(a) Let, at any moment of time, charge on the plates be (q0-q) then current through the resistor

step 1 Have friends let at any moment charge on plates of the capacitor Q0 minus Q. So current flowing

(a) Let, at any moment of time, charge on the plates be...

(a) Let, at any moment of time, charge on the plates be $\left(q_{0}-q\right)$ then current through the resistor, $i=-\frac{d\left(q_{0}-q\right)}{d t}$, because the capacitor is discharging. or, $i=\frac{d q}{d t}$ Now, applying loop rule in the circuit, $i R-\frac{q_{0}-q}{C}=0$ or, $\frac{d q}{d t} R=\frac{q_{0}-q}{C}$ or, $\frac{d q}{q_{0}-q}=\frac{1}{R C} d t$ At $t=0, q=0$ and at $t=\tau, q=q$ So, $$ \ln \frac{q_{0}-q}{q_{0}}=\frac{-\tau}{R C} $$ Thus $$ q=q_{0}\left(1-e^{-\tau / R C}\right)=0 \cdot 18 \mathrm{mC} $$ (b) Amount of heat generated $=$ decrement in capacitance energy $$ \begin{gathered} =\frac{1}{2} \frac{q_{0}^{2}}{C}-\frac{1}{2} \frac{\left[q_{0}-q_{0}\left(1-e^{-\tau / R C}\right)\right]^{2}}{C} \\ =\frac{1}{2} \frac{q_{0}^{2}}{C}\left[1-e^{-\frac{2 \tau}{R C}}\right]=82 \mathrm{~mJ} \end{gathered} $$

(a) Let, at any moment of time, charge on the plates be $\left(q_{0}-q\right)$ then current through the resistor, $i=-\frac{d\left(q_{0}-q\right)}{d t}$, because the capacitor is discharging.
or, $i=\frac{d q}{d t}$
Now, applying loop rule in the circuit, $i R-\frac{q_{0}-q}{C}=0$
or, $\frac{d q}{d t} R=\frac{q_{0}-q}{C}$
or, $\frac{d q}{q_{0}-q}=\frac{1}{R C} d t$
At $t=0, q=0$ and at $t=\tau, q=q$
So,
$$
\ln \frac{q_{0}-q}{q_{0}}=\frac{-\tau}{R C}
$$
Thus
$$
q=q_{0}\left(1-e^{-\tau / R C}\right)=0 \cdot 18 \mathrm{mC}
$$
(b) Amount of heat generated $=$ decrement in capacitance energy
$$
\begin{gathered}
=\frac{1}{2} \frac{q_{0}^{2}}{C}-\frac{1}{2} \frac{\left[q_{0}-q_{0}\left(1-e^{-\tau / R C}\right)\right]^{2}}{C} \\
=\frac{1}{2} \frac{q_{0}^{2}}{C}\left[1-e^{-\frac{2 \tau}{R C}}\right]=82 \mathrm{~mJ}
\end{gathered}
$$

Key Concepts

Energy Dissipation in Circuits

In circuits containing resistors and capacitors, energy stored in the capacitor can be partially lost as heat through the resistor when the capacitor discharges. The difference in the energy stored in the capacitor between its initial and final states represents the energy dissipated, which is an important concept in understanding energy conservation and losses in electrical circuits.

Capacitor Charging and Discharging Dynamics

The dynamics of capacitor charging and discharging involve exponential functions that characterize how the voltage across the capacitor or the charge stored evolves over time. When charging, the capacitor's voltage increases asymptotically to the supply voltage; during discharging, it decreases exponentially, both processes being governed by the RC time constant.

Differential Equations in Circuit Analysis

The transient behavior of RC circuits is modeled using differential equations derived from Kirchhoff's loop rule. These equations relate the rate of change of charge or current to the voltage across components and are typically solved to yield expressions that describe exponential charging or discharging dynamics in the circuit.

Kirchhoff's Loop Rule

Kirchhoff's loop rule is a principle in circuit analysis stating that the directed sum of the electrical potential differences (voltage) around any closed loop in a circuit must be zero. This concept is essential for setting up the differential equations that describe the voltage drops across circuit elements, such as resistors and capacitors, during transient processes.

RC Circuits

RC circuits consist of a resistor (R) and a capacitor (C) connected in series or parallel, and they are fundamental in analyzing transient behaviors in electrical circuits. The combination of R and C defines a characteristic time constant (? = RC) that determines how quickly the capacitor charges or discharges, making RC circuits a common topic in studying time-dependent responses in electronics.

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