Determining the Best Requirements for a Bulb

Determining the Best Requirements for a Bulb

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Determining the Best Requirements for a Bulb

A torch is powered by four dry cells placed in series. There is a
choice of bulbs that can be used. The user wants a bulb that will
provide a maximum amount of power. The aim is, by performing
appropriate experiments, determine the best requirements for the bulb.

Introduction:

From previously performed experiments in lessons and personal research
I have deduced that in order to obtain the maximum amount of power in
a particular circuit the internal resistance (r) has to equal the load
resistance (R). This theory can be presented on the following graph of
load resistance (x-axis) against power (y-axis).

[IMAGE]

This graph, which can be obtained from most modern textbooks, proves
that the maximum power can be achieved when the load resistance is
equal to the internal resistance (r). I can therefore use this
knowledge to develop a method by which to conduct an appropriate
experiment that will allow me to obtain the best requirements for a
particular bulb.

In an electrical circuit, power is related to current and to potential
difference. The current (I) is the number of coulombs per second
flowing past a certain point in a circuit, and the potential
difference (V) across a load is the number of joules each coulomb
transfers to the load. Therefore the rate of energy transfer in the
load, the power (P), is the product of current and voltage. This can
be written in the form of the simple equation:

P = IV

By taking this simple equation into consideration it would be possible
to set up a simple circuit in which the current and voltage could be
measured across the bulb. By altering the resistance of the circuit I
would obtain several sets of measurements of voltage and current which
would allow me, by using the equation P = IV to find the power from
the voltage and current readings, to plot a graph similar to the one
above.

Theoretically, I would then be able to read off the peak of the graph

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to obtain the internal resistance and therefore discover the best
requirements for the bulb to achieve maximum power. However, this
method would not be entirely accurate due to the fact that the peak
lies on a curve and the mathematical process of differentiation would
be required to obtain a more precise reading.

There is however, a much more simple but accurate method of finding
the internal resistance. A simple circuit can be set up where the load
resistance can be altered giving a set of different readings of
voltage and current. After a sufficient amount of results (probably 7
sets repeated 3 times), have been recorded a graph of the average
results can be plotted of potential difference (Volts) against current
(Amps). The graph should appear similar to the one below.

[IMAGE]

By drawing a line of best fit it is then possible to tell what the
internal resistance will be as it will equal the gradient of the line.
This will give the best requirements for the bulb to work at maximum
power because I already know that maximum power occurs when the load
resistance (R) = internal resistance (r).

To prove that this information could be represented as a graph and
therefore show the internal resistance as the gradient, I took the
following equation and rearranged it so as each factor represented a
part of the (y = mx + c) equation for the straight line:

EMF = Ir + IR

EMF = Ir + V

V = -rI + EMF

y = mx + c

This proves that by using the resistance equation, applying it to the
whole circuit and then rearrange it to the form 'y = mx + c', we can
find the internal resistance as it is represented by the gradient of
the line graph.

I will record 7 different sets of readings of potential difference and
current by exchanging 7 resistors into the circuit, however, if as I
draw my graph during the experiment I find that a clear trend is not
shown I will have to take more readings by using more resistors. I
will be using a set of standard resistors in my experiment and for
each I will note the potential difference and the current. I will
repeat the experiment 3 times to get a suitable average and therefore
a more accurate result.

If I find that a clear trend is not presented by my graph, I may have
to make my own resistors by connecting a combination of two resistors
in series and parallel. If I obtained and plotted my results on a
graph, and they appeared to be correlated towards the top end of the
graph, I could make some lower resistors by adding two or more in
parallel. This would give me a lower resistance that would give lower
readings of voltage and current and allow me to plot points at the
lower end of my graph giving a more accurate and obvious pattern. For
example, if I have two 20 Ohm resistors and combined them in parallel
it would produce a resistance of 10 Ohm. I worked this out using the
following formula:

[IMAGE]1/R = 1/R1+1/R2

1/R = 1/ 20 + 1/20

1/R = 2/20

R = 20/2

R = 10 Ohms

Although this is the most appropriate experiment to perform in order
to fulfil my aim there are some variables which may lead to certain
inaccuracies. For example I will have to be aware of the sensitivity
of the equipment available. In order to minimise any human error in
the recording of readings from the voltmeter and ammeter, I will be
using equipment with digital displays rather than analogue equipment
that require a dial to be analysed in order to obtain the reading. In
my experiment I have chosen to use a multimeter to record the current
and the potential difference obtained in the experiment. I have chosen
this particular because it features the facility to measure current
and potential difference, and it allows the scale of the reading to be
increased or decreased accordingly in order to obtain a more accurate
result, for example it can measure current in both amps (A) and
milliamps (mA). I will be taking down all my recordings to 3
significant figures as I feel that this is an appropriate degree of
accuracy in order to plot a precise graph.

