5CL Pre-Lab 6 - W24v3

Physics 5CL Pre-Lab 6 - Circuits III: RC Circuits Winter 2024, UCLA Department of Physics & Astronomy Directions: As you read through the pre-lab, follow along and complete the Google Slides pre-lab submission template to submit your responses to questions below on each slide as indicated. Key Equations: 𝛕 = ?𝐶 ● - RC time constant. Time it takes to charge/discharge ~⅓ of the capacitor 𝛕 ● - Resistance (resistance measured in Ohms) ? ● - Capacitance measured in Farads F or microFarads = μF 𝐶 𝑉 𝐶 (?) = 𝑉 ?? (1 − ? −? ?𝐶 ) ● - Voltage across the capacitor 𝑉 𝐶 ● - Voltage of the power supply 𝑉 ?? ● - Elapsed time since the voltage started being applied to the circuit ? ● - Time constant 𝝉 (see above for information about units of time constant) ?𝐶 ? = 𝐶𝑉 ● - charge ? ● - capacitance 𝐶 ● V - voltage Notes: ● If R is measured in Ohms and C is measured in microFarads, then the time constant has units of microseconds, not seconds! Be careful with units when calculating. ● Technically the “time constant” is the time it takes to charge about ⅔ of the capacitor, as defined by , which is roughly . This is related to the exponential 1 − 1/? ( ) 1 − 1/2. 73 ( ) ≈ 2/3 nature of the curve. ● For capacitors in series: 1/C total = 1/C 1 + 1/C 2 ● For capacitors in parallel: C total = C 1 + C 2 Background: This week’s lab will introduce capacitors. A capacitor is a circuit element capable of storing electrical energy. Two parallel plates separated by an insulator is an example of a configuration that can store energy in the form of a separation of charge and therefore be used as a capacitor. You can load positive charge onto one plate which will induce an equivalent negative charge on the other plate, creating a potential difference between the two plates. The capacitance is defined as the proportionality constant between the charge on a single plate ( ) and the voltage produced ? between the plates ( ): . As you load more charge onto the plates, the 𝑉 ? = 𝐶𝑉 voltage will increase. The capacitance tells you how much the voltage will increase per unit of charge added. Much as Ohm’s Law defines the resistance ( ), this equation defines the 𝑉 = 𝐼? capacitance. A circuit with only power supplies and resistors is always in a state of equilibrium once turned on, meaning that the current at any point in the circuit is constant (until your power supply overheats). When capacitors are thrown into the mix, the current in many circuits will pick up a 1

Physics 5CL Pre-Lab 6 - Circuits III: RC Circuits Winter 2024, UCLA Department of Physics & Astronomy time dependence. This is due to the fact that the voltage on a capacitor is proportional to the charge on the capacitor. The current is just the time derivative of charge . When you 𝐼 = ?? ?? write down Kirchhoff’s loop rule for the voltage of your circuit elements, you will have capacitor terms that depend on the charge as well as resistor terms that depend on the current . This ? ?? ?? will produce a differential equation. These types of equations are generated whenever a quantity (charge) depends upon its own rate of change (current). We won’t be solving any of these equations but will use the solutions which will involve exponential time dependence. An RC circuit is any that contains a resistor and a capacitor. If you put a charged capacitor in series with a resistor, charge will flow from one side of the capacitor, through the resistor, and to the other side of the capacitor. This will eventually deplete all charge on the capacitor over time. The resistor slows this discharge. If the capacitor of capacitance initially had charge 𝐶 at time and the resistor has resistance , the amount of charge at time t is: ? 0 ? = 0 ? Charge on discharging capacitor: ?(?) = ? 0 ? − ? ?𝐶 Then since , the voltage on the capacitor at time t is: ? = 𝐶𝑉 Voltage on discharging capacitor: 𝑉(?) = ? 0 𝐶 ? − ? ?𝐶 This is the time dependence of the voltage for a capacitor discharging through a resistor. In order to charge the capacitor in the first place, you’ll need to add a power supply into the equation. For a power supply with constant voltage and a resistor with resistance 𝑉 ?? , the voltage on a capacitor being charged as a function of time is: ? Voltage on charging capacitor: 𝑉(?) = 𝑉 ?? (1 − ? − ? ?𝐶 ) In lab this week, you will measure the voltage across a capacitor as it is charged and discharged and fit this data to the equations above in order to determine it’s time constant . It is important to note that when we charge and discharge our capacitor this week, the 𝛕 = ?𝐶 resistor in the charging circuit will be different from the resistor in the discharging circuit. This means that the R values you measure between charging and discharging will not be the same. When more than one capacitor is present in a circuit, it can be wired either in series or in parallel. When capacitors are parallel, the total capacitance is simply the sum of the two capacitance values. Capacitors in Parallel: 𝐶 ?𝑞 = 𝐶 1 + 𝐶 2 When capacitors are wired in series, the inverse of the total capacitance is equal to the sum of the inverse of the individual capacitances: 2

