Electricity

What is moving?

For a current to flow, something charged must be free to move. In the activity there are two kinds of moving charge.

• In metal wires: free electrons. An atom has a positive nucleus with electrons around it. In a metal, the outermost electron of each atom is held only weakly by its nucleus, so it wanders easily from one atom to the next. These are called free electrons. A copper wire is full of them, drifting about at random. Connect a cell, and they all start drifting the same way, towards the positive terminal.
• In the salt solution: ions. Common salt (sodium chloride) breaks up in water into positive sodium ions (Na⁺) and negative chloride ions (Cl⁻), which move freely through the liquid. When the ends A and B are dipped in it, the positive ions move towards the wire joined to the negative terminal, and the negative ions towards the wire joined to the positive terminal. A solution that conducts like this is an electrolyte.

A flow of charge is an electric current. More exactly, electric current is the rate of flow of charge: how much charge passes a point in the circuit every second.

If a charge $Q$ passes a point in a time $t$, the current is

$$\text{current } (I) = \frac{\text{charge } (Q)}{\text{time } (t)} \qquad\text{or}\qquad I = \frac{Q}{t}$$

The units of charge and current

The SI unit of charge is the coulomb (C). One electron carries a tiny charge, 1.6 × 10⁻¹⁹ C, so one coulomb is the charge of about 6.25 × 10¹⁸ electrons (6 250 000 000 000 000 000 of them).

The SI unit of current is the ampere (A). One ampere is a flow of one coulomb per second:

$$1\ \text{A} = \frac{1\ \text{C}}{1\ \text{s}}$$

Small currents are measured in smaller units:

• 1 milliampere (mA) = 10⁻³ A = 0.001 A
• 1 microampere (μA) = 10⁻⁶ A = 0.000 001 A

Current is measured with an ammeter, which is always connected in series in the circuit: the circuit is broken and the ammeter is put into the gap, so that all the current has to pass through it (Figure 11.6).

Which way does the current go?

In a metal wire, the electrons flow from the negative terminal of the cell, round the circuit, to the positive terminal. But a flow of negative charge one way does the same job as a flow of an equal positive charge the other way. Long before electrons were discovered, scientists had agreed to draw current as a flow of positive charge, from the positive terminal to the negative terminal. This direction, used in every circuit diagram, is called the conventional current.

Fill two plastic bottles with water to different levels and join them at the bottom with a pipe. Water flows from the fuller bottle to the emptier one, until the levels are the same. Then it stops. To keep it flowing, you would have to keep lifting water back up into the first bottle: you would need a pump.

A circuit works the same way. Charge flows only when there is a “difference in level” between two points, and a cell is the pump that keeps the difference going.

The water circuit	The electric circuit
Water flows from a high level to a low level.	Charge flows from the positive terminal (high potential) to the negative terminal (low potential).
When both levels are the same, the water stops. To keep it flowing, a pump uses energy to lift water from the low level back to the high level.	To keep charge flowing, the cell uses chemical energy to keep one terminal at a high potential (+) and the other at a low potential (−).
The pump provides the energy for the flow.	The chemical reaction in the cell provides the energy for the flow.
When the pump stops, the levels become equal and the flow stops.	When the chemical reaction stops (the cell is “dead”), there is no difference in potential, and the current stops.

When water flows down a pipe, it loses potential energy, and that energy can do work: put a water wheel or turbine in the pipe, and it turns. In the same way, when positive charge moves from a point at high potential to a point at low potential, it gives up energy in the wires and in the appliances (the electric load), and work is done: the bulb lights, the fan turns.

How much energy a cell can give to each unit of charge tells us its potential. The positive terminal of a cell is at a higher potential and the negative terminal at a lower potential. So, for a current to flow through an appliance, one end of it must be joined to the higher potential and the other end to the lower potential of the source.

Electromotive force

Charge needs energy to flow round a circuit. Electric sources, such as generators, batteries and solar panels, provide it. A source changes some other form of energy into electrical energy: a battery changes chemical energy, a solar panel light energy, a generator kinetic energy.

The energy that a source gives to each coulomb of charge to drive it round the whole circuit is its electromotive force (e.m.f.).

$$\text{e.m.f.} = \frac{\text{energy given by the source } (W)}{\text{charge } (Q)}$$

Potential difference

The e.m.f. creates a potential difference between the two ends of the electric load. If end A of a bulb is joined to the positive terminal of a battery and end B to the negative terminal, then A is at a higher potential than B, and positive charge flows through the bulb from A to B. As it does, electrical energy is changed into other forms in the bulb.

The potential difference (p.d.) between two points is the energy changed into other forms when one coulomb of charge moves from one point to the other. It is commonly called the voltage.

$$\text{potential difference } (V) = \frac{\text{energy transformed } (W)}{\text{charge transferred } (Q)}$$

Both e.m.f. and p.d. are energy per coulomb, so both have the same SI unit, the volt (V): 1 V = 1 J/C. Both are measured with a voltmeter, which is always connected in parallel: across the part being measured, without breaking the circuit.

