In the circuit of Figure P 28.39 , the switch S has been...

In the circuit of Figure $P 28.39$ , the switch $S$ has been open for a long time. It is then suddenly closed. Take $\varepsilon=10.0 \mathrm{V}$ , $R_{1}=50.0 \mathrm{k} \Omega, R_{2}=100 \mathrm{k} \Omega,$ and $C=10.0 \mu \mathrm{F}$ . Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at $t=0$ . Determine the current in the switch as a function of time.

Key Concepts

Initial Conditions in Capacitive Circuits

Initial conditions, particularly the initial charge or voltage on the capacitor, play a critical role in determining the transient response of a circuit. Because a capacitor cannot change its voltage instantaneously, the pre-switching condition directly affects how the voltage and current behave immediately after the switch is closed, ensuring that the analysis of the circuit's behavior accurately reflects its physical constraints.

Exponential Response

The exponential response in RC circuits refers to the characteristic behavior of the capacitor voltage and circuit current, which change exponentially with time after a sudden circuit change. This exponential function, governed by the time constant, accurately models the rate at which the capacitor charges or discharges, and is used to predict circuit performance at any moment during the transient phase.

Time Constant

The time constant, usually denoted by ? (tau), is a crucial parameter in RC circuits and is defined as the product of the effective resistance seen by the capacitor and the capacitance (? = RC). It quantifies the rate at which the capacitor charges or discharges, indicating how quickly the circuit reaches steady-state after a sudden change, such as when a switch is closed.

RC Circuits

RC circuits are electrical circuits containing resistors and capacitors, and they are fundamental to understanding how circuits respond to changes over time. In these circuits, the capacitor charges or discharges through the resistor, leading to time-dependent behavior that is critical for analyzing both transient and steady-state responses in a wide range of applications.

Transient Analysis

Transient analysis deals with the behavior of circuit variables such as voltage and current immediately after a change in circuit conditions, for example, when a switch is closed or opened. This analysis is essential for understanding how circuits evolve from one steady state to another, and it typically involves solving differential equations that describe exponential charging or discharging processes in RC circuits.

Transcript

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

00:03 So, and the switch is open for a long time.

00:08 And then suddenly the switch is closed.

00:10 So when the switch is closed, the circuit becomes the one at the right.

00:16 We are given that the emf of the battery is 10 volts.

00:19 R1 is 50 kilo -ooms, r2 is 100 kilo -ooms, and the capacitor, capacitance is 10 micro -farrid.

00:28 So in part a, we want to determine the time constant before the switch is closed.

00:38 So the power before is r equivalent before times c.

00:50 So this is just r1 plus r2, the total resistance.

00:55 So you can see that the battery, the r1 and r2 is series with the capacitor.

01:05 Okay, so 150 times 10 to the 3 times 10 to the minus 6.

01:16 So we get 1 .5 seconds.

01:21 Okay, so this is the answer for pi a.

01:24 And then in part b, we need to calculate the current, the time constant after the switch is closed.

01:35 So after the switch is closed the capacitor will discharge through r2 okay so after the switch is closed the capacitor will discharge through r2 okay so after the switch is closed okay capacitor discharges through r2 okay so the time constants after is r2 times c so this is equal to times 10 to the tree times 10 to the minus 6 so you get one second okay okay then in part c so let the switch be closed at t goes to zero determine the current in the switch as a function of time okay so so just look at the circuit again okay, so the switch is here...

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