Electricity and its applications in real life situation T
There is no denying the fact that Electricity is the flow of electric charge. We can
describe the flow of electric charge in several ways. These include the
quantities Current, Voltage and Power and as such Current (I) is the rate of flow of Charge
Carriers, such as electrons. Current is usually thought of
as moving in the direction of positive
charge, so from the positive power
supply to the negative.
However, since in metals it is electrons that
carry electric charge, the actually flow is opposite to the way in which we
think of it
Current, Voltage and Power: Electricity is the flow of electric charge. We can describe
the flow of electric charge in several ways. These include the quantities Current,
Voltage and Power.
Current : Current (I) is the rate of flow of Charge
Carriers, such as electrons. Current is usually thought of as
moving in the direction of positive charge, so from the positive power
supply to the negative. However, since in metals it is electrons that
carry electric charge, the actually flow is opposite to the way in which we
think of it.
·
Current it the amount of Charge, Q that passes a
point in a set time, t. It is measured in Amps (A),
and charge is measured in Coulombs (C). Since Amps are
SI base units, Coulombs are defined as A×s, As.
Voltage: Voltage (V) or Potential Difference (p.d.)
is a measure of the Energy transferred per Charge Carrier between
two points.
·
Voltage is the Energy, E per Charge, Q.
Voltage is measured in Volts (V), which is defined as one
Joule per Coulomb. Voltage can be defined in base units as Kgm2s-3A-1.
Power
·
Power (P) is the rate of Energy transfer. It is
measured in watts (W), where one watt is defined as one
Joule per Second. Hence watts can be expressed in base units as Kgm2s-3
·
From this definition of Power, we can substitute the algebraic
definitionsabove to produce a variety of other formulae, including 'Power
= Current × Voltage'
·
Ohm's Law states that 'Voltage = Current × Resistance'. We can
use this to produce two more definitions of Power.
Charge (symbol Q)
On the sub-atomic level we have a unit of
charge of 'e' , the charge on the electron. This has the value −1.602×10-19C.
Similarly, a proton has a charge of +1.602×10-19C .
The macro unit of charge in electricity is
the Coulomb (C). It is quite a large unit and has
approximately the same charge as 6.2×1018 electrons(or protons).
By definition the Coulomb is defined as:
the charge passing an arbitrary fixed point
when a current of 1 Amp. flows for 1 sec.
Electric current can be thought of as a flow
of charged particles. In the normal case, when charge flows through wires these
particles are electrons. However, when current flows in a vacuum, a solution or
a melt, the charge carriers are ions. In semiconductors the charge carriers are
exotic particles called 'holes'.
By definition the Ampere (A) is defined as:
the current passing an arbitrary fixed point
when a charge of 1 Coul. flows for 1 sec.
Since,
amount of fluid = rate of flow x
time
by analogy,
charge = electric current x time
(Coulombs) = (Amperes) x (seconds)
Q = It
Potential difference is short for potential energy difference.
Our definition of the volt relates charge and the work needed to move that
charge between two points.
In our definition of the volt, the charge is
1 Coulomb and the work done is 1 Joule.
Hence the definition :
Two points A & B are at a potential
difference of 1 volt if the work required to move 1 coulomb of charge between
them is 1 Joule.
A simple analogy is gravity. Compare the
energy difference between a rock at the bottom of a cliff and the energy in
moving it to the top of the cliff. An energy difference exists between the top
of the cliff and the bottom. The amount of energy difference depends on the
size of the rock and the height of the cliff (P.E. = mgh) .
In our analogy, height relates to potential
difference and rock weight relates to the amount of charge. Work is done on the
rock against the force of gravity. Work is done on the charge in moving it
against an electrostatic force.
The volt has unit of joules per coulomb (JC-1 ).
We now have an equation linking work/energy W, charge Q and potential difference V :
work =
charge x potential difference
(Joules) = (Coulombs) x (volts)
By definition, the electrical resistance (R)
of a conductor is the ratio of the p.d. (V) across it to the current (I)
passing through it.
From this equation, by making each quantity
unity, we define the Ohm:
The resistance of a conductor through which a
current of 1 ampere flows when a p.d. of 1 volt exists across it.
