Unit 2 — Electricity and Magnetism

• Metals such as copper and aluminium are good conductors because each of their atoms loses some of its outer electrons.
• These free electrons are not attached to any one atom; they wander at random inside the metal (Figure: random motion), and since they move in all directions equally, there is no net flow of charge.

• An analogy helps.
• Imagine a bicycle chain.
• When you press the pedals, all the links of the chain move together; the chain carries motion from the pedals to the rear wheel; and if something rubs against the chain, it slows down.
• In a simple electric circuit:

• the cell (source) is like the pedals: it provides the push;
• the wires are like the chain: the path the charges follow;
• the bulb is like the resistance of the rear gear: it opposes the flow and turns electrical energy into light and heat.

• When a metal wire is connected to a cell to form a closed circuit, the cell pushes the free electrons so that they drift in one direction along the wire.
• The ordered flow of electric charges through a conducting material is called electric current.

Two conditions are needed for a current to flow:

1. an electric source to push the charges (cell, battery, generator);
2. a closed circuit, a complete path with no breaks (this is why a switch can turn a lamp on or off).

Direction of current.

• By convention, current flows outside the source from the positive terminal (higher electric potential) to the negative terminal (lower potential).
• Before the electron was discovered, scientists chose this direction.
• In fact the free electrons move the opposite way, from the negative pole to the positive pole.
• In this unit we always use the conventional direction.

• Imagine watching cars cross a line on a race track.
• In one lap 10 cars cross the line in one minute; in another lap 5 cars cross in one minute.
• The first lap has a higher "rate of crossing".
• The flow of charges in a wire is similar.

Electric current intensity (I) is the quantity of electric charge that passes through a cross-section of a conductor in one second.

I = Q / t

where Q is the charge in coulombs (C) and t is the time in seconds (s). The unit of current is the ampere (A), and 1 A = 1 C/s.

An ammeter measures current. It is connected in series with the part of the circuit whose current is to be measured.

Worked example 1

A charge of 90 C passes through a lamp filament in 3 minutes. Find the current. t = 3 × 60 = 180 s. I = Q/t = 90 / 180 = 0.5 A.

Worked example 2

How much charge flows through a wire carrying 2.5 A for 20 s? From I = Q/t, Q = I × t = 2.5 × 20 = 50 C.

Meaning of a statement. "The current in a wire is 2 amperes" means that a charge of 2 coulombs passes through any cross-section of the wire each second.

Water analogy.

• Two connected containers X and Y, with a closed valve between them.
• The water level in X is higher than in Y, so the water pressure at the valve is higher on X's side.
• When the valve is opened, water flows from X to Y until the levels become equal, then the flow stops.

• In the same way, charges flow through a conductor from a point of higher electric potential to a point of lower electric potential.
• If the two points have the same potential, there is no current between them.
• A potential difference (cause) produces a current (effect).

The electric potential difference between two points (voltage) is the work done to move one coulomb of electric charge between the two points.

V = W / Q

W in joules (J), Q in coulombs (C), V in volts (V). So 1 volt = 1 joule per coulomb.

"The potential difference across a conductor is 3 volts" means that 3 joules of work are done (3 J of electrical energy transformed) to move each coulomb of charge through it.

A voltmeter measures potential difference. It is connected in parallel across the two points.

Worked example 3

A lamp uses 120 J of electrical energy to carry 20 C through it. What is the potential difference across the lamp? V = W/Q = 120 / 20 = 6 V.

An electric source supplies electrical energy and drives charges around the circuit. There are two kinds:

• Direct-voltage sources, such as an electric cell, a battery. They produce direct current (DC), of constant intensity and in a single direction.
• Alternating-voltage sources, such as an alternating-current generator. They give alternating current (AC), whose value and direction change with time.

• The electromotive force (emf, ε) of a source is the total work done by the source to move one coulomb of charge through the entire circuit (inside and outside the source).
• It is measured in volts.

• Current also flows through the source itself, so the emf is divided into two parts: the potential difference across the battery terminals (V), which is used in the external circuit, and the potential difference lost inside the source to overcome its internal resistance.
• So ε is slightly larger than V (ε > V).

In this unit we assume the internal resistance is negligible, so V (terminal) = ε. This is why the question "the potential difference between a battery's terminals is usually less than its emf" has the answer: because some energy is used inside the battery against its internal resistance.