Method:

1. Collect apparatus: 4 Cells, Voltmeter, Ammeter, Bulb, Resistors,
Wires, Crocodile clips.

2. [IMAGE]Set up apparatus in the way the diagram below illustrates.

3. Once the circuit is set up, take the readings of the potential
difference and the current and make a note of them.

4. Replace the resistor, and take the readings of the potential
difference and the current and make a note of them.

5. Continue replacing the resistors until all have been used in the
circuit and readings have been taken from the voltmeter and ammeter
for each resistor.

6. Use results as points to plot on a graph that has the y-axis
representing potential difference (volts) and the x-axis representing
current (amps).

7. Draw a line of best fit for the points on the graph.

8. Find the gradient of the line by dividing the maximum vertical
value by the maximum horizontal value.

9. Use this value as the internal resistance and deduce what the best
requirements would be, in terms of resistance, for the bulb being
tested by considering that r = R.

Safety:

Although the voltage I am working with in this experiment is
relatively low and not likely to cause any harm, there are still a few
safety considerations that I will take into account in performing my
experiment:

§ Make sure that I am not in the way of others performing the
experiment.

§ Ensure my work area is clear of any items irrelevant to my
experiment.

§ Make sure that work surface is clean and dry.

§ Whilst changing the resistors in the circuit, ensure that the
circuit has been broken by removing an end of one of the wires from
the component it is attached to.

Results:

1st set of results

Potential Difference (Volts) +/- 0.005

Current (Amps) +/- 0.005

5.98

0.04

5.94

0.06

5.87

0.12

5.60

0.25

5.16

0.51

5.75

0.14

5.71

0.18

1st repeat

Potential Difference (Volts) +/- 0.005

Current (Amps) +/- 0.005

5.97

0.04

5.93

0.06

5.81

0.12

5.58

0.25

5.14

0.51

5.79

0.11

5.72

0.16

2nd repeat

Potential Difference (Volts) +/- 0.005

Current (Amps) +/- 0.005

5.93

0.04

5.90

0.06

5.78

0.13

5.55

1.25

5.11

0.51

5.77

0.10

5.70

0.16

On observation of these recordings it is clear that they are all very
similar sets of results, however, as I plotted a quick graph during
the experiment in order to see if there was a clear trend, I was aware
that the correlation was near the upper end and the lower end of the
graph appeared quite empty. An estimation as to what the lower end of
the graph would look like would be too inaccurate, so I decided to
make smaller resistors by combining them in parallel so as I could
plot lower points on the graph and could determine a clear trend in
the results.

I made two additional resistors by combing two 30 Ohm resistors in
parallel to create a resistance of 15 Ohms; and two 22 Ohm resistors
in parallel to create an 11 Ohm resistance. I added them o the circuit
individually and repeated three sets of results with each resistance,
as I did with the original resistances. The following tables show the
recordings I obtained:

1st set of results using additional custom resistors:

Potential Difference (Volts) +/- 0.005

Current (Amps) +/- 0.005

5.39

0.34

5.35

0.36

1st repeat of results using additional resistors:

Potential Difference (Volts) +/- 0.005

Current (Amps) +/- 0.005

5.41

0.36

5.36

0.38

2nd repeat of results using additional resistors:

Potential Difference (Volts) +/- 0.005

Current (Amps) +/- 0.005

5.38

0.34

5.36

0.38

The following table shows an average set of results of all my
recordings that are to be plotted onto a graph, they are in ascending
order of the resistance:

Potential Difference (Volts) +/- 0.005

Current (Amps) +/- 0.005

5.96

0.04

5.92

0.06

5.35

0.37

5.82

0.12

5.39

0.34

5.58

0.25

5.14

0.51

5.77

0.12

5.71

0.17

Evaluation:

On observation of my graph it is clear that there is an obvious
strong, negative correlation of the points plotted. This is as I
predicted as I said that the graph would show a negative correlation
of the points and as I worked out from rearranging the following
equation, the different factors of the rearranged resistance equation
correspond to the components which make up the equation y = mx + c.