Note in the diagram, the voltmeter is in
parallel with the resistance, while the ammeter is in series.
Some conductors/devices have variable
resistances which depend on the currents flowing through them.
These are all non-ohmic conductors
except for the bottom right graph. Ohmic conductors are all metals and follow
Ohm's Law.
The current through a resistor is varied,
while the p.d. across it is measured. The graph of V against I is a straight
line through the origin.
Hence,
Ohm's law states:
The current through a conductor is directly
proportional to the p.d across it, provided physical conditions* are constant.
ELECTRICITY
Concepts & Units 2
Resistivity ρ (rho)
By
experiment it has been found that the resistance R of a material is directly
proportional to its length l and inversely proportional to its cross-sectional
area A.
Making
the proportionality an equation, the constant of proportionality ρ (rho) is
called the resisitivity.
Rearranging
the equation and making the length and area unity, we can form a definition for
the quantity.
The resistivity of a material is the resistance of a cube of
side 1 m, across opposite faces.
The
units of resistivity can be found by substituting the units for l , A and R in
the resistivity equation.
Resistivity
is measured in units of ohm-metres(Ω.m) .
Typical values of resistivity are:
Typical values of resistivity are:
metals
~10-8
semi-conductors ~ 0.5 glass/alumina ~ 1012 |
By
definition, the conductance of a material is the reciprocal of its resistance.
The unit of conductance is the Seimens S.
By
definition, the conductivity of a material is the reciprocal of its
resistivity. The units of conductivity are Sm-1 or Ω-1m-1 .
Current
density at a point along a conductor, is defined as the current per unit
cross-sectional area.
The
units of current density J are amps per m2 or Am-2 .
Metal
atoms in crystal lattices each tend to lose one of their outer electrons. These free
electrons wander randomly inside the crystal.
If a
p.d. is applied across the metal, electrons drift in the direction of the
positive contact. Since electrostatic field direction is from positive to
negative, the direction is against the field.
The
drift velocity, as it is called, is of the order of 10-3 ms-1 . When
the electrons move through the crystal lattice they collide with metal ions.
Each collision imparts kinetic energy from an electron to an ion. The ions thus
gain vibrational energy and as a consequence the temperature rises. This
phenomenon is called ohmic heating.
The
potential difference produces a steady current through the metal. On their
journey electrons gain and lose KE. The total amount of energy involved per m3 is a
measure of the resistivity of the particular metal.
Expressions
for the current I through a conductor and the current density J can be
formed using the concept of drift velocity v .
where:
I - current flowing
through conductor
L - length of conductor A - cross-sectional area n - no. free electrons/unit vol. e - charge on the electron v - average electron drift velocity |
From the
diagram the following can be implied:
volume
of the cylindrical section = LA
no.
free electrons in the section = nLA
quantity
of mobile charge Q in the section = nLAe
|
The time t for all
the electrons in the section to travel from one face to the another is the time
for one electron (on the far right) to travel the whole length.
since,
substituting
for Q and t,
since
current density J is defined as:
it
follows that,
Kirchhoff's Laws
1st Law - The sum of the
currents entering a node/junction equals the sum of the currents leaving.
I1 + I2 +
I3 = I4 + I5
I1 + I2 +
I3 - I4 - I5 = 0
This can
also be expressed as the algebraic sum: Σ I = 0
2nd Law - Around
any closed loop in a circuit, the algebraic sum of the individual p.d.'s is
zero.
This can
also be described as:
Around any closed loop in a circuit, the sum of the emf's equals
the sum of the p.d.'s across resistive elements.
V1 + V2 +
V3 - V4 - V5 = 0
The
convention is that clockwise p.d.'s are positive.
Simply,
emf's cause a rise in pd. Other circuit elements(eg resistors) cause falls in
pd. In a closed circuit, the sum of the rises in pd equals the sum of the
falls.
Consider
three resistors, R1 R2 R3 with the
same current flowing through each.
If the
p.d. across each one respectively is, V1 V2 V3 . Then
the total p.d. Vtotal across
the arrangement is:
By Ohm's
law, V=IR, therefore:
substituting
for Vtotal V1 V2 V3 into the
equation for p.d.,
Consider
three resistors R1 R2 R3 with the
same p.d. (V) across each of them.
