PH 220: Physics for Scientists & Engineers II
Exam 2 Comprehensive Problem Set & Practice Worksheet
Unit 2 • Electric Potential (Ch 7) • Capacitance & Dielectrics (Ch 8) • Current & Resistance (Ch 9) • DC Circuits (Ch 10)
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Unit 2 Official Formula Reference Sheet
Exam 2 Permitted Equations
Electric Potential & Work (Ch 7)
$\Delta V = V_B - V_A = -\int_A^B \vec{E}\cdot d\vec{l}$
$V_{pt} = \frac{k q}{r}, \quad U_{pt} = \frac{k q_1 q_2}{r}$
$\Delta U = q \Delta V, \quad W_{\text{field}} = -\Delta U$
$\vec{E} = -\vec{\nabla} V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$
$V(x) = k \int \frac{dq}{r}$
Capacitance & Dielectrics (Ch 8)
$C = \frac{Q}{V}, \quad C_0 = \frac{\varepsilon_0 A}{d}, \quad C = \kappa C_0$
$C_{\text{parallel}} = \sum C_i, \quad \frac{1}{C_{\text{series}}} = \sum \frac{1}{C_i}$
$U_C = \frac{1}{2} C V^2 = \frac{Q^2}{2C} = \frac{1}{2} Q V$
$u_E = \frac{1}{2}\varepsilon_0 E^2, \quad E = \frac{E_0}{\kappa}$
$\varepsilon_0 = 8.854 \times 10^{-12}\text{ C}^2/(\text{N}\cdot\text{m}^2)$
Current, Resistance & DC Circuits (Ch 9–10)
$I = \frac{dQ}{dt} = n q v_d A, \quad \vec{J} = \sigma \vec{E} = \frac{\vec{E}}{\rho}$
$R = \frac{\rho L}{A}, \quad V = IR, \quad P = IV = I^2 R = \frac{V^2}{R}$
$\sum I_{\text{in}} = \sum I_{\text{out}}, \quad \sum \Delta V = 0$
$R_{\text{series}} = \sum R_i, \quad \frac{1}{R_{\text{parallel}}} = \sum \frac{1}{R_i}$
RC Charge: $q(t) = C\mathcal{E}(1 - e^{-t/\tau}), \quad \tau = RC$
RC Discharge: $q(t) = Q_0 e^{-t/\tau}, \quad I(t) = -\frac{Q_0}{\tau}e^{-t/\tau}$
PRACTICE EXAM
Practice Exam #1
— Zero Potential Position Between Point Charges
If a $+27.8\ \mu\text{C}$ charge is placed $35.0\text{ cm}$ from a $-18.8\ \mu\text{C}$ charge, where would an electric potential of zero exist?
Workspace / Reasoning
Selected Answer:
[ ]
PRACTICE EXAM
Practice Exam #2
— Flashlight Battery Configuration for Maximum Runtime
Which configuration would have a flashlight last the longest, assuming all batteries are created equal and connected as shown?
Workspace / Reasoning
Selected Answer:
[ ]
PRACTICE EXAM
Practice Exam #3
— Least Voltage Loss Due to Internal Resistance
Which one of the battery combinations shown has the least voltage loss due to internal resistance, assuming they are connected to identical bulbs $R$ and have same values of $r$ per battery?
Workspace / Reasoning
Selected Answer:
[ ]
PRACTICE EXAM
Practice Exam #4
— Dielectric Insertion into Isolated Charged Capacitor
A single capacitor is charged to a potential of $V_0$ then removed from a circuit. If a dielectric is then placed between the plates with $\kappa = 3$, which of the following terms are correct for this 'new' capacitor? (Select all that apply)
Workspace / Reasoning
Selected Answer:
[ ]
PRACTICE EXAM
Practice Exam #5
— Current Ratio in Parallel Resistors
If the terminal voltage of a battery is $12\text{ V}$ and resistors $R_1 = 8\ \Omega$ and $R_2 = 6\ \Omega$ are connected in parallel in this circuit, what is the ratio of the currents through these two resistors $I_1 / I_2$?
