Remove the ohmic resistor that is attached to the white sheet of paper.\u00a0 Use the BK DMM<\/a> to measure its resistance (always make sure to record the meter range that you are using whenever you measure a value with a DMM<\/strong>). Calculate the value’s uncertainty.<\/p>\n Now remove the capacitor that is attached to the white sheet of paper.\u00a0 Use the BK DMM<\/a> to measure its capacitance (the capacitor is marked with a negative sign with an arrow pointing to the negative lead). Make sure that the capacitor is discharged before placing it in the meter.\u00a0 Ask Lab Instructor for help if you do not know how to discharge the capacitor. Calculate the value’s uncertainty.<\/p>\n Calculate the circuit’s time constant<\/a> \\(\\tau\\) with uncertainty using the two measured values obtained above. Set up an RC charging circuit using the batteries (make sure the three batteries are in series to produce approximately \\(4.5\\) \\({\\rm V}\\) like last week), capacitor, and resistor.\u00a0 Add the BK DMM to measure the voltage across the capacitor.\u00a0 Draw the circuit in your notebook making sure to label all of its components.<\/p>\n Let the capacitor fully charge.\u00a0 The voltage across the capacitor when it is fully charged is V Source.\u00a0 Record V Source (without uncertainty) in your notebook. <\/span><\/p>\n Using the timer<\/a> and BK DMM, measure the voltage across the capacitor during the capacitor’s discharge cycle (make sure the capacitor is completely charged before starting).\u00a0 Record both \\(t\\) and \\(V\\) starting at \\(t=0\\) \\({\\rm s}\\) and ending at \\(t=60\\) \\({\\rm s}\\).\u00a0 Use a reasonable time interval that allows you and your partner to accurately record both \\(t\\) and \\(V\\).\u00a0 The time interval should be as small as possible while still allowing for accurate measurements.\u00a0 Calculate the uncertainty of one representative \\(V\\) value (one near the middle of the cycle).\u00a0 Use your best estimate for \\(\\delta t\\).<\/p>\n Plot \\(V\\) vs. \\(t\\).\u00a0 Place error bars on the data point for which you determined\u00a0 \\(\\delta V\\) and \\(\\delta t\\).<\/p>\n Use your plot to graphically determine the circuit’s time constant \\(\\tau\\)<\/span>.\u00a0 (Reminder: V at t=0 is Vo)<\/p>\n Now make sure the capacitor is fully discharged.\u00a0 Using the timer and BK DMM, measure the voltage across the capacitor during the capacitor’s charging cycle.\u00a0 Again record both \\(t\\) and \\(V\\) starting at \\(t=0\\) \\({\\rm s}\\) and ending at \\(t=60\\) \\({\\rm s}\\). Use the same reasonable time interval that you used above.\u00a0 Calculate the uncertainty of one representative \\(V\\) value (one near the middle of the cycle).\u00a0 Use your best estimate for \\(\\delta t\\).<\/p>\n Plot \\(V\\) vs. \\(t\\).\u00a0 Place error bars on the data point for which you determined\u00a0 \\(\\delta V\\) and \\(\\delta t\\).<\/p>\n Use your plot to graphically determine the circuit’s time constant \\(\\tau\\)<\/span>.<\/p>\n In 1130 Physics Lab we took non-linear data and made it linear thus allowing us to use straight line analysis to determine important characteristics of a system.\u00a0 It is possible to plot our data as a straight line by using a special type of graph paper called semi-logarithmic (semi-log) graph paper.\u00a0 Exponential functions plot as straight lines on semi-log graph paper.\u00a0 Look at the sheet of semi-logarithmic graph paper that was provided to you.\u00a0 Notice that the vertical scale is logarithmic while the horizontal scale is linear.<\/p>\n Exponential functions have the form:$$ y = k e^{mx} $$ or $$ \\ln y = mx + \\ln k$$<\/p>\n where \\(m\\) is any positive or negative constant.\u00a0 Notice that the second equation is that of a line.\u00a0 On semi-log graph paper \\(m = {\\rm the\\;slope\\;of\\;the\\;line}\\) and \\(k = {\\rm the\\;y-intercept}\\).<\/p>\n Earlier we learned that the equation describing the discharge of a capacitor in series with a resistor is:$$ V(t) = V_{0}e^{-t\/(RC)}$$ \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em> This is clearly an exponential function.\u00a0 Now plot \\(V\\) vs. \\(t\\) on the semi-log graph paper using the data you collected when the capacitor was being discharged.\u00a0 Note that you do not have to take the natural log of \\(V\\) before you plot it because the vertical scale is already logarithmic.\u00a0 Draw the best fit line and calculate its slope* without uncertainty.<\/p>\n *Calculating the slope of a line plotted on semi-log graph paper is a little different than that of calculating the slope of a line plotted on linear graph paper. Because semi-log graph paper uses two different scales, linear and logarithmic, it is necessary to convert the data of one axis to match that of the other before calculating the slope.