One cell gives limited emf, which may be too small. A group of connected cells is a battery. Identical cells can be connected in two ways.

In series. The positive pole of one cell is connected to the negative pole of the next. The emf of the battery = the sum of the emfs of the cells: ε(total) = n × ε(cell). In a circuit of three 1.5 V cells in series, a voltmeter reads 4.5 V and the lamp is brighter than with one cell.

In parallel. All positive poles are joined together and all negative poles are joined together. The emf of the battery = the emf of one cell (the cells share the load; the battery lasts longer, but voltage does not increase).

Worked example 4

Three identical cells of emf 3 V are connected in series to a lamp. Find the potential difference across the lamp (negligible internal resistance). V = ε = 3 × 3 = 9 V.

Worked example 5

Four identical 1.5 V cells connected in series give 6 V. What emf is obtained if the same four cells are connected in parallel? The emf = 1.5 V (equal to one cell).

Electric resistance (R)

• is the opposition that a conductor offers to the flow of electric charges through it.
• Its unit is the ohm (Ω); it is measured with an ohmmeter.

• A fixed resistor has a constant resistance.
• A variable resistor (rheostat) has a resistance that can be changed by moving a slider, which changes the length of wire through which the current flows. A longer length of wire gives larger resistance, so the total circuit resistance increases and the current decreases. Application: the volume control knob of a radio is a variable resistor: turning it changes the resistance, hence the current and the sound volume.

Factors affecting the resistance of a conductor

• (at a given temperature).
• In a light bulb, the filament is a very thin, long tungsten wire wound into a coil, while the lead-in wires are short, thick copper.
• When the same current passes through both, only the filament gets white-hot.
• The reason: the filament has far larger resistance.
• Resistance increases when:

1. the length of the wire through which the current flows increases;
2. the cross-sectional area (thickness) decreases;
3. the material has a larger resistivity (each material has its characteristic value; tungsten has a higher resistivity than copper).

• Also, the higher the resistance of a wire carrying a given current, the more heat is produced in it.
• That is the principle of heaters, irons and incandescent lamps.

• Scientist profile. Thomas Edison (1847–1931), an American inventor with 1093 US patents, including the phonograph, the motion-picture camera and projector, and a long-lasting incandescent lamp.
• He contributed to the development of electric power systems.

Engineers draw circuits with standard symbols: cell, battery, switch (open and closed), light bulb (a circle with a cross), fixed resistor (a rectangle or zigzag), variable resistor (rectangle with an arrow, or zigzag with arrow), ammeter (circle with A) and voltmeter (circle with V).

To draw a circuit diagram from a real circuit:

1. choose the correct symbol for each component;
2. connect them in the same way as the real circuit (series with series, parallel with parallel, meters in the right places);
3. draw straight lines for wires, with neat corners.

• Current = ordered flow of charge; I = Q/t; ampere = coulomb/second; ammeter in series.
• Potential difference V = W/Q; volt = joule/coulomb; voltmeter in parallel.
• emf is the total work per coulomb by the source in the whole circuit; ε is greater than terminal voltage because of internal resistance.
• Cells in series add emf; cells in parallel keep the emf of one cell.
• Resistance opposes current; increases with length and resistivity and decreases with area.

• In a closed circuit a current flows through a conductor when a potential difference is applied across it, and the conductor's resistance opposes the flow.
• How are the three quantities related?

Activity: verifying Ohm's law.

• Apparatus: battery, fixed resistor, rheostat, switch, ammeter, voltmeter, wires.
• Connect the ammeter in series with the resistor and rheostat, and the voltmeter across the fixed resistor.

Steps:

1. Set the rheostat at its maximum value so that the current is least (this protects the components).
2. Close the circuit and record the voltmeter reading V and the ammeter reading I.
3. Reduce the rheostat resistance; record the new V and I.
4. Repeat several times and tabulate the readings.
5. Plot a graph of V (horizontal axis) against I (vertical axis).

Variables: the independent variable is the potential difference across the resistor (changed by the rheostat); the dependent variable is the current through the resistor; the controlled (constant) factors are the resistor itself (its value), its temperature and the materials.

Result: the graph is a straight line passing through the origin, which means that I is directly proportional to V.

