Chapter Five
CURRENT ELECTRICITY (II)
ELECTRIC CURRENT AND POTENTIAL DIFFERENCE
- A basic electric circuit comprises electrical components, i.e., bulbs, cells, etc., connected together by copper wires to enable electric charges to flow from oneterminal of the electrical source, through the components, to the other terminal.
Electric Current
- An electric current is the rate of flow of charge through a conductor.
- Electric current is measured using anammeter.
Using an Ammeter
- Before connecting the ammeter in the circuit, ensure that the pointer is at zero mark on the scale.
If this is not the case, use the zero adjusting screw to move it to the correct position,
- The ammeter is an instrument oflow resistance.
It is thus connected in series with other components in the circuit so that conventional current enters the ammeter through its positive terminal and exits through the negative terminal.
If the terminals are interchanged, the pointer moves away from thescale in anticlockwise direction. This can damage the instrument.
- An appropriate scale should be selected to safeguard the coil of the meter from blowing up.
If 5A scale is selected, the meter can safely read up to a maximum of 5 A.
With such a scale, ten divisions represent 1A.
For scale of2.5 A, ten divisions represent 0.5A.
- The readings on the ammeter are 2.45Awhen using 0 – 5 A scale, or 1.225A for0 – 5 scale.
- It should be noted that more accurate digital ammeters are available in the market.
Potential Difference
- Electric potential difference (p. d) is defined as the work done per unit charge in moving charge from one point to another.
- The device that produces energy to do this work is called a source of electromotive force (e.m.f).
The source may be :
- a battery, which converts chemical energy to electrical energy,
- a generator, which converts mechanical energy to electrical energy.
- When the battery does the work of ‘pumping’ charges through a conductor or an electrical device, an electric potential difference (p.d.) develops between its ends.
- This potential difference is measured in volts using the voltmeter.
From the equation, one volt is equal to one joule per coulomb.
Example 1
In moving a charge of 10 coulombs from point B, 120 joules of work is done. What is the
potential difference between A and B?
Solution
Using a Voltmeter
- The pointer is adjusted to zero as with the ammeter.
- A voltmeter is always connected across (in parallel to) the device across which the voltage is to be measured.
- This is because it is an instrument with high resistance to flow of current, hence takes little current in the circuit.
- Note that the positive terminal of the voltmeter is connected to the positive terminal of the electrical power source.
- The appropriate scale should be selected, and, when taking the reading, parallax error
should be avoided.
Points at Same Electric Potential
- The two voltmeters VI and V_{2}in figure below indicate the same reading.
- This is because a good conductor of electricity joins points A and D.
- The same applies to points Band C.
- Points A and D are said to be at the same electric potential, so are points Band C.
- Each of the voltmeters VIand V 2 therefore measure the potential difference (or voltage) between points Aand B.
- Note that the electric potential at A is higher than that at B, hence, conventional current flows from point A to B through an external circuit (bulb).
To investigate the current and voltage in a parallel circuit arrangement
Apparatus
Two 1.5 V cells, 3 identical bulbs, 3 ammeters, 4 voltmeters, switch, connecting wires.
Procedure
- Connect the circuit as shown in ABOVE
- Switch on the circuit and take the readings on the ammeters A_{1}‘ A_{2}, A_{3}and A_{4}.
- Switch off the circuit and disconnect the ammeters.
- Connect the voltmeters as shown in figure BELOW
- Take the readings on V1′ V_{2}, V3 and V_{4}
Observation
- Reading on A1+ Reading on A2 + Reading onV3= Reading on A4
- Reading on VI = Reading on V_{2}= Reading on V3 = Reading on V_{4}
Note:
When components are connected in parallel:
- the sum of the currents in parallel circuits is equal to the total current. Thus, the total current flowing into a junction equals the total current flowing out.
- the same voltage drops across each of them (since their terminals are at the same electric
potential),
Example 2
Find the current passing through L1in figure below given that 0.8 A passes through the battery, 0.28 A through L2 and 0.15 A though L_{3}•
To investigate current and voltage in series arrangement
Apparatus
Three ammeters, four voltmeters, three 2.8 V torch bulbs, holder, switch, connecting wires,
two cells.
Procedure
- Connect the circuit as shown in figure above
- Switch on the circuit and record the ammeter readings.
