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Unformatted text preview: Capacitors and calculus Capacitors do not have a stable "resistance" as conductors do. However, there is a defnite mathematical relationship between voltage and current For a capacitor, as Follows: The lower-case letter "i" symbolizes instantaneous current, which means the amount oF current at a specifc point in time. This stands in contrast to constant current or average current (capital letter "I") over an unspecifed period oF time. The expression "dv/dt" is one borrowed From calculus, meaning the instantaneous rate oF voltage change over time, or the rate oF change oF voltage (volts per second increase or decrease) at a specifc point in time, the same specifc point in time that the instantaneous current is reFerenced at. or whatever reason, the letter v is usually used to represent instantaneous voltage rather than the letter e . However, it would not be incorrect to express the instantaneous voltage rate-oF-change as "de/dt" instead. In this equation we see something novel to our experience thusFar with electric circuits: the variable oF time . When relating the quantities oF voltage, current, and resistance to a resistor, it doesn't matter iF we're dealing with measurements taken over an unspecifed period oF time (E=IR; V=IR), or at a specifc moment in time (e=ir; v=ir). The same basic Formula holds true, because time is irrelevant to voltage, current, and resistance in a component like a resistor. In a capacitor, however, time is an essential variable, because current is related to how rapidly voltage changes over time. To Fully understand this, a Few illustrations may be necessary. Suppose we were to connect a capacitor to a variable-voltage source, constructed with a potentiometer and a battery: IF the potentiometer mechanism remains in a single position (wiper is stationary), the voltmeter connected across the capacitor will register a constant (unchanging) voltage, and the ammeter will register 0 amps. In this scenario, the instantaneous rate oF voltage change (dv/dt) is equal to zero, because the voltage is unchanging. The equation tells us that with 0 volts per second change For a dv/dt, there must be zero instantaneous current (i). rom a physical perspective, with no change instantaneous current (i)....
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This note was uploaded on 04/21/2008 for the course BIO 101 taught by Professor Pott-santone during the Spring '08 term at Northeastern.
- Spring '08