\u00a0 The most common method is to convert the logarithmic axis data to linear data by taking its natural log.\u00a0 Therefore you must first calculate the natural log of your two \\(Y\\) values before you use them to calculate \\(\\Delta Y\\) in your slope equation.<\/p>\n Thus: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 $${\\rm Slope} = \\frac{\\ln Y_{2} – \\ln Y_{1}}{X_{2} -X_{1}}$$<\/p>\n Note that \\(\\ln Y_{2}-\\ln Y_{1} = \\ln (Y_{2}\/Y_{1})\\), therefore it is unit-less.<\/p>\n Compare the time constants calculated from your three plots with the one calculated with the measured \\(R\\) and \\(C\\) values.\u00a0 Do they agree within reason (keeping in mind that we have not calculated the uncertainty in the graphically calculated values)? What do you think caused the largest error in your values calculated with the plots?<\/p>\n Background<\/a> Equipment<\/a> Summary Questions<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Part 1: Measuring Component Values Remove the ohmic resistor that is attached to the white sheet of paper.\u00a0 Use the BK DMM to measure its resistance (always make sure to record the meter range that you are using whenever you measure a value with a DMM). Calculate the value’s uncertainty. Now remove the capacitor that […]<\/p>\n","protected":false},"author":293,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2039","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2039","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/users\/293"}],"replies":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/comments?post=2039"}],"version-history":[{"count":0,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2039\/revisions"}],"wp:attachment":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/media?parent=2039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n<\/span><\/p>\nPart 2: Set up an RC Circuit<\/h2>\n
Part 3: Straight Line Analysis using Semi-Log Graph Paper<\/h2>\n
\n<\/em><\/p>\n
![BK DMM<\/a> to measure its resistance (always make sure to record the meter range that you are using whenever you measure a value with a DMM<\/strong>). Calculate the value’s uncertainty.<\/p>\nNow remove the capacitor that is attached to the white sheet of paper.\u00a0 Use the BK DMM<\/a> to measure its capacitance (the capacitor is marked with a negative sign with an arrow pointing to the negative lead). Make sure that the capacitor is discharged before placing it in the meter.\u00a0 Ask Lab Instructor for help if you do not know how to discharge the capacitor. Calculate the value’s uncertainty.<\/p>\nCalculate the circuit’s time constant<\/a> \\(\\tau\\) with uncertainty using the two measured values obtained above.\n<\/span><\/p>\nPart 2: Set up an RC Circuit<\/h2>\nSet up an RC charging circuit using the batteries (make sure the three batteries are in series to produce approximately \\(4.5\\) \\({\\rm V}\\) like last week), capacitor, and resistor.\u00a0 Add the BK DMM to measure the voltage across the capacitor.\u00a0 Draw the circuit in your notebook making sure to label all of its components.<\/p>\nLet the capacitor fully charge.\u00a0 The voltage across the capacitor when it is fully charged is V Source.\u00a0 Record V Source (without uncertainty) in your notebook. <\/span><\/p>\nUsing the timer<\/a> and BK DMM, measure the voltage across the capacitor during the capacitor’s discharge cycle (make sure the capacitor is completely charged before starting).\u00a0 Record both \\(t\\) and \\(V\\) starting at \\(t=0\\) \\({\\rm s}\\) and ending at \\(t=60\\) \\({\\rm s}\\).\u00a0 Use a reasonable time interval that allows you and your partner to accurately record both \\(t\\) and \\(V\\).\u00a0 The time interval should be as small as possible while still allowing for accurate measurements.\u00a0 Calculate the uncertainty of one representative \\(V\\) value (one near the middle of the cycle).\u00a0 Use your best estimate for \\(\\delta t\\).<\/p>\nPlot \\(V\\) vs. \\(t\\).\u00a0 Place error bars on the data point for which you determined\u00a0 \\(\\delta V\\) and \\(\\delta t\\).<\/p>\nUse your plot to graphically determine the circuit’s time constant \\(\\tau\\)<\/span>.\u00a0 (Reminder: V at t=0 is Vo)<\/p>\nNow make sure the capacitor is fully discharged.\u00a0 Using the timer and BK DMM, measure the voltage across the capacitor during the capacitor’s charging cycle.\u00a0 Again record both \\(t\\) and \\(V\\) starting at \\(t=0\\) \\({\\rm s}\\) and ending at \\(t=60\\) \\({\\rm s}\\). Use the same reasonable time interval that you used above.\u00a0 Calculate the uncertainty of one representative \\(V\\) value (one near the middle of the cycle).\u00a0 Use your best estimate for \\(\\delta t\\).<\/p>\nPlot \\(V\\) vs. \\(t\\).