- Switch off the current and disconnect the ammeters.
- Connect the voltmeters as shown in figure below
Switch on the circuit and record the voltmeter readings.
Observations
(i) The reading of current by the ammeters A1and1A2 and A1, is the same.
(ii) The total voltage drops across the bulbs (V 1+ V 2 + V 3) equals the total voltage V 4 acrossthe terminals of the battery.
Note:
The above statements are true even when the bulbs are not identical.
Conclusion
In a series arrangement,
- the same current flows through each component.
- The sum of the voltage drops across the components is equal to supply voltage.
Example 3
In the circuit shown below, what is the potential difference across the bulb and the switchwhen the:
(a) switch is open?
(b) switch is closed?
(a) Potential difference across the bulb is zero since no current is flowing through it, while
the p.d across the switch is 1.5 V.
(b) The p.d across the bulb is 1.5 V, since a closed switch is a conductor and has zero voltage.
OHM’S LAW
The relationship between the voltage across a conductor and the current flowing through it is
summarized in what is referred to as Ohm’s law.
To investigate the relationship between current and voltage across a nichrome wire
Apparatus
Two-metre nichrome wire, 2 dry cells, ammeter, voltmeter, connecting wires, switch, rheostat.
Procedure
- Set up the circuit as shown in figure above
- Set the current flowing in the circuit to the least possible value.
- With the help of the rheostat, vary in steps the current flowing in the circuit and note the corresponding voltage drop across the coil.
- Record the results in table
- Plot a graph of voltage against current.
Current(A) | ||||||||
Voltage(v) |
Observation
As current increases, voltage across the coil also increases.
A graph of voltage against current is a straight line, as shown in figure
Conclusion
- The graph obtained is a straight line that passes through the origin.
- Voltage is therefore directly proportional to current.
- The gradient i.e is constant known as the resistance of the wire
- The resistance
- The S1 unit of resistance is the Ohm (Q).
- This is known as Ohm’s law, which states that the current flowing through a conductor is directly proportional to the potential difference across it provided the temperature and other physical conditions are kept constant.
- Ohms’ law can be verified using the same procedure as in experiment above, with the coil being replaced by a standard resistor.
- The graph of current against voltage is a straight line through the origin.
Voltage (Volts)
- The gradient of the graph, , gives the reciprocal of resistance (conductance whose unit is Ω^{-1}or Siemens (s)
- Thus, resistance =
- Since V
- V = IR, where V is the potential difference, I the current and R the resistance.
- Thus, Ohm’s law can also be expressed as
- From Ohm’s law, an ohmis defined as the resistance of a conductor when a current of 1 A flowing through it produces a voltage drop of 1 V across its ends. The multiples of ohms in common use are:
- 1 kilohm (kΩ) = 1 000 Ω
- 1 Megohm (MΩ) = 1 000 000 Ω
Example 4
A current of 4 mA flows through a conductor of resistance 2 00Ω. Calculate the voltage across the conductor.
Solution
Example 5
Calculate the current in amperes flowing through a device of resistance 50Ω when a 10 V
source is connected to it.
Solution
Example 6
In order to start a certain car, a current of 30 A must flow through the starter motor. Calculate
the resistance of the motor given that the battery supplies a voltage of 12 V. Ignore the internal resistance of the battery.
Ohmic and Non-ohmic Conductors
- Conductors that obey Ohm’s law (i.e., voltage across is directly proportional to current) are called ohmic conductors.
- Nichrome wire in experiment above is an ohmic conductor.
- When theexperiment is repeated with a torch filament, an electrolyte, a thermistor, a semiconductor diode, a thermionic diode or discharged tubes in place of the nichrome coil, the following I-V graphs are obtain
- A material whose graph is not a straight-line graph is said to be non-ohmic.
- The resistance of such a material changes with current flow.
Electrical Resistance
- Electrical resistance is the opposition offered by a conductor to the flow of electric current.
- It occurs when a charge flowing through a conductor has its movement impeded by collisions with the atom and impurities in the conductor.
Factors that affect the Resistance of a Metallic Conductor
Temperature
- The resistance of good conductors of electricity, like metals, increases with increase temperature.
- Heating increases the vibrations of atoms thereby increasing the collisions per cross-section area of the conductor.
- The opposition to the flow of electrons thus increases as temperature is increased.