Workspace / Reasoning
Selected Answer:
[ ]
PRACTICE EXAM
Practice Exam #6
— Assembly Energy of a Three-Charge Triangle
How much energy would be required to build the following charge configuration, bringing the charges in from infinity?
Charges are arranged in a right triangle:
• $q_a = -4.00\ \mu\text{C}$ at $(0, 20.0\text{ cm})$
• $q_b = -2.00\ \mu\text{C}$ at $(0, 0)$
• $q_c = 5.00\ \mu\text{C}$ at $(10.0\text{ cm}, 0)$
Charges are arranged in a right triangle:
• $q_a = -4.00\ \mu\text{C}$ at $(0, 20.0\text{ cm})$
• $q_b = -2.00\ \mu\text{C}$ at $(0, 0)$
• $q_c = 5.00\ \mu\text{C}$ at $(10.0\text{ cm}, 0)$
Show Full Derivation & Calculations (Box Your Final Answer)
PRACTICE EXAM
Practice Exam #7
— Potential from Electric Field Line Integral
If the electric field along the x-axis follows the equation $\vec{E}(x) = \left(300 x^3 - \frac{2}{x}\right)\hat{i}\text{ N/C}$, and the potential is known to be $V_0 = -20.0\text{ V}$ at $x = 10.0\text{ cm}$, what is the electric potential at $x = 1.50\text{ m}$?
Show Full Derivation & Calculations (Box Your Final Answer)
PRACTICE EXAM
Practice Exam #8
— Parallel Plate Capacitor Area Calculation
A potential of $9.00\text{ V}$ is placed across a capacitor and eventually reaches a charge of $Q = 24.5\text{ nC}$. If the plates are separated by $1.50\text{ mm}$, what is their plate area?
Show Full Derivation & Calculations (Box Your Final Answer)
PRACTICE EXAM
Practice Exam #9
— Dielectric Insertion into Pre-Charged Circular Capacitor
A capacitor with circular parallel plates of radius $25.5\text{ cm}$ and separation distance of $0.575\text{ cm}$ has been charged up until there is an electric field of $5.76 \times 10^6\text{ V/m}$ between them. If this capacitor is separated from the circuit and a dielectric with $\kappa = 2.50$ is placed between them, what is the electric potential between the plates?
Show Full Derivation & Calculations (Box Your Final Answer)
PRACTICE EXAM
Practice Exam #10
— Redrawing Wire Under Volume Conservation
A wire is made out of gold and has a length of $11.1\text{ m}$ and a diameter of $0.222\text{ mm}$ (resistivity $\rho = 2.44 \times 10^{-8}\ \Omega\cdot\text{m}$). If the same amount of gold is used to make a new wire, but with half the resistance of the first, how long would it be?
Show Full Derivation & Calculations (Box Your Final Answer)
PRACTICE EXAM
Practice Exam #11
— Capacitance from Exponential Discharge Timing
If it takes $24.5\ \mu\text{s}$ to discharge a capacitor connected in series with a $55.5\ \Omega$ resistor to $3.00\%$ of its initial charge, what is the capacitance of this capacitor?
Show Full Derivation & Calculations (Box Your Final Answer)
PRACTICE EXAM
Practice Exam #12
— Current Distribution in Five-Resistor Bridge Network
Each resistor shown in the diagram has a resistance of $6.00\ \Omega$, and an electromotive potential of $12.0\text{ V}$ is placed across the terminals.
Find the electric current flowing through each individual resistor: $I_A$, $I_B$, $I_C$, $I_D$, and $I_E$.
Find the electric current flowing through each individual resistor: $I_A$, $I_B$, $I_C$, $I_D$, and $I_E$.
Circuit Reduction & Calculation Workspace
PRACTICE EXAM
Practice Exam #13
— Power Dissipation in Resistor Network
Using the currents found in the previous circuit ($R = 6.00\ \Omega$ each, $I_A=1.25\text{ A}, I_B=0.75\text{ A}, I_C=0.50\text{ A}, I_D=0.25\text{ A}, I_E=0.25\text{ A}$):
Calculate the electrical power dissipated by each individual resistor ($P_A$, $P_B$, $P_C$, $P_D$, $P_E$) and the total power $P_{\text{total}}$ delivered to the network.
