electricity 2
DESCRIPTION
Electricity 2TRANSCRIPT
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14141414.... Resistance and ResistivityResistance and ResistivityResistance and ResistivityResistance and Resistivity
ResistivityResistivityResistivityResistivity
Earlier, we defined resistance. We said that if a current passes through a component that has a potential difference across it, then the component's resistance () is:
=
We saw how resistance of some circuit components changed when their temperatures
changed. Now we look at how resistance can be dependent on the shape of a
conductor
Investigating how Resistance of a Conductor Varies with its Length
To do this investigation, you can use either one of the two circuits below. In each case,
you move a moveable contact along a uniform piece of conductor so that current will
have to flow through different lengths of the conductor each time (depending on
where you place the moveable contact on the conductor). Let's use to denote the length of conductor through which current flows. For each , you would then measure the resistance in the circuits.
On the circuit in the left, you measure the voltage and current and use these to calculate resistance using = . On the circuit on the right, you can directly use the ohmmeter to measure the resistance in the length of the wire.
Once you have a number of readings, plot a graph of resistance () vs. the length (). You will find that the graph is a straight line through the origin:
conductor
moveable contact
=
Resistance is directly proportional to length of
conductor.
where = some constant
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Investigating how Resistance of a Conductor Varies with the Conductor's Cross-
sectional Area
In this experiment, we use conductors of the same materialsame materialsame materialsame material and allow current to pass
through the same lengthsame lengthsame lengthsame length of the conductor each time. However, we use wires of
different thickness each time.
We measure the diameter of the wire, , using a micrometer screw gauge. Then the cross-sectional area of the wire is = =
=
.
You can use either one of the circuits below to take the measurements:
Using wires of different thicknesses, if you draw a graph of resistance () vs. 1 . You fill find that the graph is a straight line through the origin:
Now you have two relationships showing the relationships between a conductor's
shape and its resistance:
=
= ,
We can combine these two equations, letting become a new constant. Let this constant be !. Then we have:
= "#
" is called the resistivityis called the resistivityis called the resistivityis called the resistivity of the material, and it has a constant value for any material at a given temperature. Resistivity allows you to compaResistivity allows you to compaResistivity allows you to compaResistivity allows you to compare resistive properties of re resistive properties of re resistive properties of re resistive properties of
different materials directly, without having to take their shape into accountdifferent materials directly, without having to take their shape into accountdifferent materials directly, without having to take their shape into accountdifferent materials directly, without having to take their shape into account. (Like
the Young modulus for explaining elastic properties of a material). Resistivity is
measured in units ohm metres ( m).
conductor
1 =
Resistance is inversely proportional to the
cross-sectional area of conductor.
where = some constant
1/
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The higher the resistivity, the higher is the resistance.
A summary of factors that affect resistance in a conductor:
metals have low resistivities at room
temperature (of the order of 10-8 m)
insulators have high resistivities
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Power Dissipated across a ResistorPower Dissipated across a ResistorPower Dissipated across a ResistorPower Dissipated across a Resistor
Earlier, we dscussed the power dissipation across a conductor. We said that if the
currend through the component is
the power dissipated is given by:
Now we will express this in terms of resistance. All we need to do is use
substitute for or in the equation:
Substituting for potential difference:
% = =
% &
( )
We can use any of the three equations to find the power dissipated across a resistor,
depending on the values we know.
Finding the Efficiency of a Motor for a Given Load
You can use the setup below, along with a timer, to measure the efficiency of a motor.
Turn the motor on, and measure the height
mass of the load is *, %+,
Power Dissipated across a ResistorPower Dissipated across a ResistorPower Dissipated across a ResistorPower Dissipated across a Resistor
Earlier, we dscussed the power dissipation across a conductor. We said that if the
currend through the component is , and the potential difference across it is the power dissipated is given by:
( =
Now we will express this in terms of resistance. All we need to do is use
in the equation:
fference:
Substituting for current:
% = =
%
( )
We can use any of the three equations to find the power dissipated across a resistor,
s we know.
Finding the Efficiency of a Motor for a Given Load
You can use the setup below, along with a timer, to measure the efficiency of a motor.