- This does not, however, mean that the resistance of, say, copper can fall to zero at extremely low temperatures.
- Length of the Conductor
- The resistance R of a uniform conductor of a given material is directly proportional to its length l i.e
- Hence ,resistance = constant x length…………………………………… (l)
- So, for a given conductor,
- As the length of the conductor increases, so does the resistance.
- Cross-section Area
- The resistance of a wire is inversely proportional to its cross-section area A.
- A conductor with a larger cross-section area has many free electrons for conduction, hence better conductivity.
From the above
Resistance x cross-section area = constant …………………………………(2)
For a given conductor therefore, RA = constant.
Combining equation 1 and equation 2
where is the resisitivity of the material.
- The resistivity of a material is numerically equal to the resistance of a sample of the material of unit length and unit cross section area at a certain temperature.
- The unit of p is the ohm-metre (Om).
- The resistivity of a material is dependent on temperature. For metal conductors, it increases with increase in temperature while for semiconductors; it decreases with increase in temperature.
Note:
It is helpful to express all lengths in meters so as to obtain resistivity in ohm-metre units. Every material has its resisitivity
Material | Resisitivity (Qm) | Uses |
Silver | 1.6 x 10.^{8} | Contacts on some switches |
Copper | 1.7 x 10.^{8} | Connecting wires |
Aluminium | 2.8 x 10.^{8} | Power cables |
Tungsten | 5.5 x 10.^{8} | Lamp filaments |
Constantan | 49 x 10.^{8} | Resistance boxes, variable resistors |
Nichrome | 100 x 10-^{8} | Heating elements |
Carbon | 3000 x 10-^{8} | Radio resistors |
Glass | 10-^{8}_ 10^{14} | |
Polystyrene | 10^{15} |
Resistors
- Resistors are conductors specially designed to offer particular resistance to the flow of electric current.
- They are made from many different materials which include resistance wires, metal
alloys and carbon. - Most wire-wound resistors are made of nichrome wire covered with an insulating material.
- Some metals may not make good resistors because of the effect of temperature on their resistance.
- Temperature does not however show significant effect on the resistance of some other materials like constantan and manganin.
- These metals may be mixed with carbon to make standard resistors.
Types of resistors
Fixed resistor
- They include wire-wound and carbon resistors.
- They are designed to give fixed resistance.
- The symbol for a fixed resistor.
- A fixed resistor can be wire (e.g. nichrome) wound or carbon type.
Variable Resistors
These are resistors with a varied range of resistance. They include:
- Rheostat
A rheostat is a two-terminal variable resistor represented in electrical circuits by the symbol
Moving the sliding contact along the length of the resistant material varies the resistance
between points A and B.
When the contact is nearer end A, the resistance of the rheostat is lower.
- Potentiometer
The potentiometer is a variable resistor with three terminals.
Its symbol is shown above.
In potentiometers, a contact is moved to select desired proportions of the total voltage across them.
Non-Linear Resistors
The current flowing through these resistors does not change linearly with the changes in the
applied voltage.
Such resistors include the thermistor and light dependent resistor (LDR).
- Light-dependent Resistor (LDR)
The resistance of an LDR decreases when it receives light of increasing intensity. Its symbol
- Thermistor
The thermistor is a temperature-dependent resistor.
Its resistance decreases with increase in temperature.
The electrical symbol of a thermistor is
Thermistors are used in heat-operated circuits.
To determine the resistance of a resistor using the voltmeter-ammeter method
Procedure
Set up the circuit as shown in the figure 5.24.
With the switch open, record the voltmeter reading V and the corresponding ammeter reading I.
Switch on the current and, by adjusting the variable resistor, record at least five other
values of V and the corresponding I.
Record your results in the table 5.3.
Table 5.3 Compare values of i‘
Plot a graph of V (vertical axis) against I. Note the shape of the graph.
Determine the slope (gradient) of the graph.
Observation
When the switch is open, no current flows through the resistor and therefore both the ammeter
and the voltmeter reading is zero.
When the current through the resistor increases, the voltage across it also increases. An
approximate value of the resistance of the resistor is obtained by dividing the value of the
voltage across the resistor by the corresponding current flowing through it and substituting in the
equationR = i.
