In the circuit shown in Fig. 𝐄 26.49, C=5.90 μF, ℰ=28.0 V and the emf has negligible resistance

step 1 in the first part when switch S is in position 2 it would be RC circuit and after long time tube

In the circuit shown in Fig. 𝐄 26.49, C=5.90 μF, ℰ=28.0 V...

In the circuit shown in Fig. $\mathbf{E} 26.49, C=5.90 \mu \mathrm{F}, \mathcal{E}=28.0 \mathrm{~V}$ and the emf has negligible resistance. Initially the capacitor is uncharged and the switch $S$ 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 $S$ is moved to position $\underline{2 ?}$ (b) After $S$ 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 $S$ is moved to position 2 will the charge on the capacitor be equal to $99.0 \%$ of the final value found in part (a)?

In the circuit shown in Fig. $\mathbf{E} 26.49, C=5.90 \mu \mathrm{F}, \mathcal{E}=28.0 \mathrm{~V}$
and the emf has negligible resistance. Initially the capacitor is uncharged and the switch $S$ 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 $S$ is moved to position $\underline{2 ?}$
(b) After $S$ 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 $S$ 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 is an electrical circuit composed of a resistor (R) and a capacitor (C) connected in series or parallel with a voltage source. It is a fundamental configuration used to study transient responses, where the resistor controls the rate at which the capacitor charges or discharges, thereby affecting the time-dependent behavior of the circuit.

Capacitor Charging Equation

The capacitor charging equation, typically written as Q(t) = Q_final × (1 - e^(?t/RC)), describes how the charge on a capacitor increases over time as it charges through a resistor. This equation emerges from the solution to the differential equation governing the circuit's dynamics and is essential for analyzing time-dependent responses in RC circuits.

Time Constant (? = RC)

The time constant, denoted by ? (tau), is the product of the resistance (R) and capacitance (C) in the circuit. It represents the characteristic time over which the capacitor charges to about 63.2% of its final value or discharges to about 36.8% of its initial value. Understanding the time constant is crucial for predicting how quickly a circuit reaches steady state.

Exponential Behavior in Transient Analysis

The charging (and discharging) process in RC circuits is governed by exponential functions. This exponential behavior indicates that the capacitor’s voltage and charge approach their final values asymptotically rather than instantaneously, highlighting key dynamics in the transient response study of circuits.

Final Charge of a Capacitor

The final charge on a capacitor when it is fully charged is determined by the relation Q = CV, where C is the capacitance and V is the applied voltage. This concept is fundamental in electrostatics and circuit theory, as it links the stored charge with the physical properties of the capacitor and the external voltage source.

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