Physics

My goals

By the end of this unit, I will be able to:

* analyse complex electric circuits using Kirchhoff’s laws.

* identify sources of electric current.

* describe components of simple electric circuits.

* state Kirchhoff’s laws and apply them to solve problems in electric circuits.

* acquire practical skills to manipulate apparatus and evaluate experimental producers.

* explain the differences between the potential difference and electromotive forces.

Introduction

This unit is one of the most interesting units in Physics. Even if you ask someone who did not have enough studies in Physics he or she will tell you that People studying physics will be engineers specifically electricians. This Unit addresses the principles those electricians use in their career.

Review of elements of simple electric circuits and their respective role

An electric current consists of moving electric charges. Electric current must flow in electric devices connected by conductors (wires). The motion of electrons in a conductor is compared to water flow in a pipe. To move electrons, there must be a source of electric current, a cell, a battery, a generator which acts as a pump of water.

Making a simple circuit

Activity 1

Making a simple electric circuit with a bulb, a battery and wiresMaterials:

* 2 pieces of copper wire

* 1 bulb

* 1 battery

Procedure

1. Examine diagrams A-J below. Predict whether the circuit will be complete, and record your prediction on the chart below.

2. Your teacher, with a helper, will demonstrate the arrangements to test your predictions. Record their results on the chart below.

Activity 1

What makes the bulb light?

You may already understand an electrical circuit, or this may seem like magic to you. Give what your teacher demonstrated some thought. Why do you think the bulb in the diagram lights?

It would be useful here to summarize some basic electricity points which you may know already.

a) A current flows along a metal or wire when a battery is connected to it.

b) The current is due to free electrons moving along the metal.

c) The battery has a potential difference p.d or voltage between its poles due to chemical changes inside the battery. The p.d. pushes the electrons along the metal.

One pole of the battery is called the positive (+) pole, the other is called the negative (-) pole. The “conventional” current, shown by an arrow, flows in a circuit connected from the + to the – pole. The electrons carrying the current along the circuit wires actually move in the opposite direction to the conventional current but this need not to be taken in account in calculations or circuit formulae.

Any path along which electrons can flow is a circuit. For a continuous flow of electrons, there must be a complete circuit with no gaps. A gap is usually provided by an electric switch that can be opened or closed to either cut off or allow energy flow.Most circuits contain more than one device that receives electric energy from the circuit.

These devices are commonly connected in a circuit in one of two ways, series or parallel. When connected in series, the devices and wires connecting them form a single pathway for electron flow between the terminals of the battery, generator or wall socket. When connected in parallel, the devices and wires connecting them form branches, each of which is a separate path for the flow of electrons.

Making a series and parallel circuit

Activity 2

Making a series circuit

Materials:

* 1 Battery .

* 3 Bulbs.

* 3 bulb holders .

* Assembled battery holder.

* 4 Pieces of copper wire (as needed).

Procedure

1. Construct a complete circuit with a battery and a bulb.

2. Using another wire, add a second bulb as shown on the picture below.

3. What did you notice happened to the first bulb when the second bulb was added?

2. Using another two wires, add a second bulb as shown in the figure below.

3. What do you notice happened to the first bulb when the second bulb was added?_____________________________________________________

4. Look carefully at how a parallel circuit is set up. Write a prediction of what you think will happen if you unscrew one of the bulbs in the parallel circuit._____________________________________________________

Why did you make this prediction? _____________________________________________________

5. Unscrew bulb “X”. Describe what happens to bulb “Y”.

_____________________________________________________

6. Tighten bulb “X” and unscrew bulb “Y”. Describe what happens to bulb “X”.

_____________________________________________________

After carrying out experiments for series and parallel circuits,

* What advantages and disadvantages can you note for the two cases?

* What are the characteristics of a series connection and a parallel connection?

Conclusion

Series and parallel connections each have their own distinctive characteristics.