Using
Kirchhoff's 2nd law, we can write:
By Ohm's
law, V=IR and I=V/R , therefore :
A
thermistor is a bipolar* semiconductor circuit element. It is in effect
a temperature dependent resistor.
*contacts can be connected + - or - +
*contacts can be connected + - or - +
The
arrangement below is called a potential divider. The p.d.'s VR and VT are in
the ratio of the resistors they appear across.
When the
thermistor is hot its resistance is low and of the order of 100's of ohms. In
this case, most of the 5V p.d. falls across the 10kΩ resistor.
As the
temperature decreases, the resistance of the thermistor increases.
When its
resistance reaches 10kΩ the p.d. is shared equallybetween it and the series
resistor.
At
really cold temperatures the resistance increases to the order of MΩ's, when
most of the p.d. falls across it and not the series resistor.
Like the
thermistor, the LDR is also a bipolar semiconductor circuit element. LDR's are
made from high resistance semiconductor material, whose resistance decreases
with increasing incident light intensity.
Typically
effect of light on a LDR is to reduce its resistance from ~ 106 Ω to ~
102 Ω.
The
arrangement below is called a potential divider. The
p.d.'s VR and VLDR are in
the ratio of the resistors they appear across.
In the
dark, the resistance of the LDR is of the order of MΩ's. So most of the 5V p.d.
falls across it and not the series resistor.
With
more illumination, the resistance of the LDR decreases. When it reaches 10kΩ
the p.d. is shared equally with the series resistor.
In
bright light, its resistance is of the order of 100's of Ω's. Then, most of the
p.d. falls across the series resistor.
An LED
is essentially a modified junction diode (or p-n
diode) so that it gives out light when current flows through it.
Junction
diodes are made from two types of semiconductor material, which have been
'doped' to alter their properties.
p-type: rich in charge carriers called 'holes' (missing electrons)
n-type: rich in free electrons
The two
types of semiconductor meet in the middle at what is called the p-n
junction. Here, with no p.d. applied, the holes from the p-type meet
up with the free electrons from the n-type and cancel each other out.
However,
when a p.d. is applied, with n-type '-' and p-type '+' (called forward
biased), a current flows. This current is made up of free electrons
moving across to the '+' terminal and holes moving towards the '-' terminal.
When the
polarity is reversed(reverse bias), with
n-type made '+' and p-type made '-' , no current flows. So we have a device
that only allows current to flow in one direction.
On a V-I
graph the top right quadrant shows how a very small forward p.d. causes the
diode to conduct. Notice the high current for a small p.d. increase.
The
bottom left quadrant shows what happens when the diode is reverse biased ('+'
contact connected to '-' supply and vice versa). Notice for increasing p.d.
there is a constant 'leakage current' . This
is very small, being of the order of micro-amps.
There
comes a point when the p.d. is so high that 'breakdown'
occurs. A large current passes and the diode is destroyed.
It is
essential in LED circuits that the exact p.d. falls across the device. If the
p.d. is too high the LED will allow too much current to flow through it. The
result will be overheating and failure.
To avoid
this, an LED always has a 'limiting resistor' placed in series to limit the
current. The level of current designed for is just enough to trigger light from
the device.
Example:
Find the limiting resistor for an LED, where:
i)
the max. LED current required is 100mA
ii) the forward LED voltage is 0.65V
iii)the supply p.d. is 5V
ii) the forward LED voltage is 0.65V
iii)the supply p.d. is 5V
If 0.65V
falls across the LED, then 4.35V must fall across the limiting resistor.
The
current through both the LED and the limiting resistor is 100mA.
Therefore
the limiting resistance R is given by R = V/I .
R = 4.35/0.1 = 43.5 Ω
The closest commercial resistor value is 47 Ω
ELECTRICITY
E.M.F. Internal Resistance
single
cell
|
The Single Cell
E.M.F.(E) is the p.d. across a cell when it delivers no current.
It can
also be thought of as the energy converted into electrical energy, when 1
Coulomb of charge passes through it.
The internal
resistance(r) of a cell is a very small resistance. For a 'lead-acid'
cell it is of the order of 0.01 Ω and for a 'dry' cell it is about 1 Ω.
