistor? (c) What is the voltage drop across each resistor? Sample problem 3 Two light bulbs, one of 5- and the other of 10-𝝮 resistance, are connected in series across a potential difference of 12 V. (a) What is the current in each bulb? (b) What is the voltage across each bulb? (c) What powers dissipated by each bulb and by the combination? Sample problem 4 A 2000- and a 5000- Ω resistor are in series as a part of a larger circuit. A voltmeter shows the potential difference across the 2000-𝝮 resistor to be 2 V. Find (a) the current in each resistor and (b) the potential difference across the 5000-𝝮 resistor. 17 Resistors in Parallel A typical circuit with resistors connected in parallel is given in the figure below. Resistors in a parallel circuit are connected such that the top of each resistor is connected to the same point A, whereas the bottom of each resistor are connected to the same point B. The characteristics of a resistors in parallel is that the voltage is constant, meaning the terminal voltage is equal to the potential difference of each resistor connected in parallel. The reciprocal of resistance is called the conductance, thus, the total conductance is equal to the algebraic sum of the conductance in parallel circuit. Similarly, the reciprocal of total resistance is equal to the algebraic sum of the reciprocal of all resistances connected in parallel. The total current is equal to the algebraic sum of the branch current in parallel circuit. VT R1 R2 R3 Characteristics: 1. V T =V 1 =V 2 =V 3=...=V n 2. I T =I 1 + I 2 + I 3+...+ I n 1 1 1 1 1 3. = + + +...+ RT R1 R 2 R3 Rn Sample problem 5 (a) What is the equivalent resistance of three 5-𝝮 resistors connected in parallel? (b) If a potential difference of 60 V is applied across the combination, what is the current in each resistor? Sample problem 6 Two 240-𝝮 light bulbs are to be connected to a 120-V power source. To determine whether they will be brighter when connected (a) in series or (b) in parallel, calculate the power they dissipate in each arrangement. 18 A circuit has a resistance of 50 𝝮. How can it be reduced to 20 𝝮? Sample problem 7 Sample problem 8 A 120-V house circuit has the following light bulbs turned on: 40.0 W, 60.0 W, and 175.0 W. Find the equivalent resistance of these lights. Exercises 1. (a) Find the equivalent resistance of four 60-𝝮 resistors connected in series. (b) If a potential difference of 12 V is applied across the combination, what is the current in each resistor? 2. (a) Find the equivalent resistance of four 60- Ω resistors connected in parallel. (b) If potential difference of 12 V is applied across the combination, what is 3. A 100- and a 200-𝝮 resistor are connected in series with a 40-V power the current in each resistor? source. (a) What is the voltage drop in each resistor? (b) How much power does each one dissipate? 4. A 100- and a 200-𝝮 resistor are connected in parallel with a 40-V power source. (a) What is the current in each resistor? (b) How much power does each one dissipate? produce an equivalent resistance of 200 𝝮? 5. What resistance should be connected in parallel with a 1000-𝝮 resistor to 6. A 5-𝝮 is connected in parallel with 15-𝝮 resistor. When a potential difference is applied to the combination, which resistor will carry the greater current? What will be the ratio of the currents? 7. A 25-, a 40-, and a 60-𝝮 resistors are connected in series in a circuit such that the voltage across the 25-𝝮 is 18 V. Find (a) the total resistance, (b) the current in each resistor, and (c) the voltage across the other two 8. Two batteries in parallel, each of emf of 10 V and internal resistance of 0.5 𝝮, resistors. are connected to an external 20-𝝮 resistor. Find the current in the external resistor. 9. A certain “12-V” storage battery actually has an emf of 13.2 V and an internal 0.01 𝝮. What is the terminal voltage of the battery when it delivers 80 A to resistance of the starter motor of a car engine?  10.Two 12-V batteries, one with internal resistance of 0.05 𝝮 and the other with internal resistance of 0.15 𝝮, are connected in parallel with a load of 0.05 𝝮. Find the current in the load. 19 Resistors in Series-Parallel Circuit Almost all electrical circuits contain a combination of series and parallel circuits. In analyzing these circuits, the techniques of these series and parallel circuits are applied individually to produce a much simpler overall circuit. R1 R4 I1 I3 VT R2 R5 I2 R3 R6 Sample problem 1: R 1 = 1.5 k𝝮 IT R2 = 1.2 k𝝮 + VT = 25 V _ R 5 = 1.5 I2 k𝝮 I1 R3 = 1.8 k𝝮 R 4 = 1 k𝝮 Solve for: RT , I T , V 1 , V 2 ,V 3 , V 4 , V 5 , I 1 , I 2 , PT ,  20 Sample problem 2 R 1 = 390 𝝮 R 4 = 330 𝝮 IT I2 R 2 =820 𝝮 + 𝝮 VT = 50 V - R 5 = 220 I1 R 3 = 1.2 k𝝮 R6= 270 𝝮 Solve for: RT , I T , V 1 , V 2 ,V 3 , V 4 , V 5 , I 1 , I 2 , PT ,  21 Exercises: 1. Compute: I T , RT , I 1 , I 2 , P3 R1 = 5 𝝮 IT R3 = 3 𝝮 r = 0.4 𝝮 VT = 30 V R 2 = 7𝝮 I2 I1 2. Compute: Rbe∨¿ bcde , RT , I T , I 1 , I 2 , V 2 , V 5 , P T , P 2 , P 5 a R 1 = 6.0 𝝮 b R 4 = 7.0 𝝮 c IT R 2 =6.0 𝝮 VT = 20 V + 𝝮 - R 5 = 1.0 I1 I2 R 3 = 14.5 𝝮 R6= 10.0 𝝮 f e d 3. Compute: Rcde, Rce ∨¿ cde, R R 3 R ce∨¿cde , RT , I T , I 1 , I 2 , I 3, I 4. and P 4 R 3 = 15.0 𝝮 c = 19 𝝮 d a b R5 IT I2 R 1 = 9.0 𝝮 + 𝝮 R6 = 5.0 𝝮 VT = 17 V - R 4 = 8.0 r = 0.2 𝝮 I1 I3 I4  R 2 = 2.0 𝝮 g f e 22 4. Compute: I 1 , R456 , I 2 , I T , V 1, V T , RT , P 3, P 6. and P T R 1 = 390 𝝮 R 4 = 220 𝝮 IT I1 ET 220 𝝮 + R5 = - R 2 =330 𝝮 E5 =4.4 V I2 R 3 = 390 𝝮 R6= 220 𝝮 5. Compute: RT , I T , I 1 , I 2 , V 2, V 4, V 6, P 3, P 6. and P T R1 = 180 𝝮 R3 = 15.0 𝝮 c d a b IT I2 I4 + 𝝮 VT = 17 V - R 2 = 15 k𝝮 R 4 = 5.6 r = 0.2 𝝮 R6 = 5.6 k𝝮 I1 I3 R 3 = 270 𝝮 g f e R6 = 1 k𝝮  6. Compute: I 1 , R456 , I 2 , I T , V 1, V T , RT , P 3, P 6. and P T R 1 = 12 k𝝮 R 4 = 1 k𝝮 IT I2 VT + V 2 = 15 V - R 2 =3 k𝝮 R5 = 6.8 k𝝮 I1 R 3 = 33 K𝝮 R6= 2.2 k𝝮 23