Materials Included In Kit

Additional Materials Required

Prelab Preparation

1. Prepare a bag of electrical components for each student group by placing one of each resistor and two capacitors into a bag. Each bag should contain four resistors—220 Ω, 620 Ω, 1.1 kΩ and 8.2 kΩ—and two 1000 μF capacitors.
2. Photocopy enough of the Supplementary Material information for each group.

Safety Precautions

If a resistor begins to darken or smoke, immediately disconnect the circuit and do not touch the resistor—it will be hot. Allow it ample time to cool. Handle pins with caution, as they are sharp. Please follow all laboratory safety guidelines.

Disposal

All materials may be saved and stored for future use.

Lab Hints

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day, and analysis of the results may be completed the day after the lab or as homework. An additional lab period would be needed for students to complete an optional inquiry investigation (see Opportunities for Inquiry). 
• Enough materials are provided in this kit for 24 students working in groups of three or 8 groups of students with extra capacitors and resitors. This investigation can be reasonably completed in two 50-minute class periods.
• Students may require a review of how to use a multimeter. Consider drawing the following figures on the board to serve as models of how the probes of a multimeter are connected to measure potential difference and current, respectively.
• In the Opportunities for Inquiry portion of the lab, students may make a circuit involving resistors and capacitors in various combinations of series and parallel. Check the circuit designs and assembled circuits before students connect them to a battery. Students should measure the change in current over time for their circuit to assess how the time constant is affected.
• When gathering data it is important for the circuit to be in a steady state, meaning that the capacitor is either fully charged or fully discharged. The amount of time for steady-state to be reached is generally five times the RC value of the circuit.

Teacher Tips

• Encourage students to determine how the effective capacitance can be related to the formula for capacitance. Capacitance is dependent on the dielectric material used, the surface area of the plates and the distance separating the plates. Assuming the dielectric material is unchanged, then capacitance has a direction relation to the surface area of the plates and indirect relation to the separation distance. When placing capacitors in series, the effective capacitance decreases because the separation of the positive and negative plates increases. When placing capacitors in parallel, the effective capacitance increases because the surface area of the positive and negative plates increases.
• Knowledge of how to derive formulas to track the change in current and charge in an RC circuit are beyond the scope of the course. However, knowledge of how a capacitor affects the current in a circuit is integral to understanding their importance in RC circuits.

Further Extensions

Opportunities for Inquiry
Design an experiment to investigate initial and steady-state current in circuits of varying combinations of resistors and capacitors.

Alignment to the Curriculum Framework for AP^® Physics 2 

Enduring Understandings and Essential Knowledge
The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems. (4E)
4E5: The values of currents and electric potential differences in an electric circuit are determined by the properties and arrangements of the individual circuit elements such as sources of emf, resistors, and capacitors.

The energy of a system is conserved. (5B)
5B9: Kirchhoff’s loop rule describes conservation of energy in electrical circuits. (The application of Kirchhoff’s laws to circuits is introduced in Physics 1 and further developed in Physics 2 in the context of more complex circuits, including those with capacitors.)

1. Energy changes in simple electrical circuits are conveniently represented in terms of energy change per charge moving through a battery and resistor.
2. Since electric potential difference times charge is energy, and energy is conserved, the sum of the potential differences about any closed loop must add to zero.
3. The electric potential difference across a resistor is given by the product of the current and the resistance.
4. The rate at which energy is transferred from a resistor is equal to the product of the electric potential difference across the resistor and the current through the resistor.
5. Energy conservation can be applied o combinations of resistors and capacitors in series and parallel circuits.

The electric charge of a system is conserved. (5C)
5C3: Kirchhoff’s junction rule describes the conservation of electric charge in electrical circuits. Since charge is conserved, current must be conserved at each junction in the circuit. Examples should include circuits that combine resistors in series and parallel. (Physics 1: covers circuits with resistors in series, with at most one parallel branch, one battery only. Physics 2: includes capacitors in steady-state situations. For circuits with capacitors, situations should be limited to open circuit, just after circuit is closed, and a long time after the circuit is closed.)