Ohm's law: the current intensity through a conductor (fixed resistor) is directly proportional to the potential difference across its ends, at constant temperature.

I = V / R, or V = I R, or R = V / I

The resistance can be found from the graph: R = ΔV / ΔI, which is the reciprocal of the slope of an I–V graph (where I is plotted on the vertical axis, slope = I/V = 1/R, so R = 1/slope).

Scientist profile. Georg Simon Ohm (1789–1854), a German physicist who found the law named after him and the relationship between current, voltage and resistance; the unit of resistance is named in his honour.

Worked example 1

Find the current in a metal wire of resistance 2 Ω when the potential difference across it is 6 V. I = V/R = 6 / 2 = 3 A.

Worked example 2: from a graph

An I–V graph for a fixed resistor is a straight line through the origin passing through the point (V = 2.0 V, I = 0.40 A). Find R. R = V/I = 2.0 / 0.40 = 5 Ω. Check with another point: at 1.0 V the current should be 0.20 A.

Worked example 3

What potential difference is needed to drive 0.25 A through a 48 Ω resistor? V = I R = 0.25 × 48 = 12 V.

What happens if the potential difference doubles? The resistance of the conductor does not change (it depends on length, area, material and temperature); the current doubles.

Several resistors may be connected in two ways.

The equivalent resistance (R′) is the single resistance that has the same effect on the circuit as the whole group of resistors.

Life application.

• Household appliances are connected in parallel to the mains, so that each can be switched on or off independently, and each gets the full mains voltage.
• If one lamp burns out, the others keep working, since each has its own path to the source.

Worked example 4: series

Three identical resistors are connected in series to a 12 V source. The current is 2 A. Find the resistance of each. Total (equivalent) resistance R′ = V / I = 12 / 2 = 6 Ω. R′ = 3R, so R = 6/3 = 2 Ω. Check: each resistor has 4 V across it (2 A × 2 Ω), and 3 × 4 = 12 V.

Worked example 5: parallel

Three identical 6 Ω resistors are connected in parallel to a 12 V battery. Find the equivalent resistance, the total current, and the current in each resistor. R′ = R/n = 6/3 = 2 Ω. Total current I = 12/2 = 6 A. Each branch: 12/6 = 2 A, and 3 × 2 = 6 A.

Worked example 6: two ways with the same resistors

Two identical 10 Ω resistors are connected to a 10 V battery, first in series, then in parallel. In series R′ = 20 Ω, I = 0.5 A. In parallel R′ = 5 Ω, I = 2 A. The parallel arrangement gives a current four times as large. This is why, if you want the maximum reading on an ammeter with two identical resistors and two identical cells, you connect the cells in series (to double the emf) and the resistors in parallel (to halve the resistance).

• Pacemaker (integration with medicine). A pacemaker is a small electronic circuit implanted under the skin: a battery, a microchip, and wires to the heart muscle.
• If the heartbeat becomes irregular, the circuit sends tiny electric pulses to make the heart contract regularly.

Energy-efficiency label. Appliances such as refrigerators carry a label that shows the energy rating and the monthly energy consumption, so that buyers can choose appliances that consume less (the highest rating, shown by the green arrow, A, means the highest efficiency and lowest consumption).

Electric power (P) is the rate at which an appliance consumes electric energy, i.e. the energy it consumes per second.

P = E / t, so E = P × t

Unit of power: watt (W) = 1 joule per second. "A 9 W LED lamp" uses 9 joules of electrical energy every second.

The joule is a very small unit of energy, so in household billing the larger unit kilowatt-hour (kW·h) is used: 1 kW·h = 1000 W × 3600 s = 3.6 × 10⁶ J.

Cost of electric energy. Follow four steps:

1. Power in kW = power in W ÷ 1000.
2. Find the number of operating hours.
3. Energy in kW·h = power in kW × hours.
4. Cost = energy in kW·h × price per kW·h.

(The price per kW·h used in the book's example is 1.55 Egyptian pounds. For your own calculations use the price given in the question.)

Worked example 7: comparing two lamps

A CFL of 20 W and an LED of 12 W give about the same light. Compare the monthly (30-day) costs if each is used 12 h per day at 1.55 pounds per kW·h. Hours = 30 × 12 = 360 h. CFL: 0.020 kW × 360 = 7.2 kW·h; cost = 7.2 × 1.55 = 11.16 pounds. LED: 0.012 × 360 = 4.32 kW·h; cost = 4.32 × 1.55 = 6.70 pounds. Saving = about 4.46 pounds per month per lamp. The LED is more efficient.