Physics 5CL Pre-Lab 6 - Circuits III: RC Circuits Winter 2024, UCLA Department of Physics & Astronomy Additional Resources: In addition to your online textbook resources, websites here and here can help you further explore the above concepts. More detailed descriptions, including calculus-based models of charge/current for the resistors and capacitors here . Pre-Lab Preparation Questions: Pre-Lab Question 1: a. Draw a circuit diagram containing a power supply, switch, resistor, and capacitor. b. You will be given 4 resistors and 3 capacitors with nominal values as listed below. For each resistor & capacitor combination, calculate the corresponding RC time constant in seconds. There will be 12 time constants to calculate. Note that (microseconds) 𝑘Ω · µ? = 𝑚? Ideally, do this in a spreadsheet that you can bring with you to lab, as we will be directly measuring and updating these values. Spreadsheet equations will also be much less tedious than manually calculating 12 time constants. Resistors 1 kΩ 5.1 kΩ 10 kΩ 100 kΩ Capacitors 10 μf 100 μf 470 μf Pre-Lab Question 2: To start this lab, you will set up your RC circuit with an open switch, such that the capacitor is NOT charging at . After you start recording the voltage in real time (a graph will be ? = 0 generated in PASCO), you will close your switch. The voltage across your capacitor will increase with time until it eventually saturates. You will extract your RC time constant from the shape of this graph. a. When the capacitor fully charges and saturates in voltage, do you expect the voltage across the capacitor to be the same as the battery voltage, less than the battery voltage, or greater than the battery voltage? (hint: the current in the circuit will go to zero). b. When the capacitor voltage is saturated and you open your switch again, do you expect the capacitor to keep its charge or lose its charge? (hint: consider this question from the perspective that there is no current running in the circuit because with the switch open there is not a complete circuit that includes the battery) c. However, even though there is not a complete circuit that includes the battery, there is another complete circuit that you make because you have the voltmeter with an input resistance of order 1 MΩ connected to the capacitor. Draw the circuit diagram consisting of your capacitor and the voltmeter acting as a resistor. What is its RC time constant? Pre-Lab Question 3: a. Draw a circuit diagram containing a power supply, switch, resistor, and two capacitors in series . If one capacitor has a capacitance of 470μf and the other has capacitance of 100μf, what is the total capacitance when they are in series? b. Draw a circuit diagram containing a power supply, switch, resistor, and two capacitors in parallel . If one capacitor has a capacitance of 470μf and the other has capacitance of 100μf, what is the total capacitance when they are in parallel? 4

Physics 5CL Pre-Lab 6 - Circuits III: RC Circuits Winter 2024, UCLA Department of Physics & Astronomy Life Sciences Applications: RC circuits are important for life scientists to understand because the cells in our body have both resistance and capacitance, and these circuit properties affect how much and when electrical signal transfers between neurons in our brain, cardiac muscles in our heart, and ion channels in the cell soma membrane everywhere in our body. Many life science medical and research techniques rely on voltage changes in cells to measure bio/neurological properties. Because capacitance is a natural part of our body’s material properties, events related to both intrinsic biophysical phenomena and also our external measurement techniques are time limited. More information is available in your Knight textbook in sections 23.6 and 23.7, and will also be referenced in your next lab. 5