Learning Objectives
4E5.1: The student is able to make and justify a quantitative prediction of the effect of a change in values or arrangements of one or two circuit elements on the currents and potential differences in a circuit containing a small number of sources of emf, resistors, capacitors, and switches in series and/or parallel.
4E5.2: The student is able to make and justify a qualitative prediction of the effect of a change in values or arrangements of one or two circuit elements on currents and potential differences in a circuit containing a small number of sources of emf, resistors, capacitors, and switches in series and/or parallel.
4E5.3: The student is able to plan data collection strategies and perform data analysis to examine the values of currents and potential differences in an electric circuit that is modified by changing or rearranging circuit elements, including sources of emf, resistors, and capacitors.
5B9.5: The student is able to use conservation of energy principles (Kirchhoff ’s loop rule) to describe and make predictions regarding electrical potential difference, charge, and current in steady-state circuits composed of various combinations of resistors and capacitors.
5B9.6: The student is able to mathematically express the changes in electric potential energy of a loop in a multiloop electrical circuit and justify this expression using the principle of the conservation of energy.
5C3.6: The student is able to determine missing values and direction of electric current in branches of a circuit with both resistors and capacitors from values and directions of current in other branches of the circuit through appropriate selection of nodes and application of the junction rule.
5C3.7: The student is able to determine missing values, direction of electric current, charge of capacitors at steady state, and potential differences within a circuit with resistors and capacitors from values and directions of current in other branches of the circuit.

Science Practices
1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively.
2.1 The student can justify the selection of a mathematical routine to solve problems.
2.2 The student can apply mathematical routines to quantities that describe natural phenomena.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
4.3 The student can collect data to answer a particular scientific question.
5.1 the student can analyze data to identify patterns or relationships.
6.1 The student can justify claims with evidence.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.

Answers to Prelab Questions

1. Examine Figure 1 of a battery connected to two resistors in series.
   1. Explain how Kirchoff’s loop rule can be applied to solve for the potential difference (voltage drop) across resistor R₁.

      Kirchoff ’s loop rule states that the overall change in electric potential in a circuit must equate to zero. The voltage of the battery summed with the voltage drop across the effective resistance of the circuit is equal to zero. Since the resistors are in series, the effective resistance is the sum of the resistances. Therefore, the voltage of the battery summed with the voltage drops across resistor R₁ and R₂ is equal to zero. If we know the voltage drop across resistor R₂, then the voltage drop across R₁ is equal to the voltage of the battery minus the drop across R₂ or “the drop across R₂ subtracted from the voltage of the battery.”

   2. Explain how Kirchoff’s junction rule can be applied to solve for the current through the resistors, R₁ and R₂.

      Kirchoff ’s junction rule states that the amount of current flowing into a node must equal the amount of current leaving that same node. Since there are no nodes in the circuit, then the current is constant throughout and the current flowing through resistors R₁ and R₂ is equal to the voltage of the battery divided by the effective resistance of the circuit.

2. Examine Figure 1 of a battery connected to two resistors in parallel.
   1. Explain how Kirchoff’s loop rule can be applied to solve for the potential difference (voltage drop) across resistor R₁.

      Kirchoff ’s loop rule states that the overall change in electric potential in a circuit must equate to zero. The voltage of the battery summed with the voltage drop across the effective resistance of the circuit is equal to zero. Since the resistors are in parallel, the drop in voltage across them is equal. Since the parallel branch flows directly into the negative terminal of the battery we know that the voltage drop across both resistors must equal the voltage of the battery.

   2. Explain how Kirchoff’s junction rule can be applied to solve for the current through the resistors, R₁ and R₂.

      Kirchoff ’s junction rule states that the amount of current flowing into a node must equal the amount of current leaving that same node. Since the voltage drops are the same across both resistors and are equal to the voltage of the battery, one can simply divide the voltage of the battery by the respective resistance of each resistor to solve for the respective current through each resistor.

Sample Data

Introductory Activity

Resistors in Series

Resistors in Parallel Analyze the Results
In both the series and parallel arrangements, the measured resistance values for the resistors fall within the 5% tolerance set by the manufacturer (R₁ = 220 ± 11, R₂ = 620 ± 31). The effective resistance values for the series and parallel arrangements agree with Equations 2 and 3, respectively. The effective resistance value for resistors in series is the sum of the component resistance values and has greater resistance than either resistor in the arrangement. The effective resistance value for resistors in parallel is equal to the sum of the inverse of the resistance values and has less resistance than either resistor in the arrangement.

Guided-Inquiry Activity

Part B. Capacitors in Series
Analyze the Results

• Present your data in a table that shows the capacitance of each individual capacitor, the effective capacitance of the circuit, and the voltage readings for the resistors and capacitors used.