In a series circuit, the current is the same at all points; it is not used up. In a parallel circuit the total current equals the sum of the currents in the separate branches.

Generators and receptors

Generators: Sources of electric current

Activity 4

Searching on internet about sources of electric energy

Search on internet about different sources of electric current and answer the following questions.

a) What is a source of electric current?

b) What is another name of electric source of energy?

c) List some of electric sources you have found.

d) On the picture below are some sources of electric energy. Name them and tell what the common role they have is.

Tell what energy is changed in electric energy for each device.

The pictures above show the danger of electric current. So be careful when you are in their presence.

Electromotive force

Activity 5

Electromotive force of a generator

Materials

* Battery of 6V,

* Rheostat

* Voltmeter

* Ammeter

* Connecting wires

Procedure

1. Make the connection as shown in the following figure.

2. Write down the voltage and the current indicated by the voltmeter and the ammeter when the switch is open.

3. Close the switch and vary the current in the circuit by varying the values of the rheostat and every time write down values of voltage and current indicated respectively by voltmeter and Ammeter.

4. Fill in the following table the obtained data:

5. When you vary the value of the resistance of the rheostat, does the intensity of current remain constant? Why?

6. Does the voltage remain constant?

7. What is the maximum voltage that you have got? How is this voltage called?

8. In general, if a charge Q (in coulombs) passes through a source of emf E (in volts) which relation will give the electrical energy W supplied by the source (in joules)?

9. Which relation will give the total power of the source?

Interpretation

A voltmeter connected to terminals of a battery measures the voltage between terminals of battery. When the switch was closed, we have noticed that there was a current across the circuit and the value of the voltage has been changed.

By varying the value of the resistance of the rheostat, current in the circuit is changed; voltage indicated by the voltmeter changes also, it decreases when current increases. Its maximum value is reached when the switch is open. Such voltage is called electromotive force E (emf) of the battery.

The electromotive force emf E of a source (a battery, generator, etc) is the energy transferred to electrical energy when unit charge passes through it. In other words, we can say that the emf of a source of electrical energy is its terminal p.d. on open circuit.

The emf of a battery is the maximum possible voltage that the battery can provide between its terminals.

The unit of emf like the unit of p.d; is the volt [V] and equals the emf of a source which transfers 1 Joule of energy when 1 coulomb passes through it.

In general, if a charge Q (in coulombs) passes through a source of emf E (in volts), the electrical energy supplied by the source W (in joules) is:

Then the electric energy provided to the circuit by the source is given by the relation above. The electric power P, in this case is given by: where I is the electric current in the circuit.

Examples

1. What is the power supplied by a cell of emf 4.5V, knowing that a current of 0.5A flows in the circuit?

2. Find the emf of a generator of power 12W sending a current of 1A in an external circuit.

Internal resistance

Activity 6

Existence of internal resistance in a generator

* Consider a certain number of cells which you put in an electric apparatus, like a radio...

* With your cheek, feel their temperatures before use.

* Put the cells in your apparatus and let them work for a certain time.

* Remove the cells and again with your cheek, feel the new temperature of the cells then answer the following questions:

• Are the two temperatures of the cells equal? (Before and after use)

• If not, what do you think is the cause of different temperatures?

• Is one part of the current produced by the generator consumed by it? Why?

* The same observations can be made by feeling the temperature of the battery of a telephone before a call and after a call of about 10 minutes. Have you felt the increasing of temperature of a phone after using it? If yes, you think it’s due to what?

The term internal resistance refers to the resistance within an emf. The terminal p.d. of a cell on closed circuit is also the p.d. applied to the external circuit.

In an external circuit electrical energy is changed onto other forms of energy and we regard the terminal p.d. of a cell on closed circuit as being the number of joules of electrical energy changed by each coulomb in the external circuit.

