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```EXERCISE 6
Electricity – II
Objectives: In the present laboratory exercise students will measure the equivalent resistance of two and three
resistors connected in series and in parallel and they will compare the results with those predicted from the
theoretical equations. The students will determine the unknown resistance of a resistor using the ammeter and
voltmeter and they will use to more complicated circuits.
Theoretical background
Ohm’s law
As a reminder from the previous lab, the Ohm’s law states that the current 𝐼 flows through a conductor, like a wire,
is the ratio of the voltage 𝑉 applied across the conductor over its resistance.
𝐼=
𝑉
𝑅
The current is measured in Amperés (𝐴), the voltage in Volts (𝑉) and the resistance in Ohms (𝛺). The Ohm’s law
can also be applied for the whole circuit. In this case, the 𝑉 in equation (1) represents the voltage of the battery and
𝐼 the current flows through the circuit. Then, the 𝑅 represents the equivalent resistance 𝑅𝑒𝑞 of the circuit.
𝑅𝑒𝑞 =
𝑉
𝐼
Equivalent resistance of resistors connected in series
Let us assume three resistors connected in series as shown in figure 1. We are going to calculate the equivalent
resistance of such a circuit.
Figure 1: Three resistors connected in series. A voltmeter is connected parallel to the battery to measure the voltage
and an ammeter is connected in series to the resistors to measure the current through the circuit.
1
With the term equivalent resistance, 𝑅𝑒𝑞 , we mean a resistor of appropriate value of resistance which replaces the
existing three resistors. As shown in figure 1, when the switch is closed the battery provides to the circuit a current
𝐼. This current 𝐼 flows through all the resistors and the voltage 𝑉 of the battery is applied to the circuit. The voltage
of the battery 𝑉 equals to the sum of the voltages across each individual resistor, 𝑉𝑅1 , 𝑉𝑅2 , and 𝑉𝑅3 . We get,
𝑉 = 𝑉𝑅1 + 𝑉𝑅2 + 𝑉𝑅3
(1)
We use the Ohm’s law for the circuit, 𝑉 = 𝑅𝑒𝑞 𝐼, and for each resistor, 𝑉𝑅𝑖 = 𝑅𝑖 𝐼, where 𝑖 = 1, 2, 3. Equation (1) is
now written as
𝑅𝑒𝑞 𝐼 = 𝑅1 𝐼 + 𝑅2 𝐼 + 𝑅3 𝐼 ⟺ 𝑅𝑒𝑞 𝐼 = (𝑅1 + 𝑅2 + 𝑅3 )𝐼 ⟺ 𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3
(2)
The result in equation (2) can be generalized for more resistors connected in series. If in an electric circuit there are
𝑁 resistors connected in series the equivalent resistance is given by the formula
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3 + ⋯ + 𝑅𝑁
(3)
Therefore, the equivalent resistance of resistors connected in series equals to their sum.
Equivalent resistance of resistors connected in parallel
We connect two resistors in parallel and parallel to the battery, too, as shown in figure 2.
Figure 2: Two resistors are connected parallel to the battery. The voltage across each resistor is identical to the battery’s
voltage. The current 𝐼 the battery provides to the circuit splits to two currents on the node A.
Note, there is no current through the voltmeter because of its very high resistance.
2
When the switch S is closed, the battery provides to the circuit a current 𝐼 which is measured by the ammeter. When
the current 𝐼 reaches the point A, where three wires coincide 1, it splits into two currents 𝐼1 and 𝐼2 . The current 𝐼1
flows through the resistor 𝑅1 and the current 𝐼2 flows through the resistor 𝑅2 . Due to the charge conservation
principle, it holds,
𝐼 = 𝐼1 + 𝐼2
(4)
We apply now the Ohm’s law for the whole circuit in figure 2 and we get 𝐼 = 𝑉 ⁄𝑅𝑒𝑞 , and for the two resistors and
we get 𝐼1 = 𝑉𝑅1 ⁄𝑅1 and 𝐼2 = 𝑉𝑅2 ⁄𝑅2 respectively.