  For a simple circuit containing a single resistor and capacitor:

  Resistance of resistor used, R = 8.2 kΩ
  Capacitance of capacitor used, C₁ = 990 μF

  {14014_Data_Table_3}

  For a circuit containing a single resistor and two capacitors in series:

  Resistance used = 8.2 kΩ
  Capacitance of first capacitor, C₁ = 990 μF
  Capacitance of second capacitor, C₂ = 999μF
  Measured effective capacitance C_eff = 494 μF

  {14014_Data_Table_4}

  Charge on capacitors used:

  {14014_Data_Table_5}
• Summarize your findings in several sentences while discussing charge, voltage, resistance and effective capacitance versus individual capacitance of the resistors.

  The sum of the potential differences across C₁ and C₂ is nearly equal to the potential difference across both in series. The sum of the potential difference for the resistor and two capacitors used never surpassed the potential difference provided by the power supply, which aligns with Kirchoff ’s loop rule. The charge on the separate capacitors, C₁ and C₂ and across the effective capacitor is nearly identical, which aligns with Kirchoff ’s junction rule. Because there are no junctions, the current does not split in the circuit and all capacitors accumulate the same quantity of charge. The measured effective capacitance of the circuit compared to the individual capacitances of each capacitor measured aligns almost perfectly with the relationship derived in step 9 for capacitors in series. The calculated effective capacitance is 497.23 μF. The resistor appears to have no effect on the capacitance in the circuit.

Part C. Capacitors in Parallel

Analyze the Results

• Present your data in a table that shows the capacitance of each individual capacitor, the effective capacitance of the circuit, and the voltage readings for the resistors and capacitors used.

  For a circuit containing a single resistor and two capacitors in parallel:

  Resistance used = 8.2 kΩ Capacitance of first capacitor, C₁ = 980 μF
  Capacitance of second capacitor, C₂ = 960 μF
  Measured effective capacitance C_eff = 1960 Μf

  {14014_Data_Table_6}
  Charge on capacitors used:
  {14014_Data_Table_7}
• Summarize your findings in several sentences while discussing charge, voltage, resistance and effective capacitance versus individual capacitance of the resistors.

  After an extended time and once the circuit is closed, the potential differences across the individual capacitors is equal to the potential difference across both in parallel. This aligns with Kirchoff ’s loop rule in that the change in potential across both loops is equivalent to the battery. The sum of the charge on C₁ and C₂ is equal to the charge on the effective capacitor which agrees completely with the relationship derived in step 7. There is a junction where the current must split. The charge is divided according to the values of capacitance, the larger the capacitance, the larger the charge stored. The charges stored on the individual capacitors, although different, must combine to be equal to the charge stored due to the effective capacitance. The observations show that this is the case and align with Kirchoff ’s junction rule. The calculated effective capacitance is 1940 μF, nearly identical to the measured effective capacitance. The resistor does not appear to affect the capacitance of the circuit.

Answers to Questions

Part A. Capacitor Overview

1. Recall that current is measured in amperes and is a derived unit. What SI units is the ampere derived from? Explain what a measurement of 2.0 A means with respect to the SI units.

   Amperes are derived from coulombs (the unit of electrical charge) and seconds (a unit of time). An ampere is a measure of coulombs per second. A measurement of 2.0 A means that 2.0 coulombs of charge are moving through a given point in a circuit every second.

2. Figure 9 shows a simple circuit made with a battery, a resistor and a capacitor. As the capacitor C₁ charges, it creates an electric field between the two plates that opposes the voltage applied across it.
   {14014_Answers_Figure_9}
   Figure 10 shows the relationship between the charging current, I, and the charge on the capacitor, Q. In general, describe what happens to the current and charge, respectively, on the capacitor after t = 0?
   {14014_Answers_Figure_10}
   The charge on the capacitor starts at zero, increases steadily and then begins to slow. Eventually, the charge on the capacitor reaches a maximum (asymptote) at Q_max. The current starts at a maximum, I_max, decreases steadily, and eventually goes to zero (asymptote).
3. Figure 11 shows a typical circuit used for investigating the properties of capacitors. Consider the following conditions and list the electronic components encountered by a charge moving in the circuit and identify its motion as clockwise or counter-clockwise.
   {14014_Answers_Figure_11}
   1. Assume the charge starts at point A and follows convention in the following instances:
      1. The switch is not in contact with wire W nor wire Z.

         The charge does not move from point A. There is no complete circuit for the charge to follow.