Not all the electrical energy supplied by a cell to each coulomb is changed in the external circuit. The “lost” energy per coulomb is due to the cell itself having resistance. Each coulomb has to “waste” some energy to get through the cell itself and so less is available for the external circuit. The resistance of a cell is called its internal resistance [r] and depends among other things on its size.The electric power dissipated as heat in a cell is given by: P_i = I² r

Examples

3. The power dissipated as heat in a cell is of 7W, find its internal resistance if a current of 2A flows through it.

4. Find the power dissipated as heat in a generator of internal resistance 0.6Ω crossed by a current of 3A.

Remark: Any electrical generator, then, has two important properties, an emf E and an internal resistance r. E and r may be represented separately in a diagram, though in practice they are together between the terminals. To represent a cell, we can write (E, r). So we can think of the battery as an “electric pump”, with its emf E pushing the current round the circuit through both the external (outside) resistor R and internal resistance r

In an electric circuit, a generator is then represented by the following symbol:

Activity 7

Experiment to find the emf (E) and the internal resistance (r) of a cell

Materials

* 1.5V (approx) cell,

* Resistance box,

* Push switch,

* Digital Ammeter (0-1A).

Procedure

1. Check the scale on the digital Ammeter by comparing to other Ammeters.

2. Set R at 10 Ohms. Reduce in steps of 1 Ohm, recording resistance and current. Read the Ammeter as accurately as possible. Release switch (not a tap switch) after each reading, otherwise the cell will run down during the experiment.

3. Repeat the readings, increasing R back up to 10 Ohms. Obtain average values of I, the current for each value of the resistance.

4. Calculate 1/1 .

5. Plot a graph of R against 1/1. Draw the best possible straight line through the points (they might be quite scattered) in Excel, put the equation of the line on the graph.

The gradient of this graph is the emf of the cell. The negative intercept on the y-axis is the internal resistance.

Relationship between the P.d and the emf at terminals of a cell of a closed circuit

Activity 8

Relation between the Emf and the P.d Materials

* Dry cell,

* Analogy multimetres (2),

* Rheostat and switch.

Procedure

1. Set up the circuit as shown.

2. Set the resistance of the rheostat to a large value to protect the circuit before switch on the circuit.

3. Set the milliammeter to the range 0-1 A or suitable range.

4. Set the voltmeter to the range 0-5V or suitable range.

5. Switch on the circuit. Record down the readings of the Ammeter and voltmeter.

6. Slide the rider of the rheostat to another position. Record the readings of the ammeter and voltmeter again. Tabulate the results.

Interpretation

The electrical energy provided by the generator is consumed by the generator itself and by the external circuit. Let be:

Pi: The power supplied by a cell in the internal resistance;

P_e: The power supplied by a cell in the external circuit

If P is the total power supplied by a cell, we can write: P= P_i + P_e

I being the intensity of the current provided by the cell, we write:

EI = rI2+ RI²

EI = I (rI+ RI)

E = rI+ RI

E = rI+ V

Then: V = E − rI

The previous relation V = E − rI

shows that V < E

If I =0, we have V = E, means the p.d. is equal to emf if the generator does not work.

Examples

1. What is the voltage at terminals of a battery of emf 3V and internal resistance 0.3 Ω when sending a current of 1.5 A in a circuit?

2. Knowing that the voltage at terminals of a cell is 1.5V and the current crossing the circuit is 1.2 A. Find its emf if its internal resistance is of 0.4 Ω.

Efficiency of a cell

Activity 9

Analysis of the relation P= P_i + P_e

Study the relation P= P_i + P_e and answer the following questions.

a) In the relation, there are three quantities representing power. The power supplied to the external circuit, the one dissipated in the cell by Joule’s effect and the one which is the total power supplied by the cell. Match each power by its symbol in the relation.

b) Is the total power supplied by the cell consumed by the external circuit? If yes, why? And if not, why?

c) In general, how do you call the ratio of the amount of power produced by a machine and the power put into it?

d) What is the unit of the physical quantity described in the question above?

e) The ratio of the power supplied by the cell to the external circuit and the total power supplied by the cell has a special name. Have you ever heard it? If yes what is it? What is its unit?

f) Is there another way to write that ratio using the power supplied by the generator in the internal circuit? If yes find it.

g) Is there a way to write the same quantity using the emf and voltage at terminal of a cell?