Worked example 8: a kettle

A 1500 W kettle is used 0.5 h per day. What is the weekly energy and cost at 1.55 pounds per kW·h? P = 1.5 kW. Hours per week = 0.5 × 7 = 3.5 h. Energy = 1.5 × 3.5 = 5.25 kW·h. Cost = 5.25 × 1.55 = 8.1375, about 8.14 pounds.

Worked example 9: equal energy

Appliance X has power 2000 W, appliance Y has 1000 W. X runs for 3 hours. How long must Y run to use the same energy? E = 2 kW × 3 h = 6 kW·h; time for Y = 6 / 1 = 6 hours. The lower the power, the longer it takes to use the same energy.

• Typical powers (for reference): electric heater 1–2 kW; iron 1–2.5 kW; air conditioner 1.5–2.5 kW; laptop about 50 W; fan 50–100 W; television about 100 W.
• Heating appliances and air conditioners consume the most energy.
• Increasing the operating hours (cause) increases the energy and the cost (effect).

• Avoid overloading: do not plug several high-power appliances into one socket; the large current can overheat the wires and cause a fire.
• Insulation: wires must be properly insulated so that you cannot get a shock.
• Dryness: make sure hands, wires and components are dry; tap water conducts electricity.
• Disconnect the source before changing anything in a circuit.
• Never use household mains in school experiments; use batteries.

• Ohm's law: I ∝ V at constant temperature; R = V/I; the graph is a straight line through the origin.
• Series: same current, voltage divides, R′ = nR. Parallel: same voltage, current divides, R′ = R/n.
• P = E/t; E (kW·h) = P (kW) × t (h); cost = E × price.

• Scrapyards use cranes with huge magnets to lift iron scrap and drop it by switching the magnetism off.
• How does it work, and how can the strength be controlled?
• The answer lies in the connection between electricity and magnetism.

Oersted's experiment.

• Place a compass on a table and wait until its needle settles pointing north.
• Put a straight wire above the compass, parallel to the needle.
• Connect the wire in a circuit with a battery and a switch.
• When the switch is closed, the needle deflects from its original direction; when the circuit is opened, the needle returns.

• Hans Oersted concluded that an electric current in a wire produces a magnetic field around the wire.
• This is the magnetic effect of electric current, and it is the basis of the electromagnet, the electric motor and the electric generator.

• Revision. A magnetic field is the region around a magnet where the effect of its magnetic force can be detected.
• A compass needle is a small, freely pivoted magnet that points north because of the earth's magnetic field.
• Magnetic field lines are imaginary lines that show the direction and strength of a magnetic field.

• An electromagnet consists of an insulated copper wire wound as a coil around a soft iron core.
• When current flows in the coil, a magnetic field is produced which magnetises the soft iron, and the whole device acts as a strong magnet.

Activity.

• Wind insulated copper wire evenly around a long soft-iron nail.
• Connect the ends in series with a battery, switch, rheostat and ammeter.
• Hold the nail above iron paper clips.
• Close the circuit: clips are attracted.
• Open it: the clips fall.
• So an electromagnet is a temporary magnet: it can be switched on and off by the current.
• A permanent magnet cannot be switched off.
• That is why a crane uses an electromagnet to pick up iron scrap and release it.

Factors affecting the strength of an electromagnet.

• Use the number of paper clips lifted as the indicator.
• Change one factor at a time (independent variable), measure the number of clips (dependent variable), and keep the others constant (controlled variables).

The strength increases with:

1. increasing the number of turns of the coil;
2. increasing the current intensity (reduce rheostat resistance, or add cells in series);
3. using a soft iron core (it is easily magnetised and loses its magnetism easily when the current stops; steel would stay magnetised).

Structure and function. The core is soft iron because it magnetises easily and demagnetises easily, which is what makes the electromagnet a temporary magnet.

(Both attract iron; they have north and south poles.)

Application: electric bell.

• When the button is pressed, current flows and the electromagnet magnetises its soft iron core and attracts a flexible metal strip, which brings the hammer against the gong.
• When the strip moves, it breaks the circuit at the contact screw, so the current stops and the electromagnet loses its magnetism.
• The strip springs back (elastic), closing the circuit again, and the cycle repeats rapidly, giving a continuous ring.