      2. The switch is in contact with wire W. The charge moves in a clockwise manner.

         The charge first encounters the resistor R, then capacitor C, and finally the battery, V_batt.

      3. The switch is in contact with wire Z. The charge does not move from point A.

         There is no complete circuit for the charge to follow.

   2. The charge now starts at the top plate of capacitor C, the capacitor is fully charged and convention is followed:
      1. The switch is not in contact with wire W nor wire Z.

         The charge does not move from the top place of C. There is no complete circuit for the charge to follow.

      2. The switch is in contact with wire W.

         The charge does not move from the top plate of C. The fully charged capacitor completely opposes the applied voltage from the battery and no charge moves.

      3. The switch is in contact with wire Z.

         The charge moves in a counter-clockwise manner. It first encounters the resistor R and then the bottom plate of capacitor C.

Part B. Capacitors in Series

1. Consider a capacitor connected directly to a battery. If the top plate accumulates a charge of +Q, what charge is accumulated on the bottom plate? Explain.

   If the top plate accumulates +Q charge, the bottom plate must accumulate –Q charge. The amount of charge accumulated on the plates must be equal, however the signs of the charge will be opposite.

Figure 7 is a schematic of a circuit with capacitors C₂ and C₃ in series. Examine Figure 7 and answer the following questions.

2. Which plates in the capacitors are directly connected to the battery terminals? Predict the expected charge (+Q or –Q) on these plates after an extended time once the switch is closed.

   The top plate of C₂ is connected directly to the positive terminal of the battery and will likely carry a +Q charge. The bottom plate of C₃ is directly connected to the negative terminal of the battery and will likely carry a –Q charge.

3. Note which plates are not directly connected to the battery terminals (i.e., there are no wires from the plates to the battery). Predict the expected charge on these plates after an extended time once the switch is closed.

   The bottom plate of C₂ and the top plate of C₃ do not have wires connecting them to the battery terminals. Despite not being connected to the battery terminals, these plates will respond to the accumulated charges on their partner plates. The bottom plate of C₂ will accumulate a –Q charge to match the +Q charge on the top plate of C₂ . The top plate of C₃ will accumulate a +Q charge to match the –Q charge on the bottom plate of C₃.

4. Treating the plates that are not connected to the battery as a unit, what would you notice about the net charge on the plates? Does this align with the law of conservation of charge?

   The two-plate unit is charge neutral. The positive and negative charges are equal in amount and cancel out. This agrees with the law of conservation of charge because there is no source of charge or a method of discharge for the plates, so the amount of charge on the plates cannot be increased or decreased.

5. What is the relationship between the charge accumulated on the effective capacitor and the charge on the individual capacitors, C₂ and C₃? Explain.

   The charge on the effective capacitor is equal to the charge on the individual capacitors. Whatever amount of charge accumulates on the top plate of the first capacitor, the remaining plates will carry the same amount of charge (not necessarily the same sign) because the law of conservation of charge does not allow for charge to be created or lost in the process. Therefore, Q = Q₂ = Q₃.

6. What is the voltage drop (potential difference) between points D and F? Explain how you determined this in terms of Kirchoff’s loop rule.

   The potential difference between points D and F is 6.00 V. The battery provides a potential of 6.00 V. The potential must decrease by an equal amount so that the sum of the voltages is zero. Therefore, the decrease across points D and F, and across the capacitors C₂ and C₃, must be equal to –6.00 V.

7. Write a mathematical relationship between V_battery and the potential differences caused by the components of the circuit in Figure 7.

   V_battery = V₂ + V₃

8. What is the effective capacitance, C_eff, of the circuit according to Equation 5?

   C_eff = Q/V_battery

9. Considering your answers to the previous questions, write a formula that relates the effective capacitance, C_eff, to the individual capacitors.

   V_battery = V₂ + V₃ = Q /C_eff
   Q = Q₂ = Q₃
   Q /C_eff = Q /C₂ + Q /C₃
   1/C_eff = 1/C₂ + 1/C₃

Part C. Capacitors in Parallel
Figure 12 is a schematic of a circuit with capacitors C₁ and C₂ in parallel.

1. How many paths (loops) can a charge follow from the positive terminal of the battery to the negative terminal?

   Charges can follow two paths in the circuit. The charges can flow from point D to I through points E and G or through points F and H. Either path will let the charges flow from the positive terminal battery to the negative terminal.