The graph of V against I is a straight line whose gradient gives resistance, see figure
5.12. The resistance obtained cannot be accurate since the voltmeter takes some little current,
thus not all of it flows through the resistor.
The Wheatstone Bridge Method
The Wheatstone bridge consists of four resistors and a galvanometer, as shown in figure 5.25.
The operation of the bridge involves making adjustments to one or two of the resistors until
there is no deflection in the galvanometer.
• |
Fig. 5.25: The Wheatstone bridge
The four resistors K, L, M and N are joined as shown. If K is the unknown resistance, the
values of L, M and N, or the ratio ofM to N must be known. A galvanometer G and a dry cell
are connected as shown. .
The variable resistor L (commonly a resistance box) is adjusted until there is no deflection
in G. The bridge is then said to be balanced. No current flows through G at balance and
therefore, the p.d. across BD is zero. At the same time, the potential difference across AB is
then equal to that across AD. Also, the same current I) flows through K and L and current 12
flows through M and N. Then; .
11 = 13 and 12 = 14
Therefore, I) K = 12 M (from V = IR)
Similarly, ~L = I_{4}N.
So, IlL=’I_{2}N
11K _ 12M
IlL – I2N
Therefore, ~ = ~ when the bridge is balanced.
The Wheatstone bridge is more accurate in measuring resistance than the voltmeter-
ammeter method because the value obtained by using the wheatstone bridge method does not
depend on the accuracy of the current-measuring instrument (galvanometer) used.
The Metre Bridge
Figure 5.26 shows a practical form of the Wheatstone bridge known as the metre bridge.
Fig. 5.26: Meter bridge
The wire AC of uniform cross-section area and length 1 m with a resistance of several ohms
and made of an alloy such as constantan. The length AD represents resistor M while length CD
represents resistor N. The ratio of M to N is altered by changing the position on the wire of the
movable contact D called ‘jockey’. The other arm of the bridge contains the unknown resistor
K and a known resistor L. The copper strips of low resistance connect the various parts. The
position of D is adjusted until there is no deflection in G. Then;
K = M = resistance of AD
L N resistance of DC
Since the wire is uniform cross-section, its resistance will be proportional to its length and
Hence’ K= AD _ XI
• L DC – X_{2}
Thus K= LXI
, X_{2}
The resistor L should be chosen to give balance points near the centre of the wire. This gives
a more accurate result. After obtaining the balance. K and L should be interchanged and a
second pair of values for XI and X_{2} obtained. This average of the value eliminates errors due to
non-uniformity of the wire and end corrections. In finding the balance point, the cell key or
switch should be closed before the jockey makes contact with the wire. This is necessary
because of the effect known as ‘self-induction’ in which the currents in the circuit take a short time to grow to their steady values. A high resistance should always be joined in series with
the galvanometer to protect it from damage whilst the balance is being sought.
Example 10
In an experiment to determine the resistance of a nichrome wire using the metre bridge, the
balance point was found to be at 38 em mark. If the value of the resistance in the right hand
gap needed to balance the bridge was 25 Q, calculate the value of the unknown resistor.
Resistor Networks
Resistors Connected in Series
Figure 5.28 shows three resistors connected in series.
Fig. 5.28: Resistorsin series
Since this is a series arrangement, V T = VI + V_{z}+ VyThe same current I flows through each of
the resistors. Using Ohm’s law and the fact that same current flows through the resistors;
I~ = I(R_{I}+ R_{2}+ R_{3})
Diving through by I;
Thereforfor resistors connected in series, the equivalent resistance is equal to the sum of individual resistances.
Example 11
Three resistors of 2.5 Q, 12 Q, and 3.5 Qare connected in series. What single resistor can
replace them in a circuit?
Example 12
Figure 5.29 shows three resistors in series connected to a power source. A current of 2 A flows
through the circuit.
Calculate:
(a) the voltage drop across each resistor.
(b) the voltage across the source.
(c) the total resistance in the circuit.
Resistors Connected in Parallel
Fig. 5.30 shows three resistorsRj, R, and R3 connected in parallel.
Example 13
The circuit diagram in figure 5.31 shows four resistors in parallel connected across a3 V supply.
Calculate:
(a) the effective resistance.
(b) the current through the 80 resistor.