Since the relation above involves the total power in the circuit, which is the sum of the power supplied in the external circuit and the power dissipated in the cell itself, this one is useful. It’s used to find the efficiency of the cell and how to calculate the voltage at terminals of a cell or battery.

Ohm’s law for a circuit having a cell and a resistor

Activity 10

From the observation of the following diagram and analysing different elements deduce a relation Questions:

a) State the Ohm’s law.

b) Observe the following diagram and list constituting elements.

c) Do these elements make an electric circuit? If yes, why?

d) Are these elements connected in series or in parallel? Why?

e) How can you find the total resistance of a series connection?

f) Having the total power supplied by a cell, the power supplied by a cell in an external circuit and the power distributed by the internal resistance. Write the relation between them.

g) Write down relations for each type of power and substitute them in the relation above. (Powers dissipated in internal and external resistances must be written in terms of resistance)

h) From the relation found, deduce the emf E. The relation found expresses the Ohm’s law for a circuit having a cell and a resistor.

i) Express the intensity of the current for this specific case.

Examples of application

1. The total power of a battery is of 9V and its internal resistance 3Ω. Knowing that the current crossed is of 0.4A. Find the efficiency of the battery.

2. A generator of internal resistance 2Ω sends a current of 4A in a resistor of resistance 10Ω. Calculate its power.

3. An external resistance of 4Ω is connected to an electric cell of emf 1.5V and internal resistance 2Ω. Calculate the intensity of the current flowing the external resistance.

4. An electric cell of emf 1.5V and internal resistance 2Ω is connected in series with a resistance of 28Ω. Calculate the power dissipated as heat in the cell.

Combination of cells

Activity 11

Cells wired in parallel and in series

Materials:

* 3 batteries

* 2 bulbs

* 3 assembled battery holders

* 2 bulb holders

* 6 pieces of copper wire

Procedure

1. Construct a complete circuit with one battery and one bulb.

2. Observe the brightness of the bulb.

3. Construct the circuit below. Are these batteries in series or parallel?_____________________________________________________

How can you tell? _____________________________________________________

4. Observe the brightness of this bulb. Is the bulb brighter than it was with one battery?_____________________________________________________

5. If you added a third battery to this circuit in series, what do you think would happen to the brightness of the bulb? _____________________________________________________

Why do you think this?_____________________________________________________

6. Add a third battery to this circuit. Describe what happens to the bulb as this battery is added to this circuit in series and why you think the bulb is acting in this way._____________________________________________________

7. Construct another complete circuit with one battery and one bulb. Record again what the brightness of the bulb is using your brightness metre.

8. Look at the pictures below, are the batteries in the picture in series or parallel?_____________________________________________________

How can you tell?_____________________________________________________

Construct the circuit in 8. Is the bulb brighter with two batteries than it was with one battery?_____________________________________________________

9. Add one more battery to this circuit in parallel. Describe what happens to the bulb as one more battery is added to this circuit in parallel and why you think the bulb is acting this way._____________________________________________________

10. Connect then two batteries in opposition to mean the positive (negative) terminals of batteries are connected together and the two free negative (positive) terminals are connected to the bulb. What happens to the bulb?_____________________________________________________

11. Connect two batteries in series in opposition with one battery. When the two free ends of the combination are connected to terminals of the bulb, what happens to the brightness of the bulb?_____________________________________________________

Interpretation

Combination in series

When two or more cells are arranged in series, the total emf is the algebraic sum of their emfs and the total internal resistance is the algebraic sum of their internal resistances

For n identical cells of emf E and internal resistance r each, we have: Et = n_E and r_t = n_r

Note: A series arrangement is used to increase the voltage, also the total internal resistance of the circuit, so the energy loss due to internal resistance is greater than for a single cell

Examples

1. Four 1.5V cells are connected in series to a 12Ω lightbulb. If the resulting current is 0.45A, what is the internal resistance of each cell, assuming they are identical and neglecting wires?

