Chapter 10, RLC and AC Circuits Video Solutions, Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Problem 1

Find the equivalent capacitance between the two left-most terminals in the following infinite ladder of capacitors.
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Vishal Gupta

Numerade Educator

03:51

For $t<0$, the switch in the set-up on the right is open and the capacitor stores no charge. At $t=0$, the switch is closed. Determine the current through the inductor as a function of time. The relevant emf, resistance, capacitance and inductance are $\varepsilon, R, C$ and $L$ respectively, with $L>4 R^2 C$.
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Keshav Singh

Numerade Educator

05:28

Problem 2

Determine the equivalent capacitance of the circuit, shown in the figure, across terminals A and B. Determine the charge stored by each capacitor when a battery with an emf of $21 \mathrm{~V}$ is connected between A and B, with its positive terminal pointing towards $\mathrm{A}$.
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Josh Broderick Phillips

Numerade Educator

01:10

Problem 3

The switch $\mathrm{S}$ is initially closed towards terminal A until the system has reached a steady state. Afterwards, the switch is changed to terminal B. Find the final charges on each of the capacitors with capacitances $4 F, 6 F$ and $3 F$.
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Hunza Gilgit

Numerade Educator

06:03

Problem 4

Determine the charges stored by the capacitors if the switch is closed for a long time, given that the capacitors start from a configuration with zero stored charge. What if the switch is opened from the start instead?
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Dading Chen

Numerade Educator

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Problem 5

The system below has reached a steady state after a long time. Find the final charge on capacitor A. All batteries, resistors and capacitors have an emf, resistance and capacitance of $\varepsilon, R$ and $C$ respectively.
Figure 10.20: Gargantuan circuit
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Lainey Roebuck

Numerade Educator

07:51

Problem 6

Four ideal batteries of emfs $\varepsilon_1=4 \mathrm{~V}, \varepsilon_2=8 \mathrm{~V}, \varepsilon_3=12 \mathrm{~V}$ and $\varepsilon_4=16 \mathrm{~V}$, four capacitors with identical capacitances $C_1=C_2=C_3=C_4=1 \mathrm{~F}$, and four identical resistors are connected in the form of a cube as shown in the figure. Compute the total energy $U$ stored by the capacitors after a steady state has been attained. Now, suppose points $\mathrm{H}$ and $\mathrm{B}$ are connected by an ideal wire. Find the charge stored by capacitor $C_2$ in the new steady state configuration. (International Physics Olympiad)
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Andrew Duncan

Numerade Educator

07:37

Problem 7

The capacitors on the right all have the same surface area, $A$. The separations between the two plates of the capacitors with capacitance $C$ are $d$. Now, a new capacitor plate, of total charge $Q_0$ and surface area $A$, is inserted at a distance $x$ from the left plate of the $\frac{C}{2}$ capacitor. After the system has equilibrated, what is the final charge on the left plate of the capacitor (labeled as B on the diagram) that had an original capacitance $\frac{C}{2}$ ? (Chinese Physics Olympiad)
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Vishal Gupta

Numerade Educator

04:07

Problem 8

Determine the potential difference $V(t)$ across the capacitor as a function of time $t$ for $t \geq 0$ if the capacitor does not store any charge at $t=0$.
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Anthony Ramos

Numerade Educator

03:15

Problem 9

Determine the potential difference $V_3(t)$ across resistor $R_3$ and the current $I_1(t)$ through resistor $R_1$ for $t \geq 0$ if no current flows through the inductor at $t=0$.
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Nick Johnson

Numerade Educator

03:51

Problem 10

Before $t=0$, the circuit is in steady state with the switch $\mathrm{S}$ open. At $t=0$, the switch $\mathrm{S}$ is closed. Determine the current $I_L$ through and voltage $V_L$ across the inductor at $t=0^{+}$. Next, find $I_L(t)$ and $V_L(t)$ for $t \geq 0$.
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Keshav Singh

Numerade Educator

04:05

Problem 12

Determine the current through the inductor in the below figure for $t \geq 0$ if it is $1 \mathrm{~A}$ (from the left to right end) at $t=0$. Furthermore, the potential difference across the capacitor at $t=0$ is $2 \mathrm{~V}$, with the left plate having the higher potential.
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Khoobchandra Agrawal

Numerade Educator

02:36

Problem 13

For $t<0$, the switch in the set-up below is open for a long time. At $t=0$, the switch is closed. Determine the charges stored on the two capacitors as functions of time. Note that you will have to consider three regimes.
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Varsha Aggarwal

Numerade Educator

06:06

Problem 14

Two capacitors are arranged as shown in the circuit on the next page. The bottom capacitor has capacitance $C_1$ while the top capacitor has initial plate separation $d_0$ and area $A$ (the gap is filled by vacuum). The capacitors are initially held fixed and each store an equal amount of charge $Q_0$ such that there is no net charge in the portion containing the left plates of the capacitors. Determine $Q_0$. Now, suppose that the top capacitor is released such that it is free to move - with the mass of each plate being $m$. The massless wires are coiled into two heaps such that the wires are slack. Determine time as a function of the charges on the capacitors (it is difficult to invert this relationship). Warning: heavy math ahead.
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Keshav Singh

Numerade Educator

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Problem 15

Consider a circuit where a resistor, inductor and capacitor of resistance $R$, inductance $L$ and capacitance $C$ are connected in parallel to an AC source with $\operatorname{emf} \varepsilon=\varepsilon_0 \cos \omega t$. Suppose that we forgot the impedance of a capacitor but know that the impedance of the inductor is $i \omega L$. Determine the current through the $\mathrm{AC}$ source as a function of time by determining the rate of energy stored or lost by each component.
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Lainey Roebuck

Numerade Educator

06:02

Problem 16

Determine the current through the AC source as a function of time in the long run. The capacitor has a capacitance $C=\frac{1}{2 \omega^2 L}$.
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Suman Saurav Thakur

Numerade Educator

10:26

Problem 17

Consider the circuit on the next page. The resistors have resistances $R$ while the left and right inductors have self-inductances $L_1$ and $L_2$. The mutual inductance between the inductors is $M$ and their polarities are indicated by the dot convention. Finally, the capacitor has capacitance $C=\frac{1}{\omega^2 L_2}$. By applying Kirchhoff's laws and substituting complex exponential trial solutions, determine the currents through each loop in the long run. From the perspective of impedances, what is the effect of the capacitor in this set-up? Determine the phase difference between the currents.
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Linda Winkler

Numerade Educator

02:42

Problem 18

A resistor $R$ and two parallel inductors $L_1$ and $L_2$ are connected as shown in the circuit below. The two inductors have a mutual inductance $M$ and are constructively coupled. Determine the current through inductor $L_1$ as a function of time by deriving the effective impedance of each inductor.
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Shoukat Ali

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