2. What is the voltage drop (potential difference) between points D and I? Would this potential difference be observed between any other points in the circuit? Explain how you determined this in terms of Kirchoff’s loop rule.

   The potential difference between points D and I is equal to 6.00 V, the potential of the battery. Following a path through any closed-loop circuit, the sum of the changes in voltage must equal zero. Between points D and I, there must be a decrease in electric potential equal to that of the battery. This difference in electric potential would also be measured across points E and G and points F and H.

3. After point D, there is a junction where current can flow in C₁ or C₂ and splits into I₁ and I₂. Will the current split equally between C₁ and C₂? Explain why or why not.

   The current will only split evenly if the two capacitors have the same capacitance. An equal split of current will supply the same amount of charge per second to the top plates of the capacitors, charging them at an equal rate. If the capacitance values are unequal, then the current will split unequally. However, the current through each of the capacitors will be such that charge accumulates at the same rate.

4. Write a mathematical relationship between V_battery and the potential differences caused by the components of the circuit in Figure 12.

   V_battery = V₄ = V₅

5. What is the overall effective capacitance, C_eff, of the circuit according to Equation 5?

   C_eff = Q / V_battery

6. Consider the distribution of charge between C₁ and C₂ and its relation to the charge on C_eff. How do these values compare?

   The charge on C_eff must be equal to the charge distributed on C₁ and C₂. Therefore, Q_total = Q₁ + Q₂ . There cannot be more charge accumulated on the capacitors than accumulated on the effective capacitor because charge must be conserved at the junction after point D. The total charge entering point D must be accounted for on C₁ and C₂.

7. Considering your answers to the previous questions, write a formula that relates the effective capacitor, C_eff, to the individual capacitors, C₁ and C₂.

   C_eff = Q_total/ V_battery = (Q₁ + Q₂) / V_battery C_eff = Q₁ /V_battery + Q₂ /V_battery C_eff = C₁ + C₂

Review Questions for AP^® Physics 2

1. The capacitance through one branch of a circuit is measured to be 200 μF. A capacitor is added to the circuit. The effective capacitance is now measured to be 120 μF.
   1. Was the new capacitor added in series or in parallel to the circuit? Explain how you made your determination.

      The new capacitor was added in series to the 200-μF capacitor. When capacitors are added in series, the effective capacitance is decreased because the inverse capacitances are summed. If the capacitor was added in parallel, the effective capacitance would increase because the capacitances are summative.

   2. What was the value of the added capacitor?

      1/C_eff = 1/C₁ + 1/C₂
      1/120 μF = 1/200 μF + 1/C₂
      1/C₂ = 1/120 μF – 1/200 μF
      1/C₂ = 2/600 μF
      C₂ = 300 μF

2. The time constant for any given RC circuit is the value of the effective resistance of the circuit multiplied by the effective capacitance of the circuit. What is the value of the time constant for the circuit?
   {14014_Answers_Figure_13}
   {14014_Answers_Figure_14}
   The circuit can be simplified down by using the known relationships for resistors and capacitors in series and in parallel:
   • The 20 kΩ and 5 kΩ resistors in parallel equal a 4 kΩ resistor by 1/20kΩ + 1/5kΩ = 1/4kΩ.
   • The 500 μF and 2000 μF capacitors in series become 400 μF by 1/500μF + 1/2000μF = 1/400μF.
   • The pair of 800 μF capacitors in series in the parallel branch following the first 1 kΩ resistor become 400 μF by 1/800 μF + 1/800 μF = 1/400 μF.
   • The calculated 400 μF capacitor is in parallel with the 2000 μF capacitor, which add to become 2400 μF.
   • The two remaining 1 kΩ resistors are in series with the 4 kΩ resistor, which add to become 6 kΩ.
   • The calculated 400 μF capacitor is in series with the calculated 2400 μF capacitor and become 342.857 by 1/400 μF + 1/2400 μF = 1/342.857 μF.
   Therefore, R x C = 6,000 Ω x (342.857 x 10^–6 F) = 2.057 seconds.
3. For the following circuit, the switch is closed and steady-state conditions have been reached, meaning the capacitor is fully charged.
   {14014_Answers_Figure_15}
   1. What is the voltage across the 4 Ω resistor?

      The voltage is zero because there is no current running through the branch since the capacitor is fully charged.

