Electricity 2

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

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/

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

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


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:

% = =

%

( )


s we know.

Finding the Efficiency of a Motor for a Given Load


Turn the motor on, and measure the height - by which the load rises in a time and %./0 1230 . % efficiency =

4567489 100


, 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

;<

)



by which the load rises in a time =. If 100%

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.

? @ + )

?

@

)

@

)

?

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"

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 +

@ )

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

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.

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

@

)

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.

)

)

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.

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.

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

=

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

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.


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 L2, , should be equal to potdifference across the length of material, so , = .


,

A

B PPPP

QQQQ

, B

PPPP

QQQQ

L1L1L1L1

L2L2L2L2


he total resistance.

, will be smaller. In fact, if you

will be 0.

. We can "get" this by


should be equal to potential


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

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)

Electricity 2

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Transcript of Electricity 2