Turn the motor on, and measure the height - by which the load rises in a time and %./0 1230 . % efficiency =
4567489 100
Earlier, we dscussed the power dissipation across a conductor. We said that if the
, and the potential difference across it is then
Now we will express this in terms of resistance. All we need to do is use and
;<
)
We can use any of the three equations to find the power dissipated across a resistor,
You can use the setup below, along with a timer, to measure the efficiency of a motor.
by which the load rises in a time =. If 100%
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15151515.... Electric CircuitsElectric CircuitsElectric CircuitsElectric Circuits
In a circuit, there is an emf device, such as a cell or a generator, that gives energy to
charges passing through it. As these charges flow across various components in a
circuit, they lose their energy.
As these charges flow around the circuit, they conserve both energy and chargeconserve both energy and chargeconserve both energy and chargeconserve both energy and charge.
Conservation of Charge: Kirchoff's First Law (Conservation of Charge: Kirchoff's First Law (Conservation of Charge: Kirchoff's First Law (Conservation of Charge: Kirchoff's First Law (Kirchoff's Junction Rule, Kirchoff's Junction Rule, Kirchoff's Junction Rule, Kirchoff's Junction Rule,
Kirchoff's Current Law, or "KCL")Kirchoff's Current Law, or "KCL")Kirchoff's Current Law, or "KCL")Kirchoff's Current Law, or "KCL")
As charges flow around a circuit, they are neither created, nor destroyed. So, if a
certain number of charges enter a junction in a circuit per second, it would be same
as the number of charges that leave that junction per second.
This is stated as Kirchoff's First LawKirchoff's First LawKirchoff's First LawKirchoff's First Law: The sum of currents entering a junction in a : The sum of currents entering a junction in a : The sum of currents entering a junction in a : The sum of currents entering a junction in a
circuit is equal to the sum of currents leaving that junctioncircuit is equal to the sum of currents leaving that junctioncircuit is equal to the sum of currents leaving that junctioncircuit is equal to the sum of currents leaving that junction.
? @ + )
?
@
)
@
)
?
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Conservation of Energy: Kirchoff's Second Law (Kirchoff's Loop Rule, Conservation of Energy: Kirchoff's Second Law (Kirchoff's Loop Rule, Conservation of Energy: Kirchoff's Second Law (Kirchoff's Loop Rule, Conservation of Energy: Kirchoff's Second Law (Kirchoff's Loop Rule,
Kirchoff's Voltage Law, or "KCL")Kirchoff's Voltage Law, or "KCL")Kirchoff's Voltage Law, or "KCL")Kirchoff's Voltage Law, or "KCL")
Energy is also conserved in a circuit. This means that the energy that the charges
lose as they travel through the circuit is equal to the energy that they gain in the emf
device. Remember, emf is the energy emf is the energy emf is the energy emf is the energy transferred transferred transferred transferred totototo a coulomb of charge as it goes a coulomb of charge as it goes a coulomb of charge as it goes a coulomb of charge as it goes
ththththrough a cell or a generatorrough a cell or a generatorrough a cell or a generatorrough a cell or a generator and potential energy is the energy potential energy is the energy potential energy is the energy potential energy is the energy transferred transferred transferred transferred bybybyby a a a a
coulomb of charge as it travels through the circuitcoulomb of charge as it travels through the circuitcoulomb of charge as it travels through the circuitcoulomb of charge as it travels through the circuit.
This can be expressed as Kirchoff's Second Law: Around a closed loop in a circuit, the Kirchoff's Second Law: Around a closed loop in a circuit, the Kirchoff's Second Law: Around a closed loop in a circuit, the Kirchoff's Second Law: Around a closed loop in a circuit, the
sum of emf's is equal to the sum ofsum of emf's is equal to the sum ofsum of emf's is equal to the sum ofsum of emf's is equal to the sum of potential differences.potential differences.potential differences.potential differences.
Look at the Circuit below:
If we go around the loop, the sum of emf's must be equal to the sum of potential
differences. So in this case,
? @ + ) If you know, for example, the voltage across the first lamp B= 4 V then the voltage
across the second lamp has to be 2 V.
?
@ )
"Loo"Loo"Loo"Loop"p"p"p"
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Resistors in SeriesResistors in SeriesResistors in SeriesResistors in Series
If current does not pass through a junction as it flows between two components, then
the two components are said to be connected in series. Components connectedComponents connectedComponents connectedComponents connected in in in in
series will have the same current flowing through themseries will have the same current flowing through themseries will have the same current flowing through themseries will have the same current flowing through them.
The bulbs in the circuit below are connected in series:
Let's take two resistors, B and , connected in series. Let's imagine replacing these two resistors with a single resistor, that acts just like the two resistors acted
together. Let this "replacement" resistor have an equivalent resistance, CD.