This
means that a lead-acid cell will deliver a higher current than a dry cell.
We can
obtain important equations for E and r by considering a cell with a resistance
in a circuit.
The total
resistance Rtotal is the
sum of the series resistor and the internal resistance of the cell.
by
summing p.d. around the circuit ,
substituting
for Rtotal
by Ohm's
law, substituting IR = VR
Note, VR is
called the terminal p.d. . That is the p.d.
across the cell when it is delivering current.
Information
can be obtained from the graph by manipulating the equation obtained for E and
r:
transposing
the I and r, turning the equation around,
comparing
with the equation of a straight line,
Therefore
the gradient is '- r' and the intercept
on the vertical axis is 'E' .
but
and
where r is the internal
resistance of the combination
therefore,
So to
sum up, two cells in series are equivalent to one cell with an EMF equal to the
sum of the two cells.
The
internal resistance of the combination is the sum of the internal resistances
of the two cells.
The
arrangement dealt with here is only for cells that are similar. For disimilar cells
the relationship is complicated, but can be resolved using Kirchhoff's Laws.
For similar cells
the EMF's are equal and the internal resistances are also equal.
Therefore
the combined EMF, E is given by,
and the
internal resistance of the combination is calculated from the two internal
resistances in parallel:
ELECTRICITY
Power & Energy
Energy
To
understand how energy is converted consider a simple circuit with a 'load' *
resistor and a D.C. source.
* this could be a heater element, a motor or indeed any component with resistance
* this could be a heater element, a motor or indeed any component with resistance
Charge
loses potential energy(PE) as it moves through the resistor. This electrical PE
is transformed mostly into heat energy dissipated in the resistor.
The PE
is defined as*:
work = charge x potential
difference
W = QV
(Joules) = (Coulombs) x (Volts)
Since,
charge = (current) x
(time current flows)
Q = It
(Coulombs) = (Amps) x (seconds)
therefore,
substituting for Q in the work equation,
W = (It)V
rearranging,
W = VIt
By
definition, 'power' is the rate of working and is equal to the work done
divided by the time taken.
substituting
for W
cancelling
the 't'
(Watts) = (Volts) x
(Amps)
note: 1 Watt is a rate of
working of 1 Joule per second.
The
equation for power can be modified if we make substitutions using Ohm's
Law.
substituting
in the power equation for V ,
substituting
in the power equation for I,
A
kilowatt-hour is a unit of energy.
By
definition, a kilo-watt hour is the amount of energy consumed when a 'rate of
working' (power) of 1 kilowatt is used for 1 hour.
conversion
of 1 kWh to Joules:
1 kWh = 1 kW x 1 h = 1000 W x 3600 s = 3600000
J
1 kWh = 3.6 x 106 J
Direct
current does not vary with time and it is always in one direction.
On a
plot of power against time, D.C. is a horizontal line.
The area
under the plot gives the total work done/energy used.
This is
simply the product of the constant power( Pconst.)and the
time interval that the power is used for( t' ) .
However,
for A.C. the situation is more complex.
Here not
only does the current value vary, but its direction varies too.
The
power through the resistor is given by:
But we
must take the average of this power over time 't' to calculate the energy/work.
So the
energy/work done is given by,
The Root
Mean Square ( RMS) current is
defined as:
IRMS is the
square root of the average of the current squared.
Therefore
energy/work done is given by,
IRMS is the
equivalent D.C. current having the same effect on a resistor as the A.C.
Here is
a graph of an A.C. sinusoidal waveform:
where,
Io is the maximum current
|
ω is the angular
frequency, ω = 2πf ( π pi , f frequency)
|
Recalling
the A.C. energy/work done equation,
and
substituting for IRMS
Therefore
at time t' the energy W
dissipated in resistor R is given by:
note: to avoid confusion between W in
equations and W on the graph
P (W) on the graph means power P in watts.
|
W in equations is the
energy/work done
|
ELECTRICITY
Cathode Ray Oscilloscope
construction
|
Construction
The
cathode ray oscilloscope consists of three main elements:
electron gun
Electrons
are produced by thermionic emission.
Essentially
a cathode(negative electrode) is heated and electrons boil off the surface to
be attracted by a series of anodes (positive electrodes).
