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Once you have your experimental data, the process below will walk you through the theoretical calculation of how to determine the time constant for the charging capacitor. While this kind of circuit is beyond the scope of what we've covered in lecture, we do have the to be able to understand it analytically. When you through this analytic derivation, don't plug in any numbers. 1. Write down a Kirchhoff's voltage law loop equation that starts behind the battery and travels through both resistors. tools R 11 12 go 1 IC C 2. Write down a Kirchhoff's voltage law loop that goes through just the capacitor and R2. R 2 2 3. Write down a Kirchhoff's current law equation to relate the three currents (11, 12, and le) to each other. dQ dt 4. Write down leas . We can do this because the current that passes through that path in the circuit is what causes the capacitor to become charged up. 5. We now have three equations and three unknowns (11, 12, and Q). Using substitutions, cancel out the dQ currents to end up with a differential equation whose only unknowns are Q and dt a. The differential equation should end up in the form of: V=MQ+Ndo, where M and N are dt constants that depend on the capacitance and resistances used in the circuit. 6. Based on our experimental results, the charge Q(t) for a charging capacitor follows the same pattern as it did in a linear RC circuit. Because of that, we can try a solution of the same form. a. Plug this solution into your equation: qt=A+Be 7. When you do so, you will have two different kinds of terms: a. Terms that are constant (without t in them). b. Terms that are time-dependent (with t in them). 8. Because the differential equation is always true, the time-dependent terms must cancel out for any time t. (Essentially, the left and right sides of the equation are still equal if you look only at the terms with t.) From there, you can solve for your time constant T. It should only depend on R1, R2, and C. 9. You don't need to worry about solving for A and B, as we only care about the time constant.

Once you have your experimental data, the process below will walk you through the theoretical calculation of how to determine the time constant for the charging capacitor. While this kind of circuit is beyond the scope of what we've covered in lecture, we do have the to be able to understand it analytically. When you through this analytic derivation, don't plug in any numbers. 1. Write down a Kirchhoff's voltage law loop equation that starts behind the battery and travels through both resistors. tools R 11 12 go 1 IC C 2. Write down a Kirchhoff's voltage law loop that goes through just the capacitor and R2. R 2 2 3. Write down a Kirchhoff's current law equation to relate the three currents (11, 12, and le) to each other. dQ dt 4. Write down leas . We can do this because the current that passes through that path in the circuit is what causes the capacitor to become charged up. 5. We now have three equations and three unknowns (11, 12, and Q). Using substitutions, cancel out the dQ currents to end up with a differential equation whose only unknowns are Q and dt a. The differential equation should end up in the form of: V=MQ+Ndo, where M and N are dt constants that depend on the capacitance and resistances used in the circuit. 6. Based on our experimental results, the charge Q(t) for a charging capacitor follows the same pattern as it did in a linear RC circuit. Because of that, we can try a solution of the same form. a. Plug this solution into your equation: qt=A+Be 7. When you do so, you will have two different kinds of terms: a. Terms that are constant (without t in them). b. Terms that are time-dependent (with t in them). 8. Because the differential equation is always true, the time-dependent terms must cancel out for any time t. (Essentially, the left and right sides of the equation are still equal if you look only at the terms with t.) From there, you can solve for your time constant T. It should only depend on R1, R2, and C. 9. You don't need to worry about solving for A and B, as we only care about the time constant.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Once you have your experimental data, the process below will walk you through the theoretical calculation of how to determine the time constant for the charging capacitor. While this kind of circuit is beyond the scope of what we've covered in lecture, we do have the to be able to understand it analytically. When you through this analytic derivation, don't plug in any numbers. 1. Write down a Kirchhoff's voltage law loop equation that starts behind the battery and travels through both resistors. tools R 11 12 go 1 IC C 2. Write down a Kirchhoff's voltage law loop that goes through just the capacitor and R2. R 2 2 3. Write down a Kirchhoff's current law equation to relate the three currents (11, 12, and le) to each other. dQ dt 4. Write down leas . We can do this because the current that passes through that path in the circuit is what causes the capacitor to become charged up. 5. We now have three equations and three unknowns (11, 12, and Q). Using substitutions, cancel out the dQ currents to end up with a differential equation whose only unknowns are Q and dt a. The differential equation should end up in the form of: V=MQ+Ndo, where M and N are dt constants that depend on the capacitance and resistances used in the circuit. 6. Based on our experimental results, the charge Q(t) for a charging capacitor follows the same pattern as it did in a linear RC circuit. Because of that, we can try a solution of the same form. a. Plug this solution into your equation: qt=A+Be 7. When you do so, you will have two different kinds of terms: a. Terms that are constant (without t in them). b. Terms that are time-dependent (with t in them). 8. Because the differential equation is always true, the time-dependent terms must cancel out for any time t. (Essentially, the left and right sides of the equation are still equal if you look only at the terms with t.) From there, you can solve for your time constant T. It should only depend on R1, R2, and C. 9. You don't need to worry about solving for A and B, as we only care about the time constant.
Transcribed Image Text:Once you have your experimental data, the process below will walk you through the theoretical calculation of how to determine the time constant for the charging capacitor. While this kind of circuit is beyond the scope of what we've covered in lecture, we do have the to be able to understand it analytically. When you through this analytic derivation, don't plug in any numbers. 1. Write down a Kirchhoff's voltage law loop equation that starts behind the battery and travels through both resistors. tools R 11 12 go 1 IC C 2. Write down a Kirchhoff's voltage law loop that goes through just the capacitor and R2. R 2 2 3. Write down a Kirchhoff's current law equation to relate the three currents (11, 12, and le) to each other. dQ dt 4. Write down leas . We can do this because the current that passes through that path in the circuit is what causes the capacitor to become charged up. 5. We now have three equations and three unknowns (11, 12, and Q). Using substitutions, cancel out the dQ currents to end up with a differential equation whose only unknowns are Q and dt a. The differential equation should end up in the form of: V=MQ+Ndo, where M and N are dt constants that depend on the capacitance and resistances used in the circuit. 6. Based on our experimental results, the charge Q(t) for a charging capacitor follows the same pattern as it did in a linear RC circuit. Because of that, we can try a solution of the same form. a. Plug this solution into your equation: qt=A+Be 7. When you do so, you will have two different kinds of terms: a. Terms that are constant (without t in them). b. Terms that are time-dependent (with t in them). 8. Because the differential equation is always true, the time-dependent terms must cancel out for any time t. (Essentially, the left and right sides of the equation are still equal if you look only at the terms with t.) From there, you can solve for your time constant T. It should only depend on R1, R2, and C. 9. You don't need to worry about solving for A and B, as we only care about the time constant.
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