𝑉𝑅
𝑉𝑅
𝑉
= 1+ 2
𝑅𝑒𝑞
𝑅1
𝑅2
(5)
Because the wires have no resistance 2, from figure 2 we get 𝑉𝑅1 = 𝑉𝑅2 = 𝑉. Equation (4) is now written as
𝑉
𝑉
𝑉
1
1
1
=
+
⟺
=
+
𝑅𝑒𝑞 𝑅1 𝑅2
𝑅𝑒𝑞 𝑅1 𝑅2
(6)
If in an electric circuit there are 𝑁 resistors connected in parallel, the equivalent resistance is calculated by the
formula
1
1
1
1
1
=
+
+
+ ⋯+
𝑅𝑒𝑞 𝑅1 𝑅2 𝑅3
𝑅𝑁
(7)
Experimental – Data analysis
Measurement of an unknown resistance of a resistor
Equipment needed
•
Three battery sockets
•
One switch
•
The resistor module
•
One voltmeter
•
One ammeter
•
Some wires
Procedures
Hook up the electric circuit shown in figure 3, according to the following steps:
1
The point A where three wires coincide is called node of the electric circuit. In general, any point in an electric circuit
where three or more wires coincide is called electric node.
2
The resistance of the wires is very low and is considered zero.
3
Connect the positive terminal of the battery to one terminal of the switch. The switch is open.
Connect the other terminal of the switch to the red terminal of the ammeter
Connect the other terminal of the ammeter to one terminal of the unknown resistance
Connect the other terminal of the resistor to the negative terminal of the battery.
Connect also the negative terminal of the battery to the black terminal of the voltmeter
Connect the positive terminal of the battery to the red terminal of the voltmeter, as shown in figure 3
Close the switch and register the readings from the voltmeter and ammeter on the worksheet.
Determine the resistance of the resistor using the Ohm’s law and register the result in the appropriate cell of the
worksheet.
Figure 3: Measurement of the unknown resistance of a resistor using an ammeter and a voltmeter
Measurement of the equivalent resistance of three resistors connected in series
Equipment needed
•
Three battery sockets
•
One switch
•
The resistor module
•
One voltmeter
•
One ammeter
•
Some wires
Procedures
Hook up the electric circuit shown in figure 4, according to the following steps:
•
Connect the positive terminal of the battery to one terminal of the switch. The switch is open.
•
Connect the other terminal of the switch to the red terminal of the ammeter
•
Connect the black terminal of the ammeter to the red terminal of the 15 𝛺 resistor
4
•
Connect the black terminal of the 15 𝛺 resistor to the black terminal of the 10 𝛺 resistor
•
Connect also the red terminal of the 10 𝛺 resistor to the red terminal of the 5 𝛺 resistor
•
Connect the black terminal of the 5 𝛺 resistor to the negative terminal of the battery, as shown in figure 4
•
Connect the black terminal of the voltmeter to the negative terminal of the battery
•
Connect the red terminal of the voltmeter to the positive terminal of the battery
•
Close the switch and register the readings from the voltmeter and ammeter on the worksheet.
•
Determine the equivalent resistance of three resistors connected in series using the Ohm’s law and register the
result in the appropriate cell of the worksheet.
•
Calculate the equivalent resistance according to the values given on the resistor module. Use formula (3).
•
Compare the experimental equivalent resistance with that given by equation (3).
•
Calculate the quantity |𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 − 𝑅𝑒𝑞,𝑚𝑒𝑎𝑠 |⁄𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 ∙ 100%
Figure 4: Measurement of the equivalent resistance of three resistors connected in series
Measurement of the equivalent resistance of two resistors connected in parallel
Equipment needed
•
Three battery sockets
•
One switch
•
The resistor module
•
One voltmeter
•
One ammeter
•
Some wires
5
Procedures
Hook up the electric circuit shown in figure 5, according to the following steps:
•
Connect the positive terminal of the battery to one terminal of the switch. The switch is open.
•
Connect the other terminal of the switch to the red terminal of the ammeter
•
Connect the black terminal of the ammeter to the red terminal of the 15 𝛺 resistor
•
Connect the red terminal of the 15 𝛺 resistor to the red terminal of the 10 𝛺 resistor
•
Connect also the red terminal of the 10 𝛺 resistor to the red terminal of the 5 𝛺 resistor
•
Connect the black terminal of the 15 𝛺 resistor to the black terminal of the 10 𝛺 resistor
•
Connect the black terminal of the 10 𝛺 resistor to the black terminal of the 5 𝛺 resistor, as shown in figure 5
•
Connect the black terminal of the 5 𝛺 resistor to the negative terminal of the battery
•
Connect the black terminal of the voltmeter to the negative terminal of the battery
•
Connect the red terminal of the voltmeter to the positive terminal of the battery
•
Close the switch and register the readings from the voltmeter and ammeter on the worksheet.