2. A certain number of cells of emf 1.5V and internal resistance 2Ω are connected in series. When connected this combination to an external resistance of 10Ω, a current of 500m. A flows in this resistance. Find the number of cells used.

Combination in opposition

Let us see also what the result could be if the cells were associated in opposition

Let us combine two cells in opposition. Two terminals of same sign are connected together. The direction of the current in the circuit will be determined by the direction of the current produced by the cell having the higher emf. For internal resistances they are in series. So we write:

Note: You might think that connecting batteries in opposition would be wasteful. For more purposes, that will be true. But such an opposition arrangement is precisely how a battery charger works.

Example

1. A cell of emf 2V and internal resistance 0.2Ω is associated in opposition with another cell of emf 1.5V and internal resistance 1.2Ω. Calculate the intensity of the current knowing that the external resistance is 1.1Ω.

Combination in parallel

Consider a parallel arrangement of n identical cells having (E, r) as characteristics each. The total emf of the arrangement is the emf of one cell and the total internal resistance is found considering the parallel arrangement of resistors.

Then, we have:

Note: The parallel arrangement is useful normally only if the emfs are the same. A parallel arrangement is not used to increase the voltage, but rather to provide large currents. Each of the cells in parallel has to produce only a fraction of the total current, so the energy loss due to internal resistance is less than for a single cell; the batteries will be exhausted less quickly.

Examples

1. We have 8 cells of emf 1.5V and internal resistance 2Ω. Calculate the intensity of the current flowing in an external resistance of 1Ω connected to the terminals of the 8 cells combined in parallel.

2. Six cells of unknown emf and internal resistance of 2Ω are associated in parallel. When an external resistance of 1Ω is connected to this combination a current of 1.5A is produced. Calculate the emf.

Mixing series and parallel combinations

Consider a combination having p series having q cells each.The total emf at terminals of the combination is the emf of one series, that means

Example

1. Four cells of emf 4.5V each and internal resistance 2Ω are combined in series. The combination is connected to an external resistance of 24Ω

a) What is the intensity of the current?

b) Same question if the cells are combined in parallel.

c) Same question if the combination has two parallel series of two cells each.

Receptors

Activity 12

Distinguishing a receptor from a passive resistor

a) Observe the following devices and name them

b) What is the use of each one?

c) The flowing of the current in them produces the same effect? Explain.

d) Among them, which ones transform the whole electric energy consumed in heat and which ones transform a part of electric energy consumed in another kind of energy which is not heat?

e) As we had in the case of generators, what are characteristics of these apparatuses?

Conclusion: Among the apparatuses above, there are some which transform the total electric energy consumed into heat and some transform just a part into heat, other part transformed into another type of energy which is not heat. Those which transform the whole quantity of electric energy consumed into heat are passive resistors or passive receptors and those transforming a part of the consumed electric energy in another form of energy which is not heat are called receptors or active receptors.

The main characteristics are back electromotive force and internal resistance.

Back electromotive force

Back electromotive force (emf) is normally used to refer to the voltage that is developed in electric motors. This is due to the relative motion between the magnetic field from the motor's field windings and the armature of the motor!

Internal resistance

The internal resistance of a receptor r’ is its ability to oppose electric current. When a receptor is traversed by an electric current, part of the energy consumed is transformed into heat. The power dissipated in the receptor by joule effect is: P_J = I ²r

The p.d at terminals of a receptor

Activity 13

Find the P.d at terminals of a motor

Materials

* Electric motor

* Ammeter

* Voltmeter

* Power supply

Procedure

1. Make the connection as shown in the figure below.

Measure the voltage (V) between terminals of the motor (M) and the current I in the circuit.