From Kirchoff's second law for the first circuit, we know that
B + B + The current that flows in the equivalent circuit is the same as in the first circuit.
For the equivalent circuit,
CD CD B +
cancels off, leaving: CD B +
@ )
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This works not only for two, but any number of resistors connected in series. So if F resistors B, , G, , I are connected in series, they can be replaced by an equivalent resistor having a resistance CD given by:
JK @ + ) + L ++ N
for resistors connected in seriesin seriesin seriesin series.
ResistorResistorResistorResistors in Parallels in Parallels in Parallels in Parallel
The bulbs shown below are connected in parallelin parallelin parallelin parallel:
Parallel connections involve junctions where current divides or gets collected up.
Look at loops 1 and 2. They are both closed loops. According to Kirchoff's second law,
the emf in each of these loops should be equal to the sum of potential differences
across each loop's bulbs. For each loop, the emf is the same (it's the emf provided by
the battery). This follows that the potential difference across each bulb is the same.
When componeWhen componeWhen componeWhen components are connected in parallel, they will have the same potential nts are connected in parallel, they will have the same potential nts are connected in parallel, they will have the same potential nts are connected in parallel, they will have the same potential
difference across themdifference across themdifference across themdifference across them.
This time, let's try to replace two parallel resistors with a single equivalent resistor.
junction junction
loop 2
loop 1
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Because B and are connected in parallel, at the junction O current branches out, so that @ + ). Because B and both make complete loops with the battery, the potential difference across each of them is . i.e. @ )
Equivalent resistor CD will have the total current flowing through it. For the circuit on the left, we can write:
B B
and
for the equivalent circuit,
CD
since B + , we can write: CD
B +
This time the cancels, leaving 1CD
1B +
1
Again, we can generalise this to F resistors, giving: @JK
@@ +
@) ++
@N
for resistors in parallelparallelparallelparallel.
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Simplifying Resistor CombinationsSimplifying Resistor CombinationsSimplifying Resistor CombinationsSimplifying Resistor Combinations
We can use these results to replace several resistors in a circuit with one. This makes
calculations involving resistors much easier. Before we get into doing this in full
swing, let's look at two special cases where there is a "shortcut" for calculating
equivalent resistance. (You don't have to remember these shortcuts, but let's go
through them anyway):
Two Resistors in Parallel
For equivalent resistor CD, 1CD
1B +
1
1CD B + B
CD B B +
For For For For twotwotwotwo resistors in parallel, the equivalent resistor has a resistance equal to the resistors in parallel, the equivalent resistor has a resistance equal to the resistors in parallel, the equivalent resistor has a resistance equal to the resistors in parallel, the equivalent resistor has a resistance equal to the
productproductproductproduct of the tof the tof the tof the two resistances divided by the wo resistances divided by the wo resistances divided by the wo resistances divided by the sum ofsum ofsum ofsum of the resistances.the resistances.the resistances.the resistances.
A Special Combination
@
)
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In this special combination, note that each resistor has the same resistanceeach resistor has the same resistanceeach resistor has the same resistanceeach resistor has the same resistance. Let's
now try to work out the equivalent resistance, CD. We can replace each pair of resistors (highlighted in the dashed box) in series by a
single resistance of resistance 2. Then our circuit would turn into two resistors in parallel, each with a resistance of 2:
Equivalent resistance of these two resistors (let's use the shortcut we learned before
this):
CD 2 22 + 2
CD 4
4
For this For this For this For this specificspecificspecificspecific combination, the equivalent resistance is thcombination, the equivalent resistance is thcombination, the equivalent resistance is thcombination, the equivalent resistance is theeee same as the resistance same as the resistance same as the resistance same as the resistance
of an individual resistor.of an individual resistor.of an individual resistor.of an individual resistor.
Two "Rules"
The following two rules will also help a great deal in combining resistances.