Solution
(b) Current through the 8 0 resistor
Example 14
Two resistors
of 300 and 70 0 are connected in parallel. Calculate their equiva.ent resistance.
Example 15
Several 150 Q resistors are to be connected so that a current of 2 A flows from a 50 Y source.
How many resistors are required and how should they be connected?
Solution
Example 16
Calculate the current through each resistor in the figure 5.32.
Series – Parallel Arrangement
To find the effective resistance of a series – parallel arrangement, the network is systematically
reduced into a single resistor.
Example 17
Determine
the equivalent
resistance for the resistors in figure 5.33.
Since the 30 ,Qand 70 ,Q resistors are in parallel, the two can be replaced by a single one whose
value is;
The 21 ,Qresistor is now in series with the 19,Q resistor, see figure 5.34 (b). The two resistors
can be replaced by a single resistor R_{AC}= 19 + 21 = 40 ,Q, see figure 5.34 (c).
Example 21
Calculate the effective resistance in the figure 5.35.
Solution
The reduction begins by combining the 5 Q and 7 Q resistors, which are in series, to get 12 Q.
The circuit is then re-drawn as in figure 5.36 (a). The 12 Q resistor in parallel with the 8 Q
resistor may be replaced ~c == fft! == 4.8 Q. The circuit is re-drawn as in figure 5.36 (b).
Finally.the 4.8 Q is in series with the 4 Q resistor. giving an equivalent resistance of;
R_{AC}== 4 +4.8
== 8.8Q
Example 19
Two resistors of 6 Q and 3 Q in parallel are connected in series to a 4 Q resistor and a cell of
e.m.f. 1.5 V. Calculate:
(a) the equivalent resistance of the circuit.
(b) the current through each of the resistors and the p.d. across each.
Solution
(b) Total current flows through the 4 Q resistor.
Now 16Q + 13Q = 0.25 A…………….. : (1)
But, voltage across 6 Q = voltage across-3Q
6 X 16Q = 3 X 13Q
13Q = 216Q …………………….•… ~ ·i •••••• (2)
Substituting (2) in (1);
16Q + 216Q = 0.25 A
3 x 16Q= 0.25
16Q = 0.0833 A
Substituting in (2);
13Q = 2 x 0.0833
= 0.167 A
V6Q = 0.0833 x 6
=,0.5V
This is also the voltage across the 3 Q resistor, since they are in parallel.
Alternatively; .
,To calculate current ,through either 6,Q or 3 Q .resistor, U1e p.d, across them.must be
found first. Thus, p.d. across 6Q& 3′ Q’ + p.d. across 4· Q ~ voltage’ ofthesupply
Voltage across 412 resistor == ‘0.25 x 4
= 1 V
Hence, the voltage across (6Q and 3Q) is;
V = 1.5-1
V=0.5V
CUrrent through 6 Q | = | 05 |
6 | ||
= | 8.33 x 1O-2A | |
Current through 3 Q | = | 0.5 |
3 | ||
= | 1.67 X 10-^{1} A |
Exampk20
Four resistors of 5 0, 13 0, 3 0 and 6 0 are connected to 6 V supply, as shown in figure 5.38.
Fig. 5.38 Calculate: (a) the current through the 13 0 resistor. (c) voltage V AB and V BO. Solution (a) From figure 5.38 (a), current from the supply divides into two parts at junction A. Part of The 13 0 and 2 O’inseries”f<:nw a resistor that is in parallel with the 5 ,Q resistor. The p.d. across the 13 ,Qand 2 ,Q resistor, is equal to the supply of 6 V. 6 Current through 13.Q = 13 + 2 = 6 15 = O.4A |
(b) Total current = current through 13 .Q resistor + current through 5 .Q resistor
Current through 13 .Q = 0.4 A
Current through 5,Q
p.d. across 5,Q
= 5Q
= 6
:5
= 1.2A
Therefore, current in the circuit = 0.4 + 1.2
= 1.6A
(c) V_{AB}= 0.4×13
= 5.2V
V_{BD}= V_{AC}– V_{AB}
= 1.2 x 5 – 5.2
= 6.0-5.2
= O.8V
Alternatively;
V_{BD}= 0.4 x 2
= O.8V
Example 20
Figure 5.39 shows a current of 0.8 A passing through an arrangement of four resistors.
Find the current through the 10 Q resistor.