Questions

a) What is the net electrical power received by the motor?

b) What becomes this power and how is it transformed?c) What is the relation between the voltage and the back electromotive force?

d) From the relation found, how do you calculate the intensity of the current flowing?

Exercises

1. A circuit has in series a generator of emf 6V and internal resistance 0.1Ω, a receptor of back emf 1.5V and internal resistance 0.4Ω and a passive resistor of 8.5Ω. Calculate:

a) The intensity of the current flowing in the circuit.

b) The power supply by the generator.

c) The quantity of heat produced in the resistor in one minute.

2. A battery has an emf of 12.0V and an internal resistance of 0.05Ω. Its terminals are connected to a load resistance of 3.00Ω.

(a) Find the current in the circuit and the terminal voltage of the battery.

(b) Calculate the power delivered to the load resistor, the power delivered to the internal resistance of the battery, and the power delivered by the battery.

3. Calculate the terminal voltage for a battery with an internal resistance of 0.9Ω and an emf of 8.5V when the battery is connected in series with

(a) an 81Ω resistor, and (b) 810Ω.4. A 9V battery whose internal resistance r is 0.5Ω is connected in the circuit shown in the figure.

a) How much current is drawn from the source?

b) What is the terminal voltage of the battery?

c) What is the current in the 6Ω resistor?

5. What is the internal resistance of a 12V car battery whose terminal voltage drops to 8.4V when the starter draws 75A? What is the resistance of the starter?

6. A 1.5V dry cell can be tested by connecting it to a low-resistance Ammeter. It should be able to supply atleast 22A. What is the internal resistance of the cell in this case, assuming it is much greater than that of the Ammeter?

7. A cell whose terminals are connected to a wire in nickel silver of resistivity 30 x 10 -6 Ωcm and cross sectional area 0.25mm2 and length 5m sends a current of 160mA. When the length is reduced to a half, the intensity of the current is of 300mA. Calculate:

a) The internal resistance.

b) The emf of the cell.

8. A cell (E = 1.5V, r = 1.3Ω) sends a current in an external resistance of 3Ω. Calculate:

a) The intensity of the current in the circuit.

b) The p.d at terminals of the cell.

c) The power of generator.

d) The efficiency of the cell.

9. A battery is composed by 120 cells in series. Each element has an emf of 2V and an internal resistance of 0.001Ω. The combination is connected to an external resistance of 4.8Ω. Calculate:

a) The intensity of the current in the circuit.

b) The voltage at terminals of the battery.

c) The energy dissipated by joule effect when the current flows in the circuit in one hour.

Kirchhoff’s rules

Activity 14

Find the equivalent Resistance In this experiment, you will investigate three ideas using combinations of resistors in series and in parallel. Remember that the total or equivalent resistance in a series circuit is given by: Req = R1 + R2 + . . .

For a parallel circuit, the equivalent resistance is given by: 1/Req = 1/R1 + 1/R2 +...

In this experiment you will be using a digital multimeter (DMM) which can function as either a voltmeter or an Ammeter.

The voltmeter must always be wired in parallel with the resistor whose voltage you are measuring. The ammeter, used to measure current, must always be wired in series. Disconnect the meter from the circuit before you change the function setting. Failure to follow these procedures can result in serious damage to the meter. Be sure that you use the correct units with your data.

Materials*

1 multimeter.

* 1 330 Ω or 240 Ω resistor.

* 1 1000 Ω (1KΩ) resistor.

* 1 2000 Ω (2KΩ) resistor.

* 1 3000 Ω (3KΩ) resistor or 1 3300 Ω (3.3KΩ) resistor.

* 1 0- 10K resistor substitution box.

* 2 spade lugs.

* 2 2’ red banana wires.

* 2 2’ black banana wires.

* 4 4” black banana wires.