When no current flows in the outside circuit, the p.d. between the cell’s terminals equals its e.m.f. That is why the first voltmeter in Figure 11.5 reads the e.m.f. When a current flows, some electrical energy is used up inside the battery itself, so the p.d. across the load is less than the e.m.f. For example, a battery of e.m.f. 3 V might give a p.d. of 2.5 V across a bulb. That means the battery gives each coulomb 3 J of energy, but the bulb changes only 2.5 J of it into heat and light; the other 0.5 J warms the battery.

Electromotive force	Potential difference
The energy the source gives to one coulomb of charge to drive it round the whole circuit.	The energy changed into other forms when one coulomb flows between two points of the outside circuit.
It doesn’t depend on the resistance connected in the circuit.	It depends on the resistance connected in the circuit.
It is the cause of the potential difference.	It is the effect of the e.m.f.
It belongs to a source (cell, generator).	It exists between any two points of a circuit.

The law

In 1827 a German schoolteacher, Georg Simon Ohm, published the relation between the potential difference across a conductor and the current through it. It is now called Ohm’s law:

The current flowing through a conductor is directly proportional to the potential difference across its ends, provided its temperature and other physical conditions stay the same.

If a current $I$ flows when the p.d. is $V$, then

$$I \propto V \qquad\text{so}\qquad \frac{V}{I} = \text{constant}$$

This constant is called the resistance $R$ of the conductor:

$$\frac{V}{I} = R \qquad\text{or}\qquad V = IR$$

So resistance is the ratio of the potential difference across a conductor to the current through it.

Join a thick copper wire and a thin copper wire of the same length, one at a time, to the same cell. Charge flows much more easily through the thick wire. Every conductor gets in the way of the current a little: the moving electrons keep bumping into the atoms of the metal, which vibrate in their places. This opposition to the flow of current is called electrical resistance.

The SI unit of resistance is the ohm, named after Georg Simon Ohm, with the symbol Ω (the Greek capital letter omega).

From Ohm’s law, $R = V/I$, so

$$1\ \Omega = \frac{1\ \text{V}}{1\ \text{A}}$$

One ohm is the resistance of a conductor through which a current of 1 A flows when a potential difference of 1 V is applied across its ends.

The ohm in fundamental units

The ohm is a derived unit. To see which fundamental units it is made of, write $V$ and $I$ in terms of simpler quantities:

$$R = \frac{V}{I} = \frac{\text{energy} / \text{charge}}{\text{charge} / \text{time}} = \frac{\text{force} \times \text{displacement}}{\text{charge}^2 / \text{time}}$$

Force = mass × acceleration, and charge = current × time ($Q = It$), so

$$R = \frac{m \times a \times s}{(I t)^2 / t} = \frac{m \times a \times s}{I^2 \times t}$$

Acceleration has units m s⁻², so putting in the units:

$$\Omega = \frac{\text{kg} \times \text{m s}^{-2} \times \text{m}}{\text{A}^2 \times \text{s}} = \frac{\text{kg m}^2}{\text{A}^2\ \text{s}^3} = \text{kg m}^2\ \text{A}^{-2}\ \text{s}^{-3}$$

The ohm contains the units of mass (kg), length (m), current (A) and time (s), so it is a derived unit.

Factors affecting resistance

The resistance of a conductor depends on four things:

1. Length. Of two wires of the same thickness, the longer one has more resistance. Double the length, and the electrons must get past twice as many atoms: twice the resistance.
2. Thickness (cross-sectional area). A thin wire has more resistance than a thick one. A thick wire is like a wide road: more charge can flow side by side.
3. Temperature. For a metal wire, heating it increases its resistance: the hotter atoms vibrate more and get in the electrons’ way more often.
4. The material. Copper and nichrome wires of exactly the same length and thickness have very different resistances: the nichrome wire’s is about 60 times larger.

For a given conductor kept in the same conditions, though, the ratio $V/I$ stays the same, however large $V$ is. That is Ohm’s law again.

A resistor is anything in a circuit that has resistance: a bulb, a heater element, or the small striped component inside a radio. Two or more of them can be joined in two basic ways, in series and in parallel.

Series combination of resistors

In a series combination, the resistors are joined end to end, one after another, and the first and the last are connected to the source.

• There is only one path for the current, so the same current flows through every resistor.
• The p.d. of the source is shared among the resistors.
• If one resistor breaks (a bulb fuses), the circuit is broken and they all stop working.
• Separate switches can’t be used for each one.

Older strings of small decoration bulbs, the kind hung on houses for Tihar (Deepawali), were joined in series: if one bulb fused, the whole string went dark, and you had to test bulb after bulb to find the dead one.

