In the circuit shown in Fig. E 26.50, C=5.90 μF ℰ=28.0 V, and the emf has negligible resistanc

step 1 So in this question this is a circuit that we're given. Right, in position, we have two possible

In the circuit shown in Fig. E 26.50, C=5.90 μF ℰ=28.0 V,...

In the circuit shown in Fig. $\mathrm{E} 26.50, \quad C=5.90 \mu \mathrm{F}$ $\mathcal{E}=28.0 \mathrm{V},$ and the emf has negligible resistance. Initially the capacitor is uncharged and the switch $\mathrm{S}$ is in position $1 .$ The switch is then moved to position $2,$ so that the capacitor begins to charge. (a) What will be the charge on the capacitor a long time after the switch is moved to position 2$?$ (b) After the switch has been in position 2 for 3.00 $\mathrm{ms}$ , the charge on the capacitor is measured to be 110$\mu \mathrm{C}$ . What is the value of the resistance $R ?$ (c) How long after the switch is moved to position 2 will the charge on the capacitor be equal to 99.0$\%$ of the final value found in part (a)?

Key Concepts

RC Circuit

An RC circuit consists of a resistor and a capacitor connected either in series or parallel, which is used to analyze how voltages and currents change over time in the circuit. It is particularly important in studying transient behaviors, such as how a capacitor charges or discharges when the circuit configuration is altered.

Capacitor

A capacitor is an electrical component that stores energy in an electric field created between two conductive plates separated by an insulating material. Its ability to store charge is defined by its capacitance (C), and the relationship between the stored charge (Q) and the voltage (V) is given by Q = C × V.

Time Constant

The time constant (?) of an RC circuit, defined as ? = R × C, provides a measure of how quickly the circuit responds to changes. It characterizes the rate of charging or discharging, indicating that after a duration of one time constant, the capacitor voltage reaches about 63.2% of its final value in a charging scenario.

Exponential Charging Equation

The exponential charging equation, Q(t) = Q_final (1 - e^(-t/RC)), describes how the charge on a capacitor builds over time in an RC circuit. This equation shows that the charging process is gradual, with the capacitor’s charge asymptotically approaching its maximum value, and emphasizes the role of the time constant in governing the speed of the charging process.

Transcript

00:01 So in this question this is a circuit that we're given.

00:04 Right, in position, we have two possible positions for the switch over here, switch as, can be position one, where it'll be connected to just the resistor.

00:17 So this is kind of like the discharging position.

00:21 And the other position would be number two, which is, across the emf battery.

00:29 So across the battery, we would be, charging our capacitor.

00:34 So these are the two different modes.

00:36 So by given that at first it starts off at 1, where the capacitor is completely uncharged, and then we move on to 2.

00:44 And we allow it to stay for very, very long time on 2.

00:49 We want to find what is the total charge that has accumulated on the capacitor for a very long time.

00:56 So basically, we can use the capacitor equation q is equals to cv where v would be the emf of the battery because after a very very long time the emf of the battery will be just the potential drop across the capacitor because the capacitor will be acting like an open circuit after a very long time so the voltage across it just the emf this case the capacitance given to be 5 .9 micro ferrets and the voltage emf is 28 volts.

01:34 Therefore, the charge is given as such.

01:41 Now of course, what we want to find next is actually the charge, but only after 3 milliseconds.

01:51 It is considered to be 110 times 10 power minus 6 columns.

01:57 This is for t equals to 3 milliseconds.

02:03 You want to find what is the resistance r based on this.

02:09 So what we actually need is the equation that governs the increase in charge, where q is equals to q0, multiplied by 1 minus exponential negative t over rc.

02:28 So this tells us what is the amount of charge at time t for here...

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