•
Determine the equivalent resistance of three resistors connected in parallel using the Ohm’s law and register
the result in the appropriate cell of the worksheet.
•
Calculate the equivalent resistance according to the values given on the resistor module. Use formula (7).
•
Compare the experimental equivalent resistance with that given by equation (7).
•
Calculate the quantity |𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 − 𝑅𝑒𝑞,𝑚𝑒𝑎𝑠 |⁄𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 ∙ 100%
Figure 5: Measurement of the equivalent resistance of three resistors connected in parallel
6
Measurement of the equivalent resistance of a mixed combination of resistors
Equipment needed
•
Three battery sockets
•
One switch
•
The resistor module
•
One voltmeter
•
One ammeter
•
Some wires
Procedures
Hook up the electric circuit shown in figure 6, according to the following steps:
•
Connect the positive terminal of the battery to red terminal of the resistor 𝑅𝑥 .
•
Connect the black terminal of the resistor 𝑅𝑥 to the black terminal of the switch
•
Connect the red terminal of the switch to the red terminal of the ammeter
•
Connect the black terminal of the ammeter to the red terminal of the 10 𝛺 resistor
•
Connect the red terminal of the 10 𝛺 resistor to the red terminal of the 15 𝛺 resistor
•
Connect also the black terminal of the 10 𝛺 resistor to the black terminal of the 15 𝛺 resistor
•
Connect the black terminal of the 10 𝛺 resistor to the negative terminal of the battery
•
Connect the black terminal of the battery to the black terminal of the voltmeter
•
Connect the positive terminal of the battery to the red terminal of the voltmeter
•
Close the switch and register the readings from the voltmeter and ammeter on the worksheet.
Figure 6: Measurement of the equivalent resistance of three resistors connected in a mixed way
7
•
Determine the equivalent resistance of the three resistors in the electric circuit using the Ohm’s law and register
the result in the appropriate cell of the worksheet.
•
Calculate the equivalent resistance according to the values given on the resistor module. Use formulae (3) and
(7) appropriately.
•
Compare the experimental equivalent resistance with that given by the combination of equations (3) and (7).
•
Calculate the quantity |𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 − 𝑅𝑒𝑞,𝑚𝑒𝑎𝑠 |⁄𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 ∙ 100%
In the last section of your report, you describe briefly the task(s) of this laboratory session. Then you have to write
comments on what you measured and how the theory is compared to the data. As an option, you may also write
some thoughts for the improvement of the accuracy of the measurements.
8
Electricity II
Student’s worksheet – Lab 6
Student’s name: ____________________________________________ Grade: _________ Date: __________
Lab partner(s): ____________________________________________________________________________
Experiment 1: Measurement of an unknown resistance
Battery’s voltage
Current through the resistor
Resistance
𝑉 (𝑉)
𝐼 (𝐴)
𝑅 (𝛺)
Experiment 2: Measurement of the equivalent resistance of three resistors connected in series
Battery’s voltage
Current through
Equivalent
Equivalent resistance
% Percentage
𝑉 (𝑉)
the circuit
resistance
𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 (𝛺)
discrepancy
𝐼 (𝐴)
𝑅𝑒𝑞,𝑚𝑒𝑎𝑠 (𝛺)
Experiment 3: Measurement of the equivalent resistance of three resistors connected in parallel
Battery’s voltage
Current through
Equivalent
Equivalent resistance
% Percentage
𝑉 (𝑉)
the circuit
resistance
𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 (𝛺)
discrepancy
𝐼 (𝐴)
𝑅𝑒𝑞,𝑚𝑒𝑎𝑠 (𝛺)
Experiment 4: Measurement of the equivalent resistance of three resistors in an electric circuit
Battery’s voltage
Current through
Equivalent
Equivalent resistance
% Percentage
𝑉 (𝑉)
the circuit
resistance
𝑅𝑒𝑞,𝑐𝑎𝑙𝑐 (𝛺)
discrepancy
𝐼 (𝐴)
𝑅𝑒𝑞,𝑚𝑒𝑎𝑠 (𝛺)
9
```