Calculate the electrical power dissipated by each individual resistor ($P_A$, $P_B$, $P_C$, $P_D$, $P_E$) and the total power $P_{\text{total}}$ delivered to the network.
Circuit Reduction & Calculation Workspace
PRACTICE EXAM
Practice Exam #14
— Two-Body Relativistic & Classical Coulomb Kinematics
An electron and a positron ($m_e = 9.11 \times 10^{-31}\text{ kg}$, $q = \pm 1.60 \times 10^{-19}\text{ C}$), initially separated by $100\text{ nm}$, are launched directly away from each other along a line at $v_0 = 4.75 \times 10^4\text{ m/s}$ each.
(a) Find their maximum separation distance.
(b) What is the maximum separation if a proton and antiproton ($m_p = 1.67 \times 10^{-27}\text{ kg}$) are used with initial separation $0.0500\text{ nm}$ and the same initial launch speeds?
(c) Calculate the escape velocity for both particle pairs.
(d) If an electron and proton are fired away from each other from initial separation $1.00\text{ nm}$ so that momentum and energy conservation bring them to rest simultaneously at separation $1.00\text{ mm}$, find the initial speeds $v_e$ and $v_p$.
(a) Find their maximum separation distance.
(b) What is the maximum separation if a proton and antiproton ($m_p = 1.67 \times 10^{-27}\text{ kg}$) are used with initial separation $0.0500\text{ nm}$ and the same initial launch speeds?
(c) Calculate the escape velocity for both particle pairs.
(d) If an electron and proton are fired away from each other from initial separation $1.00\text{ nm}$ so that momentum and energy conservation bring them to rest simultaneously at separation $1.00\text{ mm}$, find the initial speeds $v_e$ and $v_p$.
Show Full Derivation & Calculations (Box Your Final Answers)
PRACTICE EXAM
Practice Exam #15
— Parallel Plate Capacitor Breakdown & Stored Energy
A capacitor is made from two flat parallel plates spaced $0.40\text{ mm}$ apart. When a charge of $0.020\ \mu\text{C}$ is placed on the plates, the potential difference between them is $250\text{ V}$.
(a) Find the capacitance $C$.
(b) Find the area of each plate $A$.
(c) Find the charge $Q$ on the plates when the potential difference is increased to $500\text{ V}$.
(d) What maximum potential difference can be applied before the electric field exceeds $3.0\text{ MV/m}$?
(e) How much energy is stored in the capacitor at that maximum potential?
(a) Find the capacitance $C$.
(b) Find the area of each plate $A$.
(c) Find the charge $Q$ on the plates when the potential difference is increased to $500\text{ V}$.
(d) What maximum potential difference can be applied before the electric field exceeds $3.0\text{ MV/m}$?
(e) How much energy is stored in the capacitor at that maximum potential?
Show Full Derivation & Calculations (Box Your Final Answers)
PRACTICE EXAM
Practice Exam #16
— Six-Capacitor Complex Circuit Analysis
For the capacitor circuit shown in the diagram:
(a) Find the effective capacitance of the circuit.
(b) If connected to a $9.00\text{ V}$ battery, find the potential difference across each of the six capacitors ($V_5, V_{3.5}, V_8, V_{1.5}, V_{0.75}, V_{15}$).
(c) Find the stored electrostatic energy in each capacitor and the total stored energy.
(a) Find the effective capacitance of the circuit.
(b) If connected to a $9.00\text{ V}$ battery, find the potential difference across each of the six capacitors ($V_5, V_{3.5}, V_8, V_{1.5}, V_{0.75}, V_{15}$).
(c) Find the stored electrostatic energy in each capacitor and the total stored energy.
Network Reduction, Potential Traversal & Energy Workspace
GROUP WORK
Group Work 7.1
— Speed of Charged Particle from Electron-Volt Kinetic Energy
What is the speed of a proton ($m_p = 1.67 \times 10^{-27}\text{ kg}$, $q = 1.60 \times 10^{-19}\text{ C}$) with $3,000\text{ eV}$ of kinetic energy?
