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EXPERIMENT 1
Electromotive Force and Internal Resistance of a Battery
Please, read carefully this entire guide (i.e., both the theoretical principles and the experimental procedure), PRIOR to your scheduled Lab session. For your report (10 points), you only need to deliver the questionnaire (pages 9 and 10) plus the spreadsheet with the requested graphs and tables.
Objectives
- Measure in a direct current circuit and get acquainted with the basic functions of a multimeter: voltmeter, ammeter and ohmmeter.
- Determine the characteristics of a potential difference generator (battery) experimentally: ✓ The electromotive force, emf , 𝜀. ✓ The internal resistance, 𝑟. ✓ La maximum power, 𝑃𝑚𝑎𝑥, that a circuit can supply. ✓ The efficiency, 𝜂, of the power supply in a circuit.
1. Theoretical Principles
Batteries
A battery is a device that can maintain a constant (or periodical, for alternating currents) potential difference in a circuit. Since energy is partially dissipated in any circuit, the battery requires an external energy source. Many electric batteries consist of one or more electrochemical cells that allow the transformation of chemical energy into electrical energy.
The electromotive force of a battery, ε (or emf ) is defined as the work done by the external force field
on a positive charge dq to move it from the low-potential terminal through its interior to the highpotential terminal: 𝑑𝑊𝑒𝑥𝑡 𝜀 = 𝑑𝑞 It may also be defined as the factor of proportionality between the external power consumed by the battery and the intensity it generates:
The emf is expressed in SI (MKS) units in Volts (1V = 1J/C), which correspond to the units of electric potential. Ideally, a battery should supply a constant potential difference in the circuit that would correspond exactly to its emf. However, in practice, this potential difference is a little bit smaller than the emf value. Let’s consider the electric circuit shown in Figure 1 , with a battery of emf 𝜀 feeding a variable resistor 𝑅. Measurements of the difference in electric potential between the positive and negative terminals of the battery (𝑉 = 𝑉𝑃−𝑉𝑁) and of the current that circulates throughout the circuit (𝐼) for different values of the resistance 𝑅, would yield the graph displayed in Figure 1 : Figure 1 This suggests that the battery experiences an internal voltage drop, as if basically a small internal resistor (𝑟) would dissipate a small fraction of the energy consumed through Joule’s law. The energy balance would be given by: Power consumed = Power supplied by the battery + Power dissipated (Joule’s law) 𝜀𝐼 = (𝑉𝑃 − 𝑉𝑁)𝐼 + 𝑟𝐼^2 Hence, the potential difference across the terminals of the battery will be given by: 𝑉𝑃 − 𝑉𝑁 = 𝜀 − 𝑟𝐼 Where the intensity, 𝐼, circulating in the circuit depends on the external resistance, 𝑅, following: 𝜀 𝐼 = 𝑅 + 𝑟 The electric power supplied by the battery will then be:
When the external resistance 𝑅 is zero, the intensity will be maximum: 𝜀 𝑅 = 0 → 𝐼𝐶 = 𝑟 And this is called dead short circuit. This intensity is destructive for the battery, and will result in its complete discharge.
The Graph of Terminal Potential Difference vs. Current
The equation 𝑉 = 𝑉𝑃 − 𝑉𝑁 = 𝜀 − 𝑟𝐼 corresponds to a straight line, 𝑉 = 𝑓(𝐼) (see Figure 2 ), in which:
- The absolute value of the slope corresponds to the internal resistance of the battery, 𝑟.
- The intercept on the Y-axis yields the emf, 𝜀.
- The intercept on the X-axis yields the maximum current the battery can deliver when the potential difference is zero, known as the dead short circuit: 𝜀 𝐼𝐶 = 𝑟 Figure 2
2. Experimental Procedure
Material:
- Breadboard
- Four banana – pin tip cables
- Two banana – banana cables
- Two multimeters
- Ten resistors with nominal resistances: 4,7 Ω, 10 Ω, 18 Ω, 33 Ω, 47 Ω, 56 Ω, 82 Ω, 100 Ω, 150 Ω, 470 Ω • Rechargeable battery, ~ 9V. Warning: do not keep the battery connected a long time to avoid heating the resistors and discharging the battery itself!
Multimetre: voltmetre, ammetre i ohmmetre.
A multimeter incorporates, among other functions, acting as a voltmeter and ammeter (both for direct current and alternating current) and as a ohmmeter for the measurement of resistances. The fundamental rules for the correct use are:
- The measurement of a resistance (function ohmmeter) has to be always performed disconnecting the battery that feeds the circuit.
- It is necessary to choose the option DC of the multimeter for a direct current circuit or the option AC for an alternating current circuit.
- It is necessary to choose a convenient scale to obtain the maximum possible resolution. Using the function AUTO in the multimeter the right scale is automatically selected (advisable).
- Functioning as an ammeter, it has to be connected in serial with an element through which the intensity to be measured circulates.
- Functioning as a voltmeter, it has to be connected in parallel with the element across which the potential difference has to be determined. We have to remember that an ideal ammeter has a zero internal resistance while an ideal voltmeter has an infinite internal resistance.
Part A: Direct determination of ( ε , r )
Use the multimeter operating as a voltmeter connected to the terminals of the battery, to determine the emf as the voltage between terminals.