Procedure

1. Set the DMM function switch to “Ohms” (Ω). Measure and record the resistance of the resistors R₁, R2, R₃, R₄.

2. Figure 5.29 is a sketch of the components. In this sketch, the DMM is wired in parallel with R₃ in order to measure the voltage V₃. Wire the circuit shown in Figure 1, but do not connect it to the power supply until it has been approved by your lab instructor. Once it has been approved, apply power. Set the DMM to DCV. Connect the black banana wire to COM and the red wire to the V- Ω input. Measure the power supply voltage (Vps) and the voltages across R₁(V₁), R₂ (V₂), R₃ (V₃), and R₄ (V₄) as indicated in Figure 5.30.

3. Unplug the circuit and disconnect the meter. Change the function switch on the DMM to direct current amperes. Move the red wire to the “mA/μA terminal. Study Figure 5.31 and notice the way the ammeter is wired in series with the resistors. Again, have the lab instructor approve the circuit before you plug it in. Make the current measurements I₁, I₂, and I3 indicated in Figure 4 of 5.31

4. R₁ and R₂ are in series. R₃ and R₄ are also in series. The two series circuits are in parallel. Calculate the equivalent resistance, R_eq, for the entire circuit. Show your working.

5. Set the decade resistance box to the value you calculated for R_eq for the circuit. Be sure the DMM is set on DCA and wire the DMM and the resistance box in series with the power supply. Measure Ieq.

6. Turn everything off, disconnect the components and put the equipment away neatly.

Simple circuits can be analysed using the expression V = IR and the rules for series and parallel combinations of resistors. Very often, however, it is not possible to reduce a circuit to a single loop. The procedure for analysing more complex circuits is greatly simplified if we use two principles called Kirchhoff's rules:

Kirchhoff’s first rule is a statement of conservation of electric charge. All charges that enter a given point in a circuit must leave that point because charge cannot build up at a point. If we apply this rule to the junction shown in Figure 5.31. we obtain I₁ = I₂+ I₃

Kirchhoff’s second rule follows from the law of conservation of energy. Let us imagine moving a charge around a closed loop of a circuit. When the charge returns to the starting point, the charge–circuit system must have the same total energy as it had before the charge was moved. The sum of the increases in energy as the charge passes through some circuit elements must equal the sum of the decreases in energy as it passes through other elements. The potential energy decreases whenever the charge moves through a potential drop -IR across a resistor or whenever it moves in the reverse direction through a source of emf. The potential energy increases whenever the charge passes through a battery from the negative terminal to the positive terminal.

Examples

1. A single-loop circuit contains two resistors and two batteries, as shown in figure 5.34 (neglect the internal resistances of the batteries).

(a) Find the current in the circuit.

(b) What power is delivered to each resistor? What power is delivered by the 12V battery?

2. Find the currents I1, I2, and I3 in the circuit shown in Figure 5.35

Exercises

1. In the following circuit, using the Kirchhoff’s rules find the currents I₁, I₂ and I₃.

2. (a) Determine the currents I₁, I₂ and I3 in the figure below. Assume the internal resistance of each battery is r = 1Ω.

(b) What is the terminal voltage of the 6V battery?

3. In the circuit of the figure, determine the current in each resistor and the voltage across the 200Ω resistor.

4. Using Kirchhoff’s rules,

(a) find the current in each resistor in the figure bellow.

(b) Find the potential difference between points C and F. Which point is at the higher potential?

5. Taking R = 1.00kΩ and E = 250V in the figure below, determine the direction and magnitude of the current in the horizontal wire between A and E

6. A dead battery is charged by connecting it to the live battery of another car with jumper cables as shown in the figure. Determine the current in the starter and in the dead battery.

Career guidance

Did you know that Physics is one of the subjects that will help you have a career in engineering?

In engineering, we have electrical engineering, which basically uses concepts discussed in this Unit.Start to plan your career. Be an engineer or more specifically an electrical engineer.