EMF = Ir + IR

EMF = Ir + V

V = -rI + EMF

y = mx + c

This proves that by using the resistance equation, applying it to the
whole circuit and then rearrange it to the form 'y = mx + c', I am now
able to find the internal resistance as it is represented by the
gradient of the line graph.

According to my graph the gradient of the line is:

Vertical / Horizontal = 0.925/ 0.53 = 1.75 (to 3 s.f)

= 1.75 Ohms (to 3 s.f)

This means therefore that the most suitable bulb for this circuit
would be one with a resistance of 1.75 Ohms in order for this
particular circuit to work at maximum power. However this requirement
may be quite difficult to obtain as it is a measure to 3 significant
figures, a more realistic resistance would be perhaps 1.8 Ohms in
order for this particular circuit to work at maximum power. In order
to give an idea of the error margins of the results, I added two more
possible lines of best fit and calculated the gradients of the lines,
that gives an idea of the error in the resistance values. I labelled
the lines of best it 1, 2 and 3. Line 1 is the true line of best fit
and is marked on the graph in red. Lines 2 and 3 are other possible
lines that give an idea of the error margins. The gradient for line 2
is 1.75, and line 3 is 1.88 both measurements being to 3 significant
figures, therefore the error margins are quite small suggesting a
clear trend and accurate measurements.

Overall I think my experiment was a success, my graph showed a clear
trend and I was able to plot an accurate line of best fit. However
there were some factors in my experiment that may have led to certain
inaccuracies. For example, the equipment I was using was quite
accurate but in order to be more precise I could have used a more
sensitive set of meters in order to obtain a more accurate result. I
was using meters that measured amps and volts to the nearest
hundredth, however, I could have used a more precise scale for example
one that measured to 3 or 4 decimal places instead of 2 decimal
places.

Obviously more sets of results would also give a more accurate result
as I would have a more precise average and could plot more points on
the graph and therefore obtain a more suitable line of best fit. For
example a greater range of resistors to exchange into the circuit
during the experiment would have given a wider range of results to
plot on the graph. Similarly more repeats would have given a more
precise average which would allow me to plot a more accurate line of
best fit thus producing a gradient closer to the actual value trying
to be obtained.

Also, concerning accuracy, I did come across some anomalous results in
my recordings. I have marked them on the graph by circling them in red
pen. After tracing back the points from my graph to the result table I
realised that two of the three anomalies were my additional resistors,
which I customised to get lower resistances, that were anomalous. From
this, due to the fact that the additional resistors were a combination
of two resistors added in parallel, I can deduce that the wires that
were connecting the two resistors in parallel may have had a
resistance themselves which could have caused inaccuracy as this would
of added to the overall resistance. However, there are other factors
which could have caused these anomalies, for example if the resistors
heated up whilst the recordings were being taken the results would be
inaccurate. Poor averaging could have been another reason or perhaps
just human error, however the fact that the customised resistors in
parallel were the anomalies could likely rules out these
possibilities.

A moderation to check that the final value obtained is rational would
be to take the values of current and potential difference and multiply
them bearing in mind the equation: P = IV. This would obviously give
me the power values that would then allow me plot a separate graph in
which the peak value is the internal resistance. As I mentioned, I am
aware from previously performed experiments that if a circuit is to
work at maximum power, the internal resistance (r) has to equal the
load resistance (R). I could then use the graph, which would appear
similar to the one below, to read off the peak value and get an
estimate as to whether or not my value obtained by my experiment is
rational as they should be approximately similar values.

[IMAGE]

By plotting this graph it would provide me with a check to see if my
value is accurate as the peak value of this graph should be
approximately equal to the value of the resistance obtained in the
actual experiment. This is moderation that I would consider if I were
to perform the experiment again.

In order to do this graph accurately however, I would need to use the
multimeter to obtain the actual resistance of the resistors as opposed
to using the resistances given. This would make the graph more
accurate and therefore give a more precise idea of the internal
resistance.

In conclusion, my experiment has shown the most suitable requirements
for a bulb which is to be used in a torch with four dry cells would be
one with an approximate resistance of 1.8 Ohms with a maximum of 6
Volts through the circuit.
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