   2. What is the voltage across the 6 μF capacitor?

      First, one must solve for current. We use Ohm’s law and Kirchoff ’s loop rule for each component of loop ABA to make the sum of the voltage increase due to the battery and voltage drops due to the resistors equate to zero.
      12 V – 10 Ω(I) – 3 Ω(I) – 2 Ω(I) = 0
      12 – 15(I) =0
      15(I) = 12
      I = 12/15 =4/5 or 0.8 A
      
      To find the capacitor voltage we can simply use Kirchoff ’s loop rule for loop BCB and use the known current calculated above for the 3 Ω resistor.
      4 Ω(0) – V_C + 3 Ω(I) = 0
      V_C = 3 Ω(4/5 A)
      V_C = 2.4 V

   3. How much charge is stored on the capacitor?

      Q = CV
      C = 6 μF
      V = 2.4 V
      Q = (6 × 10^–6 F)(2.4 V) = 1.44 × 10^–5 C

Teacher Handouts

14014_Teacher1.pdf

References

AP^® Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.

Introduction

Capacitors are essential for general power management in electronic circuits such as charging units. They collect charge when connected to a battery and release the charge when the battery is removed. This ensures there is no loss of current in the circuit. Capacitors can also limit current by collecting charge and building a voltage.

Concepts

• Circuits
• Capacitors
• Kirchoff’s rules
• Resistors
• Conservation of energy

Background

For any circuit containing a voltage source (e.g., a battery) and a closed path from the positive terminal to the negative terminal, charges will flow producing a current in the wire. The amount of current is proportional to the applied voltage and the amount of resistance present in the loop. For a given resistor, the applied voltage divided by the resistance gives the current, as shown by Equation 1.

where V is the voltage drop (change in potential) across the resistor, I is the current, and R is the resistance. This equation is commonly referred to as Ohm’s law, named after German physicist Georg Ohm (1789–1854).

Ohm’s law can be applied to the overall circuit—voltage source and components—or to the individual components. By measuring the voltage drop (potential difference) across a component and the current flowing through a component, the resistance of the device can be calculated.

Another German physicist, Gustav Kirchoff (1824–1827), expanded on Ohm’s work by considering how energy and charge must be conserved in a circuit. This work can be summarized into two rules:

1. Kirchoff’s loop rule states that following a closed loop from any point in a circuit, the net change in voltage (electric potential) is equal to zero. In order to follow the conservation of energy at a point in a circuit, the charges must gain and lose equal amounts of electric potential through the circuit before returning to that point.
2. Kirchoff’s junction rule states that the sum of the currents flowing into a node (i.e., a place where two or more wires meet) is equal to the sum of the currents flowing out of the node. In accordance with the conservation of charge, there will not be a build-up or depletion of charges at a junction.

These two rules guide the determination of current and voltage differences in simple and complex circuits. In order to utilize the laws effectively, an internally consistent convention for the signs of voltage changes and directions of current flow must be employed. The historical convention is to follow the flow of positive charge through the circuit. In this case, the positive charges start at the positive terminal of the battery and flow “downhill” (lose energy) until they arrive at the negative terminal. A battery can be viewed as increasing the electric potential of charges from the negative terminal to the positive terminal within the battery. A device with resistance can be viewed as decreasing the potential of the charges through the circuit.

When electrical components are connected one after another, the components are said to be in series. Current flows from one component directly to the next component along a single path (see Figure 1). Electrical components in a parallel arrangement have one lead connected to a common wire. The other lead is then connected to a different common wire. In this case, the current has separate paths it may follow to reach the battery (see Figure 1). When resistors are arranged in series, the effective resistance of the resistor group is equal to the sum of the individual resistor values (Equation 2): When resistors are arranged in parallel, the effective resistance of the resistor group is equal to the sum of the inverse resistor values: Recall that a capacitor is an electronic component used to store charge in a circuit. The plates within the capacitor accumulate electric charge for later use (acting as a voltage source) or as a means to limit current. The capacitance, C, of a capacitor is an intrinsic property that depends on the construction of the capacitor (geometry of the plates and dielectric material used). The formula for determining capacitance is: where C is capacitance, k is the dielectric constant, ε₀ is the permittivity of free space, A is the area of the plates, and d is the separation distance of the plates.

Capacitance is also the ratio of the charge on the plates, Q, and the potential difference (voltage drop), V, across the capacitor. (Equation 5). This investigation will focus on RC circuits with capacitors in steady-state situations. This means situations are limited to an open circuit, just after the circuit is closed, and an extended time after the circuit is closed (at least 5 times the value of RC of the circuit).