Part B: Graph of Terminal Potential Difference vs. Current
Experimental determination of the resistances
Use the multimeter, operating as an ohmmeter (with the cables connected to COM/Ω ) to determine the true value of the resistance of the resistors used in this activity. Press the Ω button to indicate that you are measuring resistances and the Auto button to have the multimeter choose the scale automatically. List the corresponding values in TABLE 2. Assemble the circuit in Figure 3 on the breadboard. The potential difference between terminals is given by: 𝑉 = 𝑉𝑃 − 𝑉𝑁 = 𝜀 − 𝑟𝐼 The above equation corresponds to the graph of terminal potential difference vs. current shown in Figure 2. Measure the values of 𝑉 and the current 𝐼 that flows through the circuit for each value of the resistance 𝑅. Figure 4 List the corresponding values in TABLE 2. Plot the function 𝑉 = 𝑓(𝐼) based on the data collected in TABLE 2 and perform a linear regression of the experimental data points (with the current 𝐼 in the X-axis and the voltage 𝑉 in the Y-axis). As shown in Figure 2 , the intercept on the Y-axis yields 𝑉 = 𝜀, while the intercept on the X-axis yields the maximum current the battery can deliver when the potential difference is zero (i.e., dead short circuit), 𝐼𝐶 = 𝜀⁄𝑟. Determine (𝜀, 𝑟) from the graph, and list the corresponding values in TABLE 3.
Power and efficiency of the battery
From the values of (𝜀, 𝑟) obtained from the graph of terminal potential difference vs. current, determine, for each resistor:
- The (external) power consumed by the battery: 𝑃𝑐𝑜𝑛𝑠𝑢𝑚 = 𝜀𝐼.
- The electric power supplied by the battery to the circuit: 𝑃𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 = (𝑉𝑃 − 𝑉𝑁)𝐼.
- The power dissipated by the internal resistance of the battery (Joule’s law): 𝑃𝐽𝑜𝑢𝑙𝑒 = 𝑟𝐼^2.
- The energy balance in the battery: Δ𝑃 = 𝑃𝑐𝑜𝑛𝑠𝑢𝑚 − 𝑃𝑠𝑢𝑏𝑚𝑖𝑛 − 𝑃𝐽𝑜𝑢𝑙𝑒.
- The efficiency of the battery: 𝜂𝑏𝑎𝑡𝑡 = 𝑃𝑠𝑢𝑏𝑚𝑖𝑛/𝑃𝑐𝑜𝑛𝑠𝑢𝑚 List the corresponding values on the yellow cells in TABLE 2. Plot the function 𝑃𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 = 𝑓(𝐼) , with the current 𝐼 in the X-axis and the electric power supplied by the battery 𝑃𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 in the Y-axis. Fit the experimental points to a second degree equation, indicating in the same chart the equation of the fitted curve.
Part B: Determination of ( ε, r ) from the graph of Terminal Potential Difference vs.
Current
List the values (𝜀, 𝑟) determined from linear regression of the experimental data, with the corresponding error and units. All values must be given with the appropriate number of significant figures. 𝜀 = 8,489 ± 0,000362991 (V) → 𝜀 = 8,489 0 ± 0,0004 (V) 𝑟 = 0,0526 ± 0, 033114664 (Ω) → 𝑟 = 0,05 ± 0,04 (Ω)
Power and efficiency of the battery
With the values of the three powers calculated in TABLE 2, and according to the principle of conservation of energy, we should obtain a balance Δ𝑃 ≈ 0. Is energy conserved? Comment on the results obtained. Considering the principle of conservation of energy, its balance should have a result of 0. When we consider the errors calculated during the experiment, the principle is confirmed and therefore, fully fulfilled. Discuss how the efficiency of the battery varies as the value of the external resistor increases. Provide an explanation for such a variation. The efficiency of the battery increases as the value of the external resistance increases. The efficiency is calculated using the following formula: 𝜂 =
As the exterior resistance enhances its value, smaller is the result and the intensity. Also, the value is multiplying a term with a negative sign, meaning that the smaller it is, the great the power supplied will be, and, as a result of this, the efficiency of the battery will be higher.
Determine from the chart 𝑃𝑠𝑢𝑏𝑚𝑖𝑛 = 𝑓(𝐼) , the maximum value of the power supplied by the battery, 𝑃𝑚𝑎𝑥 , and the corresponding intensity, 𝐼𝑃𝑚𝑎𝑥 ,. From this chart, we can determine that the maximum power supplied by the battery Pmax is 3 53,26 mW and the corresponding intensity IPmax is 82 ,73 mA. With the values of (𝜀,^ 𝑟)^ determined from the battery charging chart (TABLE 3), determine the theoretical values of 𝐼𝑃𝑚𝑎𝑥 and 𝑃𝑚𝑎𝑥.
= 0,0 74 (A) ≈ 74 ,20(mA)
= 0, 323 (W) ≈ 323 ,44(mW)
Do coincide the theoretical values with those determined in the chart? The value obtained is relatively similar, but not entirely the same because the result obtained by the graph doesn’t take into account the electromotive force. On the other hand, in the theoretical calculations we do consider it. Also, if we consider the errors, they do match. y = - 0,0496x^2 + 8,3787x + 3, R² = 0, 0, 50, 100, 150, 200, 250, 300, 350, 400, 0,00 20,00 40,00 60,00 80,00 100,00 120,00 140,00 160, P supplied [mW] I [mA]
P supplied = f(i)