Show Work • Calculations • Formulas
Final Answer:
[m/s]
GROUP WORK
Group Work 7.2
— Maximum Speed in Quadratic Potential with Turning Points
An electric potential along the x-axis is given by $V(x) = 4000 x^2$ (with $V$ in volts and $x$ in meters). A particle of mass $m = 1.0\text{ g}$ and charge $q = 15\text{ nC}$ moves in this potential and has turning points at $x = \pm 9.0\text{ cm}$. What is the maximum speed of the particle?
Show Work • Calculations • Formulas
Final Answer:
[m/s]
GROUP WORK
Group Work 7.3
— Distance of Closest Approach to Lead Nucleus
A proton is shot toward the nucleus of a lead atom ($Z = 82$, diameter of nucleus $= 14.0\text{ fm}$, radius $R_{\text{Pb}} = 7.0\text{ fm}$). The proton is fired at an initial speed of $v_0 = 5.0 \times 10^7\text{ m/s}$ from far away. What is its distance of closest approach to the surface of the lead nucleus? (Assume the lead nucleus remains stationary).
Show Work • Calculations • Formulas
Final Answer:
[fm]
GROUP WORK
Group Work 7.4
— Potential Difference Between Surfaces of Charged Spheres
Two uniformly charged conducting spheres are placed with their centers $125\text{ cm}$ apart:
• Left sphere $A$: $q_1 = 150\text{ nC}$, radius $r_1 = 45.0\text{ cm}$
• Right sphere $B$: $q_2 = 50.0\text{ nC}$, radius $r_2 = 15.0\text{ cm}$
What is the magnitude of the potential difference between point $a$ (on surface of $A$) and point $b$ (on surface of $B$)?
• Left sphere $A$: $q_1 = 150\text{ nC}$, radius $r_1 = 45.0\text{ cm}$
• Right sphere $B$: $q_2 = 50.0\text{ nC}$, radius $r_2 = 15.0\text{ cm}$
What is the magnitude of the potential difference between point $a$ (on surface of $A$) and point $b$ (on surface of $B$)?
Show Work • Calculations • Formulas
Final Answer:
[V]
GROUP WORK
Group Work 7.5
— On-Axis Electric Potential of a Continuous Uniform Charged Rod
A thin rod of length $L$ carries a total uniform positive charge $Q$. It lies along the x-axis centered at the origin (from $x = -L/2$ to $x = +L/2$). Which of the following is the correct mathematical expression for the electric potential at a point on the x-axis at distance $x \gt L/2$?
Workspace / Reasoning
Selected Answer:
[ ]
GROUP WORK
Group Work 8.1
— Electric Field and Potential in a Three-Electrode Array
Three square metal plates ($10\text{ cm} \times 10\text{ cm}$, area $A = 0.010\text{ m}^2$) are positioned along the x-axis: plate 1 at $x=0$ carries $-50\text{ nC}$, plate 2 at $x=10\text{ cm}$ carries $+100\text{ nC}$, plate 3 at $x=20\text{ cm}$ carries $-50\text{ nC}$. Let $V(0) = 0\text{ V}$.
Find the electric potential $V$ at:
(a) $x = 5.0\text{ cm}$
(b) $x = 10.0\text{ cm}$ (at center plate)
(c) What is the magnitude of the electric field $E$ between the first two plates ($0 \lt x \lt 10\text{ cm}$)?
Find the electric potential $V$ at:
(a) $x = 5.0\text{ cm}$
(b) $x = 10.0\text{ cm}$ (at center plate)
(c) What is the magnitude of the electric field $E$ between the first two plates ($0 \lt x \lt 10\text{ cm}$)?