Application: sorting metal waste.

• Electromagnets separate iron and steel from mixed waste for recycling.
• Recycling metal needs far less energy than making new metal from ore.

If a wire carrying current is placed in another magnetic field, such as that of a magnet, what happens?

Activity.

• Two parallel copper rails are connected to a battery and switch.
• A light cylindrical metal rod lies across the rails, between the poles of a strong U-shaped magnet.
• When the switch is closed, the rod rolls along the rails.

• Conclusion: a straight conductor carrying current and placed perpendicular to a magnetic field experiences a magnetic force in a definite direction.
• If the cell connections are reversed (so the current direction reverses), the force and movement reverse.
• Reversing the magnet's poles also reverses the direction of motion.
• This force is the basis of the electric motor.

If we put a coil instead of a straight wire between the poles of the magnet, the coil turns. Parts of a simple DC motor:

• a coil of insulated copper wire, free to rotate between the poles of a strong U-shaped permanent magnet;
• a commutator (split ring): a metal cylinder split into two halves insulated from each other, connected to the coil ends;
• two graphite brushes that touch the commutator and carry current into and out of the coil; they are the contacts through which current enters and leaves;
• a direct-current source (battery).

How it works.

• When current flows in the coil, a magnetic force acts on each of the two long sides (AB and CD).
• Because the currents in the two sides are in opposite directions, the forces are equal in magnitude but opposite in direction and act along different lines of action.
• Two such forces form a couple, producing a turning effect called torque, which rotates the coil.
• (An everyday example of a couple: turning a steering wheel with both hands.) The split-ring commutator reverses the current in the coil every half turn, so that the torque always turns the coil the same way and the rotation continues.

Factors affecting the torque (and so the speed) of a motor:

1. increasing the current through the coil;
2. increasing the number of turns of the coil;
3. increasing the area of the coil;
4. increasing the magnetic field strength, by using stronger magnets or more magnets.

Reversing the rotation.

• Direction of rotation can be reversed by (a) reversing the current direction (swap the cell connections) or (b) reversing the magnet's poles (turn the magnet round).
• Doing both together leaves the direction unchanged.

Energy conversion: an electric motor converts electrical energy into mechanical (kinetic) energy.

Applications:

• fans, blenders, washing machines, electric cars, the metro.
• A drill/screwdriver has a speed control: low speed for screws (accuracy and no damage to the screw head), high speed for drilling into walls (more power to penetrate hard material).

• What if you turn the shaft of a motor by hand, instead of supplying it with current?
• The process reverses and the motor produces electrical energy.
• This is the principle of the generator.

Activity.

• Connect a small DC motor to an LED (or small bulb), fix a small fan on its shaft and turn the fan by hand.
• The lamp lights; turning faster makes it brighter.
• (An LED lights only when current passes in one direction; if it does not light, reverse its connections.)

An electric generator converts mechanical energy into electrical energy.

Working principle.

• A coil is rotated by some source of mechanical energy between the poles of a magnet.
• As the coil cuts the magnetic field lines, an electromotive force (emf) is generated in it, driving a current through the external circuit.
• The emf generated increases when the coil's rotational speed increases (and also with a stronger magnet or more turns).

Life application: bicycle dynamo.

• A small generator pressed against the wheel is turned by friction as the wheel rotates; it gives electricity to the lamps without batteries.
• The faster the wheel turns, the larger the emf and the brighter the lamp.

Uses:

1. Power stations. Turbines driven by water falls, steam or wind turn coils in large generators to power cities and factories. Power stations use generators producing alternating current (AC), the type used in homes. (A generator with a split-ring commutator produces direct current.) Advantages of AC: it can be transmitted over long distances with small energy loss, and it can be converted to DC when needed.
2. Emergency generators in homes, hospitals and factories for power cuts.

• Current produces a magnetic field (Oersted).
• Electromagnet: coil + soft iron core; strength increases with turns, current and an iron core; temporary magnet; used in cranes and bells.
• A wire carrying current in a magnetic field experiences a force; a coil experiences torque, giving the motor.
• Motor: electrical to mechanical; generator: mechanical to electrical; emf rises with rotational speed.