Whenever a number of resistors are connected in Whenever a number of resistors are connected in Whenever a number of resistors are connected in Whenever a number of resistors are connected in seriesseriesseriesseries, the , the , the , the equivalentequivalentequivalentequivalent resistance is resistance is resistance is resistance is
greater than the resistance of the largest resistor in the combinationgreater than the resistance of the largest resistor in the combinationgreater than the resistance of the largest resistor in the combinationgreater than the resistance of the largest resistor in the combination
Whenever a number of resistors are connected in Whenever a number of resistors are connected in Whenever a number of resistors are connected in Whenever a number of resistors are connected in parallelparallelparallelparallel, the , the , the , the equivalent resistance equivalent resistance equivalent resistance equivalent resistance
is smaller than is smaller than is smaller than is smaller than the the the the resistance of the smallest resistor in the combinationresistance of the smallest resistor in the combinationresistance of the smallest resistor in the combinationresistance of the smallest resistor in the combination
Looking at More Complex Resistor CombinationsLooking at More Complex Resistor CombinationsLooking at More Complex Resistor CombinationsLooking at More Complex Resistor Combinations
We already saw a glimpse of how you can combine resistors and, step-by-step, you
can use this to replace an entire set with a single resistor. The following pages
contain some more complex examples.
)
)
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Internal ResistanceInternal ResistanceInternal ResistanceInternal Resistance
In a battery or a generator, we know that there is an "emf source" that produces
energy. At the same time, some of this energy is lost inside the battery/generator,
before the current even leaves the battery/generator.
To account for the lost energy, we imagine there is a resistor inside the battery that
uses up. We say the battery has an internal resistance (internal resistance (internal resistance (internal resistance ()))).
So if you connect a component across a battery, it can only draw the "leftover" energy,
after internal resistance used up some of the energy from emf. The "leftover" energy
of course is the potential difference across the batttery. We write:
=
where = terminal potential difference, = electromotive force, = current through the battery or generator and = internal resistance.
Experiment: Finding Internal Resistance of a CellExperiment: Finding Internal Resistance of a CellExperiment: Finding Internal Resistance of a CellExperiment: Finding Internal Resistance of a Cell
The circuit shown in the diagram above can be used to determine the internal
resistance and the emf of the cell.
The rheostat is used to control the current in the circuit, which is measured by the ammeter. As the current is varied, the terminal potential difference of the battery changes. A graph of vs. is drawn.
You can model the internal resistance of a cell as a "resistor" in series with the cell.
gradient = -intercept:
= +
The equation given above can be rearranged as:
When a graph of vs. is drawn, you get a downward-sloping line. The (negative) gradient of
the graph gives the internal resistance of the cell and
the intercept on the vertical axis gives the emf.
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Resistors in Series: Resistors in Series: Resistors in Series: Resistors in Series: The Potential DividerThe Potential DividerThe Potential DividerThe Potential Divider
Imagine two resistors in series with each other.
Remember that according to Kirchoff's voltage rule, + = . If and are the same, then would be shared equally between the two resistors. The proportion of that is used up across a resistor is equal to the proportion of its resistance compared to the total resistance. This means, for example, that a resistor responsible
for 80% of the total resistance in a series combination is going to have a potential
difference across it that is 80% of the total potential difference across the cell. We can
write, in terms of ratios:
=
We can derive this result as well. The combined resistance of the resistors is + =
. The current drawn from the battery is then
. It is this same current that flows
through . The potential difference across is =
=
Which is often written as:
=
!
which expresses in terms of the resistor's resistance in comparison to the total resistance. This is sometimes called the "potential divider method""potential divider method""potential divider method""potential divider method" of finding the
potential difference across a resistor.
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We can control the potential difference across by controlling the resistance of (you're changing in the above equation and thereby changing ). As a numerical example, let's take the following situation:
Now suppose I replace with a 50 resistor. The 100 V will divide between the two resistors as follows:
I've changed the potential difference across by just changing the resistance of . In fact, I can just put a variable resistor instead of and I can change the potential difference across easily.
= 100 V
= 30
= 30 V
= = 70
= 70 V
=
= 100 V
= 30
= 37.5 V
= = 50
= 62.5 V
=
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You can put a thermistor in place of the variable resistor. Then, when the
temperature increases the resistance across the thermistor would drop, and the
potential difference across will increase. This sort of circuit is useful in thermostats.
Note that if we connect anything parallel to the 30 resistor, the potential difference
across that component would be the same as the potential difference across the 30
resistor. For instance, consider this next circuit:
When the surroundings get brighter, the resistance across the LDR reduces. This
means the potential difference across the LDR reduces as well. However, because the
potential differences across the LDR and the 30 resistor still need to add up to 100
V, the potential difference across the 30 resistor must increase. Since the bulb is in
parallel with the 30 resistor, potential difference across the bulb would increase as
well, causing it to glow brighter.