Experiment Overview

The purpose of this advanced inquiry lab activity is to apply Kirchoff’s rules to resistor–capacitor (RC) circuits and study the effect the arrangement of circuit components has on current and changes in potential. First, you will practice the laboratory technique of using a multimeter to measure the current and potential difference across resistors in series and parallel circuits. Ohm’s law is used to confirm the resistance values printed on the resistors. The procedure provides a model for the guided-inquiry design of experiments to determine the effect of placing capacitors in series and in parallel.

Materials

Prelab Questions

1. Examine Figure 1 of a battery connected to two resistors in series.
   1. Explain how Kirchoff’s loop rule can be applied to solve for the potential difference (voltage drop) across resistor R₁.
   2. Explain how Kirchoff’s junction rule can be applied to solve for the current through the resistors, R₁ and R₂.
2. Examine Figure 1 of a battery connected to two resistors in parallel.
   1. Explain how Kirchoff’s loop rule can be applied to solve for the potential difference (voltage drop) across resistor R₁.
   2. Explain how Kirchoff’s junction rule can be applied to solve for the current through the resistors, R₁ and R₂.

Safety Precautions

If a resistor begins to darken or smoke, immediately disconnect the circuit and do not touch the resistor—it will be hot. Allow it ample time to cool. Handle pins with caution, as they are sharp. Please follow all laboratory safety guidelines.

Procedure

Introductory Activity

Resistors in Series

1. Obtain two resistors of different resistance. Hint: The colored bands should not be in the same order. Record the resistance values in a data table.
2. Use the cord connectors to connect the resistors in series according to Figure 2.
   {14014_Procedure_Figure_2}
3. Set the multimeter to measure voltage (V).
4. Attach the cord connectors to the battery electrodes.
5. Place the probes of the multimeter on both sides of resistor R₁. The multimeter is now in parallel with R₁.
6. Record the potential difference (voltage drop) in volts (V) across R₁ in a data table.
7. Repeat steps 5 and 6 for R₂ and the effective resistor, R_1,2.
8. Disconnect the circuit from the battery.
9. Set the multimeter to measure milliamps (mA).
10. Disconnect the wire between R₁ and R₂. Attach the free end of the connector to one of the multimeter probes.
11. Place the other probe on the free end of the resistor. The multimeter is now in series with the resistors.
12. Reconnect the circuit to the battery.
13. Record the current in milliamps (mA) in a data table.
14. Disconnect the resistors from the cord connectors. Let the resistors cool before handling.

Resistors in Parallel

15. Using the same resistors from the previous section, assemble the parallel resistor component according to Figure 3.
    {14014_Procedure_Figure_3}
16. Set the multimeter to measure voltage.
17. Connect the parallel resistor component to the battery.
18. Measure and record the potential difference across resistors R₁ and R₂ and the effective resistor R_1,2.
19. Disconnect the circuit from the battery.
20. Set the multimeter to measure current.
21. Arrange the multimeter probes to measure the current entering the effective resistor R_1,2.
22. Connect the circuit to the battery and record the current.
23. Disconnect the circuit from the battery.
24. Repeat steps 21–23 to measure and record the current through R₁ and R₂ and the current exiting R_1,2.
25. Disconnect all the components when all measures have been recorded.

Analyze the Results

• Using the collected voltage and current data, calculate the measured resistance for each resistor in the series and parallel arrangements. Display this data in appropriate data tables.
• Use Table 1 from the Supplementary Material in the Teacher PDF to determine the color-band resistance value of each resistor. Compare the calculated resistance values to the color-band resistance values. Do the experimental values fall within the tolerance range of the resistor?
• Calculate the effective resistances of R_1,2 in both the series and parallel arrangements using Equations 2 and 3, respectively.
• Do the measured effective resistance values for the series and parallel arrangements agree with the calculated values? Identify any possible sources of error.

Guided-Inquiry Design and Procedure
Form a working group with other students to discuss the following questions.