Solution
The network of resistors can be replaced by two resistors of 40 Q and 60 Q in parallel.
C |
n _ p.d. between P and R
urrentoug ;lot. – (30 + 10)
P.d. between P and R = 0.8 X R_{E}, where RE is equivalent resistance for the whole network.
R = 40×60
E 60+40
= 24Q
P.d. across P and R = 0.8 x 24
= 19.2
Therefore, current through 10
= 0.48A
ELECTROMOTIVE FORCE AND INTERNAL RESISTANCE
The function of a cell in a circuit is to supply electrical energy. By definition.the electromotive
force (e.m.f.) of a cell is the potential difference between its terminals when no charge is
flowing out of the cell (cell in open circuit). .
Figure 5.40 shows a circuit that may be used to demonstrate the difference between
e.m.f. of a cell and terminal voltage. The reading of the voltmeter when the switch is open is
the e.m.f. of the cell.
Once a cell supplies current to an external circuit, the potential difference across it
drops by a value referred to as ‘lost voltage’. This loss in voltage is due to the internal resistance
of the cell. The potential difference across the cell when the circuit is closed.is referred to
as the terminal voltage of the cell.
Fig. 5.40: Em.f. ofa cell
A cell or any source of e.m. f. is made up of materials that are not perfect conductors of electricity.
They therefore offer some resistance tothe flow of current that they generate. This resistance
is usually low and is called the internal resistance of the cell or battery.
Relationship between E.M.F. and Internal Resistance
If a resistor R is connected in series with a cell as shown in figure 5.41, the internal resistance
of the cell r is considered to be connected in series with the external resistor R.
Fig. 5.41: Internal resistance of a cell
Using the equation E = V + Ir and hence V = E – Ir, the gradient of the graph gives the internal
resistance r of the cell.
If the graph is extrapolated so as to cut the voltage axis, the point at which it does so
gives the e.m.f. of the cell.
Method 2
Apparatus
Ammeter, voltmeter, variable resistor, cells, connecting wires.
Fig. 5.44: Determination of internal resistance of a cell
Procedure
- Switch on the circuit and increase the current in step from a minimum value.
- Record the corresponding voltage V.
- Complete the table 5.5.
Table 5.5
- Plot a graph of i against R.
Results and Observation
The graph is a straight line with a positive gradient.
Fig. 5.45: Graph of fagainst R
The gradient of the graph gives ~. Internal resistance can be obtained in two ways:
(i) Extrapolating the graph to cut R axis gives r, see figure 5.45.
(ii) If the intercept on t axis is A, then, A = ~
Example 21
A battery consisting of four cells in series, each of e.m.f. 2.0 V and internal resistance 0.6 Q, is
used to pass a current through a 1.6 Q resistor. Calculate the current through the battery.
The e.m.f. of the battery is the sum of the e.m.f. of all the cells while the internal resistance of
the battery is the sum of all the internal resistances of the cells.
:. Current through battery =
Example 22
A cell drives a current of 2.0 A through a 0.6 Q resistor. When the same cell is connected to a
0.9 Q resistor, the current that flows is 1.5 A. Find the internal resistance and the e.m.f of the cell.
Solution
Taking E as e.m.f. ofthe cell and R the internal resistance;
E = 1R + Ir
From figure 5.46 (a);
E == (2.0 x 0.6) + 2.0r
=: 1.2 + 2r ……………………………………….. (1)
From figure 5.46 (b);
E = (1.5 x 0.9) + 1.5r
E =: 1.35 + 1.5r ……………………………….. (2)
Since the e.m.f. is the same in both circuits;
1.2 + 2r =: 1.35 + 1.5r
2r – 1.5r =: 1.35 – 1.2
0.5r = 0.15
r = 0.3 Q
Substituting for r in equation (1);
E =: 1.2 + 2r
E =: 1.2 + 2 x 0.3
E = 1.2 + O~6
= 1.8V
Exampie23
A battery consists of two identical cells, each of e.m.f. 1.5 V and internal resistance 0.6 Q,
connected in the parallel. Calculate the current the battery drives through a 0.7 Q resistor.
Solution
When identical cells are connected in parallel, see figure 5.47, the equivalent e.m.f. is equal to that of only one cell. The equivalent internal resistance is equal to that of two such resistors connected in parallel.