{"id":2027,"date":"2017-08-30T10:12:28","date_gmt":"2017-08-30T14:12:28","guid":{"rendered":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/?page_id=2027"},"modified":"2020-02-10T09:14:57","modified_gmt":"2020-02-10T14:14:57","slug":"lab-2-background-draft","status":"publish","type":"page","link":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/lab-2-background-draft\/","title":{"rendered":"Lab 2: Background"},"content":{"rendered":"

A capacitor is a device that stores charge.\u00a0 It turns out that the amount of charge stored (\\(+q\\) on one “plate”, \\(-q\\) on the other) is proportional to the voltage difference \\(\\Delta V\\) across the capacitor, where the proportionality constant \\(C\\) depends on the geometry of the device and on the details of what’s between the plates (vacuum?\u00a0 a cell membrane<\/a>?):$$ q = C \\Delta V.$$<\/p>\n

We call the proportionality constant \\(C\\) the capacitance<\/em>.\u00a0 Don’t confuse it with the abbreviation for coulomb, \\(\\rm{C}\\)!\u00a0 Capacitance is measured in farads, \\(\\rm{F}\\), where \\(1\\:\\rm{F}=1\\:\\rm{C}\/\\rm{V}\\).\u00a0 As you will see, typical capacitors have capacitance values that are only small fractions of a farad.<\/p>\n

It takes time to charge (or discharge) a capacitor.\u00a0 You can show that if you put an uncharged capacitor in series with a battery and a resistor at \\(t=0\\), the voltage across the capacitor at time \\(t\u22650\\) will be:$$ V(t) = V_{\\rm Source}(1-e^{-t\/\\tau}).$$<\/p>\n

Similarly, if the plates of a charged capacitor are connected by a resistor, the capacitor will discharge according to:$$ V(t) = V_{0}e^{-t\/\\tau},$$<\/p>\n

where \\(V_{0}\\) is the initial voltage across the capacitor.<\/p>\n

In the equations above we used the time constant \\(\\tau=RC\\), which characterizes how long it takes to charge or discharge a capacitor by some amount; \\(1\\) time constant is the time it takes to gain \\(63\\%\\) of the final charge when charging a capacitor, or to lose \\(63\\%\\) of the initial charge when discharging a capacitor.\u00a0 Measuring time constants for \\(RC\\) circuits provides us with a nice way to measure capacitance if we know resistance, or vice versa.<\/p>\n

Below are two plots.\u00a0 The first shows the voltage across a capacitor as a function of time as the capacitor is being charged.\u00a0 The second shows the voltage across the capacitor as a function of time as the capacitor is being discharged.<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Equipment<\/a> Data Collection and Analysis<\/a>
\nSummary Questions<\/a><\/p>\n

\"badge\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

A capacitor is a device that stores charge.\u00a0 It turns out that the amount of charge stored (\\(+q\\) on one “plate”, \\(-q\\) on the other) is proportional to the voltage difference \\(\\Delta V\\) across the capacitor, where the proportionality constant \\(C\\) depends on the geometry of the device and on the details of what’s between […]<\/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-2027","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2027","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=2027"}],"version-history":[{"count":0,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/2027\/revisions"}],"wp:attachment":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/media?parent=2027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}