Show Work for All Sub-Parts
(a) Potential at x = 5 cm (kV):
[kV]
(b) Potential at x = 10 cm (kV):
[kV]
(c) Field magnitude E (kV/m):
[kV/m]
GROUP WORK
Group Work 8.4
— Switch Transient Charge Redistribution Across Capacitor Network
Initially, the switch is in position $A$, charging capacitor $C_1 = 15.0\ \mu\text{F}$ to $100\text{ V}$ while $C_2 = 20.0\ \mu\text{F}$ and $C_3 = 30.0\ \mu\text{F}$ are completely uncharged. The switch is then flipped to position $B$, connecting $C_1$ in parallel across the series combination of $C_2$ and $C_3$.
Find the final charge and potential difference on each capacitor:
Find the final charge and potential difference on each capacitor:
Show Work for All Sub-Parts
Charge Q_1 (µC):
[µC]
Voltage V_1 (V):
[V]
Charge Q_2 & Q_3 (µC):
[µC]
Voltage V_2 (V):
[V]
Voltage V_3 (V):
[V]
GROUP WORK
Group Work 9.1
— CRT Television Electron Beam Power and Current Density
The electron beam inside an old cathode ray television tube has a diameter of $0.40\text{ mm}$ and carries a beam current of $50.0\ \mu\text{A}$. The electrons strike the screen at speed $v = 4.0 \times 10^7\text{ m/s}$ and transfer all kinetic energy upon impact.
(a) How many electrons strike the screen each second?
(b) What is the current density $J$ in the beam?
(c) What electrical impact power is delivered to the screen?
(a) How many electrons strike the screen each second?
(b) What is the current density $J$ in the beam?
(c) What electrical impact power is delivered to the screen?
Show Work for All Sub-Parts
(a) Electrons per second:
[e⁻/s]
(b) Current Density J (A/m²):
[A/m²]
(c) Delivered Power (W):
[W]
GROUP WORK
Group Work 9.2
— Drift Velocity and Current Density in a 10-Gauge AC Wire
The current supplied to an air conditioner is $4.00\text{ A}$ through a 10-gauge wire with diameter $d = 2.60\text{ mm}$. The free electron density of copper is $n_e = 8.48 \times 10^{28}\text{ m}^{-3}$.
(a) Find the current density $J$.
(b) Find the electron drift velocity $v_d$.
(a) Find the current density $J$.
(b) Find the electron drift velocity $v_d$.
Show Work for All Sub-Parts
(a) Current Density J (A/m²):
[A/m²]
(b) Drift Velocity v_d (m/s):
[m/s]
GROUP WORK
Group Work 9.4
— Non-Uniform Proton Beam Area Integration
A $5.00\text{ mm}$ diameter proton beam carries a total current of $1.50\text{ mA}$. The current density in the beam increases linearly with radial distance from center: $J(r) = J_{\text{edge}} \left(\frac{r}{R}\right)$, where $R = 2.50\text{ mm}$ is the beam radius. Determine the edge current density $J_{\text{edge}}$.
Show Work • Calculations • Formulas
Final Answer:
[A/m²]
GROUP WORK
Group Work 9.5
— Time-Dependent Charge and Exponential Decay of Current
The total charge that has entered a wire at time $t$ is given by $Q(t) = (20.0\text{ C})(1 - e^{-t / 2.0\text{ s}})$. At what time $t$ has the current decayed to exactly $\frac{1}{3}$ of its maximum initial value?
Show Work • Calculations • Formulas
Final Answer:
[s]
GROUP WORK
Group Work 10.1
— Extracting Battery EMF and Internal Resistance from I vs R Graph
A variable load resistor $R$ is connected across the terminals of a real battery. The graph shows the circuit current $I$ as a function of external load resistance $R$. At $R = 0\ \Omega$, $I = 6.0\text{ A}$; at $R = 10.0\ \Omega$, $I = 3.0\text{ A}$; at $R = 20.0\ \Omega$, $I = 2.0\text{ A}$.
Determine the battery EMF $\mathcal{E}$ and internal resistance $r$:
Determine the battery EMF $\mathcal{E}$ and internal resistance $r$:
Show Work for All Sub-Parts
Battery EMF (V):
[V]
Internal Resistance r (Ω):
[Ω]
GROUP WORK
Group Work 10.2
— Circuit Synthesis: Resistor Topology for Target Power
You have three resistors ($2.0\ \Omega, 3.0\ \Omega, 6.0\ \Omega$) and an ideal $6.0\text{ V}$ battery. Which arrangement of all three resistors causes the battery to deliver exactly $9.0\text{ W}$ of power?