Now consider this next setup. You have a movable contact that can slide along the
length of a resistive material. We divide the material into two parts, A and B, based
on where the contact is. The potential difference across B is ,.
,
A
B
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Imagine moving the contact further down. You'll have the following situation:
Because the length of A is
So the potential difference across B, and hence
move the sliding contact all the way down to position
On the other hand, if you move the contact to
drawing loops around the circuit:
According to the loop L1, the potential difference across the entire length of the
resistor should be . According to the loop L2, difference across the length of material, so
This is the same principle we use when we use a rheostat to control the voltage across
a circuit.
Imagine moving the contact further down. You'll have the following situation:
Because the length of A is larger, that part contributes more to the total resistance.
So the potential difference across B, and hence ,, will be smaller. In fact, if you move the sliding contact all the way down to position PPPP, , will be 0.
On the other hand, if you move the contact to QQQQ, , = . We can "get" this by drawing loops around the circuit:
According to the loop L1, the potential difference across the entire length of the
. According to the loop L2, , should be equal to potdifference across the length of material, so , = .
This is the same principle we use when we use a rheostat to control the voltage across
,
A
B PPPP
QQQQ
, B
PPPP
QQQQ
L1L1L1L1
L2L2L2L2
Imagine moving the contact further down. You'll have the following situation:
he total resistance.
, will be smaller. In fact, if you
will be 0.
. We can "get" this by
According to the loop L1, the potential difference across the entire length of the
should be equal to potential
This is the same principle we use when we use a rheostat to control the voltage across
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Current enters B and leaves via A. It starts flowing through the external circuit (the
part with the bulb, ammeter and the voltmeter) starting from the point where the
slider touches the coil. If you move the slider all the way to B, the current enters the
external circuit immediately and it will have the full potential difference applied
across it. If the contact is placed at A, the potential difference around the external
circuit would be 0.
Resistors in Parallel: How the Current DividesResistors in Parallel: How the Current DividesResistors in Parallel: How the Current DividesResistors in Parallel: How the Current Divides and Shortand Shortand Shortand Short----circuitingcircuitingcircuitingcircuiting
Consider the following circuit.
Current divides at AAAA. Note that both resistors have the same potential difference
across them. Since = - , the resistor with the least resistance is going to get
larger current to flow through it.
If the resistance of one of the branch is almost zero, then almost all the current is
going to flow through that branch. This is used to "short" circuits-- i.e. to provide a
very low resistance path to for most current to flow. For example, in the circuit below,
almost no current is going to flow through the 30 resistor, instead it would flow
around the wire that is connected across the resistor.
Finding the "Brightest Bulb" in a CircuitFinding the "Brightest Bulb" in a CircuitFinding the "Brightest Bulb" in a CircuitFinding the "Brightest Bulb" in a Circuit
The brightest bulb is one that has the greatest power dissipation across it.
SeriesSeriesSeriesSeries: If two bulbs are connected in series, then the current across them is the same.
Since . = , the bulb with the largest resistance would be the brightest.
ParallelParallelParallelParallel: The potential difference across them is the same. Since . = , the bulb with the least resistance will be the brightest.
AAAA
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Connecting Ammeters and VoltmetersConnecting Ammeters and VoltmetersConnecting Ammeters and VoltmetersConnecting Ammeters and Voltmeters
AmmetersAmmetersAmmetersAmmeters that measure current must be always connected inmust be always connected inmust be always connected inmust be always connected in seriesseriesseriesseries with the
component. This ensures that the current through the ammeter is the same as the
current through the component.
Because ammeter is simply a measuring device, it shouldn't disturb the circuit and
the power dissipated across it should be minimal. Because ammeters are connected in
series, they need to have a they need to have a they need to have a they need to have a negligiblenegligiblenegligiblenegligible resistanceresistanceresistanceresistance compared to the components, so that
the power dissipated across them would be negligible (according to . = ).
VoltmetersVoltmetersVoltmetersVoltmeters, that measure potential difference, must be connected must be connected must be connected must be connected parallelparallelparallelparallel to the
component. This way, it has the same potential difference across it as the component.
Because they are connected in parallel, to draw very little current from the cell they they they they
need to have a very high resistanceneed to have a very high resistanceneed to have a very high resistanceneed to have a very high resistance.
In most cases, the circuit in (b) is preferable.
More Complicated Circuit ProblemsMore Complicated Circuit ProblemsMore Complicated Circuit ProblemsMore Complicated Circuit Problems
(beyond Edexcel level, but better to know)