The
anodes accelerate the electrons and collimate them into a narrow beam.
deflection system
The
deflection system consists of two pairs of parallel plates called X-plates and Y-plates.
To
display a waveform, a repetitive reversing voltage is applied to the X-plates.
This
causes the electron beam to be slowly repelled from the left-hand plate and
attracted towards the right-hand plate.
On the
CRO screen this translates as an illuminated dot moving from left to right.
The
voltage is then reversed and increased rapidly. The effect is to move the dot
very quickly from right to left (fly-back).
The
applied voltage is called the time-base. The
curve has the general shape of a 'saw-tooth' and
is often referred to by this name.
The
p.d. applied to the Y-plates is the signal to be examined.
With
the p.d. across the X-plates (the time-base) switched off, a
sinusoidal signal makes the dot go up and down, executing simple harmonic
motion.
With
the time-base on, a sine wave is displayed.
display
The
display screen is coated on the inside with a very thin layer of a phosphor called cadmium
sulphide.
This
fluoresces (gives out green light) when electrons impact its surface.
A
layer of graphite is painted on the the inside of the vacuum
tube close to the fluorescent screen.
As a
result of impacting electrons, the screen acquires a negative charge. To reduce
this effect, the graphite is electrically connected to 'earth' (zero volts).
This allows excess charge to drain away.
Without
this feature, the accumulated charge would reduce the numbers of electrons
arriving at the screen, reducing brightness.
The
graphite layer also catches rebounding electrons that are back scattered from
impact with the screen.
brightness
The
brightness of a CRO display is a measure of the numbers of electrons impacting
the screen.
The
heating element heats the cathode in the 'electron gun' to produce electrons
forming a beam current.
This
beam current is controlled by having a grid over the cathode. The p.d. of the
grid can then be varied to accelerate electrons through it or repel some back
to the cathode.
focus
The
electron beam is focussed by passage through several annular anodes.
These
'collimate' the beam into a narrower, faster stream of
electrons, producing a smaller, sharper dot on the screen.
time-base
This
controls how fast the dot moves across the screen.
With
the time-base switched off, the dot appears static in the centre.
With
higher settings the dot appears as a horizontal line.
The control has units of 'time/cm' or 'time/division', with settings in the range 100 ms - 1 μs per cm/division.
sensitivity/gain
This
controls the vertical deflection of the dot. It has units of 'volts/cm' or 'volts/division'.
It is
important to know the names of vertical measurements on a waveform.
The peak
voltage (V0) is
the maximum vertical displacement measured from the time-line.
The peak-to-peak
voltage (VPP) is
the vertical displacement between the minimum and maximum values of voltage.
VPP = 2 V0
Frequency
is measured by finding the 'wavelength' of the waveform along the time-line.
In
other words to find the period (T) of
the wave (the time interval for one complete oscillation).
T = (horizontal
dist. on screen in cm) x (time-base setting)
Once T has
been measured, the frequency f can
then be found. The period T is
inversely proportional to frequency f :
Example:
In the example above, measuring from the first crest to the second
horizontally, the distance is 3.3 cm.
If the
time-base is 2ms/cm (2 x 10-3 sec./cm),
then the period (T) is:
T = 3.3 x 2 x 10-3 = 6.6
x 10-3 secs.
Hence
the frequency (f) is:
f = (6.6 x 10-3)-1 = 151.5
Hz (1 d.p.)
Another
way of measuring frequency is by Lissajous
figures.
These
images are obtained by switching off the time-base and inputting an AC signal
into it.
We
then have the case where we have two AC signals, one across the X-plates and
the other across the Y-plates.
Different
ratios of frequency give different images. In this way an unknown frequency can
be identified, provided it is in a simple ratio with the first frequency.
On a
double beam oscilloscope the phase difference in cm is be measured along the
time-line between similar points on each wave.
To
measure the phase difference in radians, the horizontal length of one wave (in
cm) must also be known (the period).
Then
the phase difference is given by:
Example:
The phase difference is 0.6 cm and the wavelength is 3.1 cm. therefore :
phase
difference (φ phi) =( 2 π x 0.6)/3.1 = 1.2
rads.
Besides
frequency, Lissajous figures can also be used to identify phase differences.
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