Workspace / Reasoning
Selected Answer:
[ ]
GROUP WORK
Group Work 10.3
— RC Charging Dynamics and Characteristic Time Constant
An initially uncharged $1.50\ \mu\text{F}$ capacitor is connected in series with a $500\ \Omega$ resistor and a $6.16\text{ V}$ battery.
(a) Find the initial current $I_0$.
(b) Find the circuit time constant $\tau$.
(c) Find the current after one time constant ($t = \tau$).
(d) Find the capacitor voltage after one time constant.
(a) Find the initial current $I_0$.
(b) Find the circuit time constant $\tau$.
(c) Find the current after one time constant ($t = \tau$).
(d) Find the capacitor voltage after one time constant.
Show Work for All Sub-Parts
(a) Initial Current I_0 (mA):
[mA]
(b) Time Constant τ (ms):
[ms]
(c) Current at t = τ (mA):
[mA]
(d) Voltage at t = τ (V):
[V]
GROUP WORK
Group Work 10.4
— Two-Battery Multi-Loop Kirchhoff Circuit Analysis
For the multi-loop circuit shown in the diagram containing two batteries ($12.0\text{ V}$ and $3.00\text{ V}$) and five resistors ($3\ \Omega, 4\ \Omega, 5\ \Omega, 12\ \Omega, 24\ \Omega$):
Find the currents flowing through the $3\ \Omega$ resistor, $4\ \Omega$ resistor, $5\ \Omega$ resistor, and $24\ \Omega$ resistor:
Find the currents flowing through the $3\ \Omega$ resistor, $4\ \Omega$ resistor, $5\ \Omega$ resistor, and $24\ \Omega$ resistor:
Show Work for All Sub-Parts
Current through 3 Ω resistor (A):
[A]
Current through 4 Ω resistor (A):
[A]
Current through 5 Ω resistor (A):
[A]
Current through 24 Ω resistor (A):
[A]
GROUP WORK
Group Work 10.5
— Parallel vs Series Total Dissipated Power Ratio
Three resistors ($R_1 = 2.50\text{ k}\Omega, R_2 = 3.50\text{ k}\Omega, R_3 = 4.50\text{ k}\Omega$) are connected to a $100\text{ V}$ DC power supply. What is the ratio of the total power delivered when connected in parallel to the total power delivered when connected in series ($\frac{P_{\text{parallel}}}{P_{\text{series}}}$)?
Show Work • Calculations • Formulas
Final Answer:
[ratio]
CONCEPTUAL
Conceptual Trap #1
— Equipotential Surfaces & Conservative Field Work
An electric field in a region is highly non-uniform, with strongly curved equipotential surfaces. A test charge $q$ is transported from point $A$ to point $B$ along an intricate, winding path. Points $A$ and $B$ are both located on the exact same equipotential surface $V = 50\text{ V}$, but along the path the particle traverses regions where the potential reaches $+500\text{ V}$ and electric fields exceed $10^6\text{ V/m}$. What is the net work done by the electrostatic field on the charge?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #2
— Conductor Cavity Shielding with Off-Center Charge
A neutral, solid conducting spherical shell of outer radius $R_2$ has an off-center spherical cavity of radius $R_1$. A point charge $+q$ is placed inside the cavity, shifted away from the center. A second positive charge $+Q$ is placed outside the shell at distance $D \gt R_2$. Which statement correctly describes the electric field and charge distribution?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #3
— Dielectric Insertion at Constant Voltage: Energy & Force
A dielectric slab ($\kappa \gt 1$) is partially inserted between the plates of a parallel-plate capacitor that remains connected to an active, ideal battery ($V = \text{constant}$). As the slab enters the capacitor, what happens to the electrostatic force on the slab and the stored electrical energy?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #4
— The 50% Energy Dissipation Paradox in Capacitor Charging
When a capacitor $C$ is charged from zero to voltage $\mathcal{E}$ through a circuit with series resistance $R$, the battery supplies total energy $W_{\text{battery}} = Q\mathcal{E} = C\mathcal{E}^2$, but the capacitor stores only $U = \frac{1}{2}C\mathcal{E}^2$. Exactly $50\%$ of the supplied energy is dissipated as heat in the resistor ($E_{\text{diss}} = \int I^2 R dt = \frac{1}{2}C\mathcal{E}^2$). If the resistor is replaced by a zero-resistance superconductor ($R \to 0$), what happens to the missing $50\%$ of the energy?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #5
