The circuit in Figure P28.37 has been connected for a long...

Name: Problem 37, Chapter 28: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2020-09-14T07:02:21-08:00
Duration: 5 min 41 s
Channel: Sheh Lit Chang
Description: The circuit in Figure P28.37 has been connected for a long time. (a) What is the potential difference across the capacitor? (b) If the battery is disconnected from the circuit, over what time interval does the capacitor discharge to one-tenth its initial voltage?

The circuit in Figure P28.37 has been connected for a long time. (a) What is the potential difference across the capacitor? (b) If the battery is disconnected from the circuit, over what time interval does the capacitor discharge to one-tenth its initial voltage?

Key Concepts

Steady-State Behavior in DC Circuits

In a DC circuit that has been connected for a long time, the capacitor reaches a steady state where it is fully charged. In this state, the capacitor behaves like an open circuit, meaning no further current flows into or out of it, and the voltage across the capacitor essentially equals the voltage of the source (taking into account any resistive voltage dividers present in the circuit).

RC Time Constant

The RC time constant, denoted by ? (tau), is defined as the product of the resistance (R) and the capacitance (C) in an RC circuit. It characterizes the rate at which the capacitor charges or discharges; specifically, it is the time it takes for the voltage across the capacitor to either rise to about 63% of its final value during charging or decrease to about 37% of its initial value during discharging.

Exponential Decay in Capacitor Discharge

During the discharging phase of a capacitor in an RC circuit, the voltage decreases exponentially over time according to the formula V(t) = V?e^(–t/RC), where V? is the initial voltage. Calculating the time it takes for the capacitor to discharge to a specific fraction of V?, such as one-tenth, involves solving this exponential equation, illustrating how the RC time constant governs the rate of decay.

Transcript

00:01 In this question, we are given this circuit.

00:03 It has been connected for a long time.

00:07 So there are two parts in this question.

00:09 In parts a, we want to find the potential difference across the capacitor.

00:14 Okay, so one thing we need to recognize is that when the switch, when the whole thing is connected for a long time, there will be no current flow in the capacitor.

00:27 So the current will be flowing in the resistor, so effectively the diagram becomes, you can kind of ignore the presence of the capacitor for the moment, okay, yeah, so you have this circuit diagram and you can kind of ignore the capacitor for a moment, okay, we just want to find a potential difference for the capacitor.

01:01 So what we need is to find out the potential at these two points.

01:15 Okay, that would be the, there will be the, then we take a difference, then we can find a potential difference across the capacitor.

01:24 Okay, so this is a parallel connection.

01:29 Okay, so here we can wots, 10 words, all right, and then you can kind of imagine that the, the current here will be 2m.

01:41 We can calculate to be 2 ms.

01:44 Here will be 1m, right? then after, so for the left branch, so after going through the 1 -oam resistor, the potential here will be 8 volts, and then here will be 2 volts for the right branch by just using v -i -r.

02:08 By r.

02:09 Okay, so here the potential difference, based on that, the potential difference across the capacitor is equal to 8 minus 2, it goes to 6 volts...

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