\u00a0 Place error bars on the data point for which you determined\u00a0 \\(\\delta V\\) and \\(\\delta t\\).<\/p>\nUse your plot to graphically determine the circuit’s time constant \\(\\tau\\)<\/span>.<\/p>\nPart 3: Straight Line Analysis using Semi-Log Graph Paper<\/h2>\nIn 1130 Physics Lab we took non-linear data and made it linear thus allowing us to use straight line analysis to determine important characteristics of a system.\u00a0 It is possible to plot our data as a straight line by using a special type of graph paper called semi-logarithmic (semi-log) graph paper.\u00a0 Exponential functions plot as straight lines on semi-log graph paper.\u00a0 Look at the sheet of semi-logarithmic graph paper that was provided to you.\u00a0 Notice that the vertical scale is logarithmic while the horizontal scale is linear.<\/p>\nExponential functions have the form:$$ y = k e^{mx} $$ or $$ \\ln y = mx + \\ln k$$<\/p>\nwhere \\(m\\) is any positive or negative constant.\u00a0 Notice that the second equation is that of a line.\u00a0 On semi-log graph paper \\(m = {\\rm the\\;slope\\;of\\;the\\;line}\\) and \\(k = {\\rm the\\;y-intercept}\\).<\/p>\nEarlier we learned that the equation describing the discharge of a capacitor in series with a resistor is:$$ V(t) = V_{0}e^{-t\/(RC)}$$ \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>\n<\/em><\/p>\nThis is clearly an exponential function.\u00a0 Now plot \\(V\\) vs. \\(t\\) on the semi-log graph paper using the data you collected when the capacitor was being discharged.\u00a0 Note that you do not have to take the natural log of \\(V\\) before you plot it because the vertical scale is already logarithmic.\u00a0 Draw the best fit line and calculate its slope* without uncertainty.<\/p>\n*Calculating the slope of a line plotted on semi-log graph paper is a little different than that of calculating the slope of a line plotted on linear graph paper. Because semi-log graph paper uses two different scales, linear and logarithmic, it is necessary to convert the data of one axis to match that of the other before calculating the slope.\u00a0 The most common method is to convert the logarithmic axis data to linear data by taking its natural log.\u00a0 Therefore you must first calculate the natural log of your two \\(Y\\) values before you use them to calculate \\(\\Delta Y\\) in your slope equation.<\/p>\nThus: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 $${\\rm Slope} = \\frac{\\ln Y_{2} – \\ln Y_{1}}{X_{2} -X_{1}}$$<\/p>\nNote that \\(\\ln Y_{2}-\\ln Y_{1} = \\ln (Y_{2}\/Y_{1})\\), therefore it is unit-less.<\/p>\nCompare the time constants calculated from your three plots with the one calculated with the measured \\(R\\) and \\(C\\) values.\u00a0 Do they agree within reason (keeping in mind that we have not calculated the uncertainty in the graphically calculated values)? What do you think caused the largest error in your values calculated with the plots?<\/p>\nBackground<\/a> Equipment<\/a> Summary Questions<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"Part 1: Measuring Component Values Remove the ohmic resistor that is attached to the white sheet of paper.\u00a0 Use the BK DMM to measure its resistance (always make sure to record the meter range that you are using whenever you measure a value with a DMM). Calculate the value’s uncertainty. Now remove the capacitor that […]<\/p>\n","protected":false},"author":293,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2039","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2039","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/users\/293"}],"replies":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/comments?post=2039"}],"version-history":[{"count":0,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2039\/revisions"}],"wp:attachment":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/media?parent=2039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}](\"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-content\/uploads\/sites\/105\/2018\/02\/bk-meter.jpg\"){kind=link}
![timer<\/a> and BK DMM, measure the voltage across the capacitor during the capacitor’s discharge cycle (make sure the capacitor is completely charged before starting).\u00a0 Record both \\(t\\) and \\(V\\) starting at \\(t=0\\) \\({\\rm s}\\) and ending at \\(t=60\\) \\({\\rm s}\\).\u00a0 Use a reasonable time interval that allows you and your partner to accurately record both \\(t\\) and \\(V\\).\u00a0 The time interval should be as small as possible while still allowing for accurate measurements.\u00a0 Calculate the uncertainty of one representative \\(V\\) value (one near the middle of the cycle).\u00a0 Use your best estimate for \\(\\delta t\\).<\/p>\nPlot \\(V\\) vs. \\(t\\).