— Drift Speed vs. Fermi Speed vs. Electromagnetic Signal Velocity
When a household wall switch is flipped, a light bulb meters away lights up in less than $10^{-8}\text{ s}$. Yet the drift velocity of conduction electrons in the copper wire is only about $v_d \sim 10^{-4}\text{ m/s}$ (sluggish snail's pace), while the random thermal/Fermi speed of the electrons is $v_F \sim 10^6\text{ m/s}$. What physical mechanism accounts for this near-instantaneous transmission of power?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #6
— Tungsten Filament Inrush Current & Temperature Coefficient
An incandescent light bulb with a tungsten filament has a temperature coefficient of resistivity $\alpha = +4.5 \times 10^{-3}\ ^\circ\text{C}^{-1}$. When glowing white-hot at steady state ($T \approx 2500^\circ\text{C}$), it draws $100\text{ W}$ at $120\text{ V}$. At the cold instant the switch is first closed (room temperature $T_0 = 20^\circ\text{C}$), how does the cold inrush current compare to the normal steady-state operating current?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #7
— Resistor Cube Equipotential Symmetry Across Body Diagonal
Twelve identical resistors of resistance $R$ are soldered together along the twelve edges of a 3-dimensional cube. What is the equivalent resistance $R_{\text{eq}}$ between two diametrically opposite corners across the main body diagonal?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #8
— Series vs Parallel Dielectric Slicing Topologies
A parallel-plate capacitor is filled half with dielectric $\kappa_1$ and half with dielectric $\kappa_2$. In Case A, the boundary is parallel to the plates (splitting thickness $d/2$ and $d/2$). In Case B, the boundary is perpendicular to the plates (splitting area $A/2$ and $A/2$). Which case yields higher capacitance, and what are their equivalent circuit models?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #9
— Non-Ideal Voltmeter & Ammeter Loading Errors
You need to measure a very large resistance $R \approx 1.0\text{ M}\Omega$ using a voltmeter with finite internal resistance $R_V = 10\text{ M}\Omega$ and an ammeter with small internal resistance $R_A = 10\ \Omega$. Which circuit configuration minimizes the fractional systematic measurement error?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #10
— Instantaneous Power Dissipation in RC Discharge
In a discharging $RC$ circuit ($Q(t) = Q_0 e^{-t/\tau}$), the resistor dissipates stored electrical energy as heat. At what time $t$ is the instantaneous rate of energy dissipation (power $P(t)$) equal to exactly half of its initial value $P_0$?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #11
— Maximum Power Transfer Theorem & Efficiency
A real DC battery has electromotive force $\mathcal{E}$ and constant internal resistance $r$. It is connected to a variable external load resistance $R_L$. Under what condition does the load resistor $R_L$ dissipate maximum electrical power, and what is the electrical efficiency $\eta = \frac{P_{\text{load}}}{P_{\text{total}}}$ at that operating point?
Workspace / Reasoning
Selected Answer:
[ ]
CONCEPTUAL
Conceptual Trap #12
— Terminal Voltage of an Opposing EMF Battery Under Charge
Two real batteries are connected in a single loop with their positive terminals opposing each other: Battery 1 has $\mathcal{E}_1 = 12.0\text{ V}$ and $r_1 = 2.0\ \Omega$, while Battery 2 has $\mathcal{E}_2 = 6.0\text{ V}$ and $r_2 = 1.0\ \Omega$. An ideal voltmeter is placed across the terminals of Battery 2. What voltage does it read?
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