A 2.36-μF capacitor that is initially uncharged is connected...

A $2.36-\mu \mathrm{F}$ capacitor that is initially uncharged is connected in series with a $5.86-\Omega$ resistor and an emf source with $\mathcal{E}=120 \mathrm{V}$ and negligible internal resistance. (a) Just after the connection is made, what are (i) the rate at which electrical energy is being dissipated in the resistor; (ii) the rate at which the electrical energy stored in the capacitor is increasing; (iii) the electrical power output of the source? How do the answers to parts (i), (ii), and (iii) compare? (b) Answer the same questions as in part (a) at a long time after the connection is made. (c) Answer the same questions as in part (a) at the instant when the charge on the capacitor is one-half its final value.

A $2.36-\mu \mathrm{F}$ capacitor that is initially uncharged is connected in series with a $5.86-\Omega$ resistor and an emf source with
$\mathcal{E}=120 \mathrm{V}$ and negligible internal resistance. (a) Just after the
connection is made, what are (i) the rate at which electrical energy
is being dissipated in the resistor; (ii) the rate at which the electrical energy stored in the capacitor is increasing; (iii) the electrical power output of the source? How do the answers to parts (i), (ii),
and (iii) compare? (b) Answer the same questions as in part (a) at a long time after the connection is made. (c) Answer the same questions as in part (a) at the instant when the charge on the capacitor is
one-half its final value.

Key Concepts

RC Circuit

An RC circuit is a fundamental electrical circuit consisting of a resistor and a capacitor connected in series or parallel with a voltage source. Its behavior is characterized by the charging and discharging of the capacitor through the resistor, leading to exponential changes in voltage and current over time. This concept is essential for understanding both the transient and steady?state responses observed in many electronic and electrical systems.

Transient Response

The transient response of a circuit refers to its behavior immediately after a change in voltage, such as when a switch is closed. In an RC circuit, the initial conditions (like an uncharged capacitor) lead to a rapid change in current and voltage, which then decays or rises exponentially toward a steady state. This analysis is crucial for determining values such as initial current, power dissipation, and energy storage rates at various times during the charging process.

Energy Dissipation in a Resistor

The resistor in a circuit converts electrical energy into heat through the process known as Joule heating. The rate of energy dissipation is given by the power formula P = I²R, where I is the instantaneous current. In transient analysis of RC circuits, this dissipation is highest immediately after the circuit is activated and decreases as the current decays over time.

Capacitor Energy Storage

Capacitors store energy in the electric field created between their plates, with the stored energy expressed as U = (1/2) C V², where C is the capacitance and V is the voltage across the capacitor. During the charging process in an RC circuit, the rate at which energy is accumulated by the capacitor changes over time, starting from zero and asymptotically approaching its maximum value as the voltage across it increases.

Conservation of Energy and Power Balance

In any electrical circuit, the conservation of energy principle dictates that the total power provided by the source must equal the sum of the power being dissipated (for instance, as heat in resistors) and the rate at which energy is stored (such as in a capacitor). This balance is essential for analyzing circuits during transient conditions, ensuring that all energy contributions and losses are properly accounted for.

Transcript

00:01 For this question, we are looking at a capacitor being attached in series, a resistor, and an emf source.

00:13 So it's quite a simple setup.

00:15 We are given all of the values such as the emf, the resistance, as well as the capacitance.

00:27 What you want to find is power dissipated by our resistor, the energy being stored by our capacitor and also the power dissipated by the source right at three different points in time so the first point in time will be just when the circuit is formed right t equals to zero when the current just started to flow number two is when t is equal to infinite so after a very long time and finally the third point in time is when the charge is equals to half of its final value.

01:20 I'm going to start off with the first one, number one when t goes to zero and just started the entire circuit.

01:31 So the capacitor can be ignored, right, can assume that it's just a piece of wire.

01:36 There's no charge stored in it yet or anything so it can just ignore and therefore the power dissipated by our resistor it's just using p equals to v square over r you can take the entire emf of the battery to be the potential drop across the resistor to give us to 460 watts now for how much the capacitor is actually charging, right, the rate of charge.

02:27 We're going to use the equation u equals to q square over 2c.

02:33 And therefore the rate of charging of this electrical energy is du dt.

02:44 D differentiating this, the only one that is dependent on time is q.

02:48 Right, c is a constant so we can keep that.

02:50 Differentiating q, q square, we get 2 q, dqdt.

02:58 You know that the qdt is just the current.

03:03 So this is q times i over c.

03:09 Now because there is no charge, that is stored in the capacitor yet, so q is 0, and therefore this entire thing is 0...