Let the resistances be $R_1, R_2$ and $R_3$, the p.d. across them $V_1, V_2$ and $V_3$, and the total resistance of the combination $R$. Since the p.d. is shared,

$$V = V_1 + V_2 + V_3$$

By Ohm’s law each p.d. is the current times the resistance, and the current $I$ is the same everywhere:

$$IR = IR_1 + IR_2 + IR_3$$

Dividing by $I$:

$$R = R_1 + R_2 + R_3$$

In a series combination, the total resistance is the sum of the individual resistances. It is always larger than the largest of them.

Parallel combination of resistors

In a parallel combination, two or more resistors are each connected separately across the source, between the same two points.

• Every resistor has the same potential difference across it.
• There is a separate path for the current through each resistor, so the current is divided among them.
• If one resistor breaks, current still flows through the others and they keep working.
• Each one can have its own switch.

This is how the lights, fans and sockets in a house are wired.

Let the currents through the resistors be $I_1, I_2$ and $I_3$, and the total resistance $R$. Since the current divides,

$$I = I_1 + I_2 + I_3$$

By Ohm’s law, $I = V/R$ for each, with the same $V$ everywhere:

$$\frac{V}{R} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$$

Dividing by $V$:

$$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$

In a parallel combination, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. The total is always smaller than the smallest of them: every extra path makes it easier for the current to get through, like opening more ticket counters at a bus park.

When a current flows through a resistance, the electrons keep colliding with the atoms of the conductor, and the energy they carry is turned into heat. The changing of electrical energy into heat in an electric load is called the heating effect of electric current.

Electric heaters, kettles, immersion rods, irons and rice cookers all work by the heating effect. Their heating element is made of nichrome, an alloy (mixture) of nickel and chromium. Nichrome suits the job because:

• its resistance is high, so a short coil turns a lot of electrical energy into heat quickly;
• its melting point is high, about 1400 °C, so it can glow red-hot without melting;
• it doesn’t oxidize (rust or burn) even when red-hot in air, at 900 °C and more, because a thin, tough layer of chromium oxide forms on its surface and protects it.

A filament bulb uses a very thin wire of tungsten, which melts at 3422 °C, the highest melting point of any metal. It is heated to about 2500 °C, so hot that it gives out light as well as heat. To stop it burning up, the glass bulb holds no oxygen: it is filled with an unreactive gas such as argon, or emptied of air.

The heating effect also depends on the current. A sudden rise in the supply’s voltage, or too many appliances on one wire, can make the current so large that the wires themselves overheat and their plastic covering melts. That is why every house has a fuse or circuit breaker (MCB): a safety device that cuts off the current if it gets too large.

An electric load changes electrical energy into other forms. The energy changed into another form in one second is the power of the appliance.

$$\text{electric power } (P) = \frac{\text{energy transformed } (E)}{\text{time } (t)} \qquad \cdots (1)$$

Its SI unit is the watt (W): 1 W = 1 J/s. 1 kilowatt (kW) = 1000 W.

The energy is changed because charge flows through the appliance across the p.d. between its ends. From the definition of potential difference,

$$V = \frac{W}{Q} \qquad\text{so}\qquad W = QV$$

which is the electrical energy changed by the appliance. Putting this into equation (1):

$$P = \frac{QV}{t}$$

and since $Q/t$ is the current $I$,

$$P = VI$$

Power = potential difference × current.

The electrical energy used by appliances is called electricity consumption. It is measured by the electric meter (energy meter) on the wall of every house.

Energy is power × time:

$$\text{energy } (E) = \text{power } (P) \times \text{time } (t)$$

In joules, the numbers get huge: a 1000 W heater uses 3 600 000 J in one hour. So electricity is sold in a bigger unit, the kilowatt-hour (kWh): the energy used by an appliance of power one kilowatt in one hour. In everyday speech, one kilowatt-hour is simply called one unit.

$$\begin{aligned} 1\ \text{unit} &= 1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} \\ &= 3\,600\,000\ \text{J} = 3.6 \times 10^{6}\ \text{J} = 3.6\ \text{MJ} \end{aligned}$$

How much energy an appliance uses depends on two things: its power and how long it is used. A 10 W LED bulb left on for 100 hours uses as much (1 unit) as a 1000 W heater in one hour.

To find out how much electricity a house used in a month, the meter reader notes the reading on the meter and subtracts last month’s reading. The difference is the number of units used. The bill is the number of units multiplied by the price of one unit.

To work it out from the appliances instead, use

$$E = P \times n \times t$$

where $P$ is the power of each appliance in kilowatts, $n$ is the number of appliances of that kind, and $t$ is the time each one is used in hours. The answer is in kWh (units).

1. Select the best answers

2. Differentiate between

3. Give reasons

4. Answer the following questions

5. Solve the following numerical problems