\u00a0 Place error bars on the data point for which you determined\u00a0 \\(\\delta V\\) and \\(\\delta t\\).<\/p>\nUse your plot to graphically determine the circuit’s time constant \\(\\tau\\)<\/span>.\u00a0 (Reminder: V at t=0 is Vo)<\/p>\nNow make sure the capacitor is fully discharged.\u00a0 Using the timer and BK DMM, measure the voltage across the capacitor during the capacitor’s charging cycle.\u00a0 Again record both \\(t\\) and \\(V\\) starting at \\(t=0\\) \\({\\rm s}\\) and ending at \\(t=60\\) \\({\\rm s}\\). Use the same reasonable time interval that you used above.\u00a0 Calculate the uncertainty of one representative \\(V\\) value (one near the middle of the cycle).\u00a0 Use your best estimate for \\(\\delta t\\).<\/p>\nPlot \\(V\\) vs. \\(t\\).\u00a0 Place error bars on the data point for which you determined\u00a0 \\(\\delta V\\) and \\(\\delta t\\).<\/p>\nUse your plot to graphically determine the circuit’s time constant \\(\\tau\\)<\/span>.<\/p>\nPart 3: Straight Line Analysis using Semi-Log Graph Paper<\/h2>\nIn 1130 Physics Lab we took non-linear data and made it linear thus allowing us to use straight line analysis to determine important characteristics of a system.\u00a0 It is possible to plot our data as a straight line by using a special type of graph paper called semi-logarithmic (semi-log) graph paper.\u00a0 Exponential functions plot as straight lines on semi-log graph paper.\u00a0 Look at the sheet of semi-logarithmic graph paper that was provided to you.\u00a0 Notice that the vertical scale is logarithmic while the horizontal scale is linear.<\/p>\nExponential functions have the form:$$ y = k e^{mx} $$ or $$ \\ln y = mx + \\ln k$$<\/p>\nwhere \\(m\\) is any positive or negative constant.\u00a0 Notice that the second equation is that of a line.\u00a0 On semi-log graph paper \\(m = {\\rm the\\;slope\\;of\\;the\\;line}\\) and \\(k = {\\rm the\\;y-intercept}\\).<\/p>\nEarlier we learned that the equation describing the discharge of a capacitor in series with a resistor is:$$ V(t) = V_{0}e^{-t\/(RC)}$$ \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>\n<\/em><\/p>\nThis is clearly an exponential function.\u00a0 Now plot \\(V\\) vs. \\(t\\) on the semi-log graph paper using the data you collected when the capacitor was being discharged.\u00a0 Note that you do not have to take the natural log of \\(V\\) before you plot it because the vertical scale is already logarithmic.\u00a0 Draw the best fit line and calculate its slope* without uncertainty.<\/p>\n*Calculating the slope of a line plotted on semi-log graph paper is a little different than that of calculating the slope of a line plotted on linear graph paper. Because semi-log graph paper uses two different scales, linear and logarithmic, it is necessary to convert the data of one axis to match that of the other before calculating the slope.\u00a0 The most common method is to convert the logarithmic axis data to linear data by taking its natural log.\u00a0 Therefore you must first calculate the natural log of your two \\(Y\\) values before you use them to calculate \\(\\Delta Y\\) in your slope equation.<\/p>\nThus: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 $${\\rm Slope} = \\frac{\\ln Y_{2} – \\ln Y_{1}}{X_{2} -X_{1}}$$<\/p>\nNote that \\(\\ln Y_{2}-\\ln Y_{1} = \\ln (Y_{2}\/Y_{1})\\), therefore it is unit-less.<\/p>\nCompare the time constants calculated from your three plots with the one calculated with the measured \\(R\\) and \\(C\\) values.\u00a0 Do they agree within reason (keeping in mind that we have not calculated the uncertainty in the graphically calculated values)? What do you think caused the largest error in your values calculated with the plots?<\/p>\nBackground<\/a> Equipment<\/a> Summary Questions<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"Part 1: Measuring Component Values Remove the ohmic resistor that is attached to the white sheet of paper.\u00a0 Use the BK DMM to measure its resistance (always make sure to record the meter range that you are using whenever you measure a value with a DMM). Calculate the value’s uncertainty. Now remove the capacitor that […]<\/p>\n","protected":false},"author":293,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2039","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2039","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/users\/293"}],"replies":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/comments?post=2039"}],"version-history":[{"count":0,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2039\/revisions"}],"wp:attachment":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/media?parent=2039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}](\"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-content\/uploads\/sites\/105\/2017\/08\/timer.jpg\"){kind=link}