Part A. Capacitor Overview

1. Recall that current is measured in amperes and is a derived unit. What SI units is the ampere derived from? Explain what a measurement of 2.0 A means with respect to the SI units.
2. Figure 4 shows a simple circuit made with a battery, a resistor and a capacitor. As the capacitor, C, charges, it creates an electric field between the two plates that opposes the voltage applied across it.
   {14014_Procedure_Figure_4}
   Figure 5 shows the relationship between the charging current, I, and the charge on the capacitor, Q. In general, describe what happens to the current and charge on the capacitor after t = 0.
   {14014_Procedure_Figure_5}
3. Figure 6 shows a typical circuit used for investigating the properties of capacitors. Consider the following conditions and list the electronic components encountered by a charge moving the circuit and identify its motion as clockwise or counterclockwise.
   {14014_Procedure_Figure_6}
   1. Assume the charge starts at point A and follows convention in the following instances.
      1. The switch is not in contact with wire W nor wire Z.
      2. The switch is in contact with wire W.
      3. The switch is in contact with wire Z.
   2. The charge now starts at the top plate of capacitor C, the capacitor is fully charged, and convention is followed.
      1. The switch is not in contact with wire W nor wire Z.
      2. The switch is in contact with wire W.
      3. The switch is in contact with wire Z.

Part B. Capacitors in Series

1. Consider a capacitor connected directly to a battery. If the top plate accumulates a charge of +Q, what charge is accumulated on the bottom plate? Explain.

Figure 7 is a schematic of a circuit with capacitors C₂ and C₃ in series. Examine Figure 7 and answer the following questions.

2. Which plates in the capacitors are directly connected to the battery terminals? Predict the expected charge (+Q or –Q) on these plates after an extended time once the switch is closed.
3. Note which plates are not directly connected to the battery terminals (i.e., there are no wires from the plates to the battery)? Predict the expected charge on these plates after an extended time once the switch is closed.
4. Treating the plates that are not connected to the battery as a unit, what would you notice about the net charge on the plates? Does this make sense with respect to the law of conservation of charge?
5. What is the relationship between the charge accumulated on the effective capacitor and the charge on the individual capacitors, C₂ and C₃? Explain.
6. What is the voltage drop (potential difference) between points D and F? Explain how you determined this in terms of Kirchoff’s loop rule.
7. Write a mathematical relationship between V_battery and the potential differences caused by the components of the circuit in Figure 7.
8. What is the effective capacitance, C_eff, of the circuit according to Equation 5?
9. Considering your answers to the previous questions, write a formula that relates the effective capacitance, C_eff , to the individual capacitors.
10. Write a detailed procedure for how to determine the relationship between voltage, capacitance, and charge on a capacitor plate in a series circuit for steady-state situations. The procedure should also be able to determine the role of resistors in an RC series circuit and be able to confirm the relationship found in step 9. Note: All circuit configurations tested should include at least one resistor.

Analyze the Results

• Present your data in a table that shows the capacitance of each individual capacitor, the effective capacitance of the circuit, and the voltage readings for the resistors and capacitors used.
• Summarize your findings in several sentences while discussing charge, voltage, resistance and effective capacitance versus individual capacitance of the resistors.

Part C. Capacitors in Parallel
Figure 8 is a schematic of a circuit with capacitors C1 and C2 in parallel.

1. How many paths (loops) can a charge follow from the positive terminal of the battery to the negative terminal?
2. What is the voltage drop (potential difference) between points D and I? Would this potential difference be observed between any other points in the circuit? Explain how you determined this in terms of Kirchoff’s loop rule.
3. After point D, there is a junction where current can flow in C₁ or C₂ and splits into I₁ and I₂. Will the current split equally between C₁ and C₂? Explain why or why not.
4. Write a mathematical relationship between V_battery and the potential differences caused by the components of the circuit in Figure 8.
5. What is the overall effective capacitance, C_eff, of the circuit according to Equation 5?
6. Consider the distribution of charge between C₁ and C₂ and its relation to the charge on C_eff. How do these values compare?
7. Considering your answers to the previous questions, write a formula that relates the effective capacitor, C_eff, to the individual capacitors, C₁ and C₂.
8. Write a detailed procedure for how to determine the relationship between voltage, capacitance, and charge on a capacitor plate in a parallel circuit for steady-state situations. The procedure should also be able to determine the role of resistors in an RC parallel circuit and be able to confirm the relationship found in step 7. Note: All circuit configurations tested should include at least one resistor.

Analyze the Results

• Present your data in a table that shows the capacitance of each individual capacitor, the effective capacitance of the circuit, and the voltage readings for the resistors and capacitors used.
• Summarize your findings in several sentences while discussing charge, voltage, resistance and effective capacitance versus individual capacitance of the resistors.

Student Worksheet PDF

14014_Student1.pdf