A circuit containing a resistor of resistance $R$ and an inductor of inductance $L$ initially carries a steady-state current $I_0$. At time $t = 0$, the power source is bypassed, allowing the current to decay through the resistor as shown in the $I(t)$ graph below. Which of the following claims correctly describes the magnitude of the potential difference $|V_L|$ across the inductor as time progresses, based on the features of the graph?(a) The potential difference $|V_L|$ is zero at $t = 0$ because the current reaches its maximum value $I_0$ at that instant. , (b) The potential difference $|V_L|$ increases over time because the magnitude of the slope of the $I(t)$ graph increases as current decays. , (c) The potential difference $|V_L|$ is maximum at $t = 0$ and decreases toward zero because $|V_L|$ is proportional to the magnitude of the slope of the $I(t)$ graph. , (d) The potential difference $|V_L|$ remains constant over time because it is determined by the area under the $I(t)$ curve, which is fixed. :

A circuit containing a resistor of resistance $R$ and an inductor of inductance $L$ initially carries a steady-state current $I_0$. At time $t = 0$, the power source is bypassed, allowing the current to decay through the resistor as shown in the $I(t)$ graph below. Which of the following claims correctly describes the magnitude of the potential difference $|V_L|$ across the inductor as time progresses, based on the features of the graph? • Nerd Notes

AP Physics

Unit 1 - Vectors and Kinematics

MCQ

Mathematical

Intermediate

Step	Reasoning
Relate the magnitude of the potential difference across the inductor $\|V_L\|$ to the current $I(t)$. \[ \|V_L\| = L \left\| \dfrac{dI}{dt} \right\| \]	The question asks about the magnitude of the potential difference across the inductor, which is governed by Faraday’s law of induction applied to self-inductance.
Connect the derivative $\dfrac{dI}{dt}$ to a visual feature of the given graph. \[ \text{Slope} = \dfrac{dI}{dt} \]	The rate of change of current with respect to time corresponds directly to the instantaneous slope of the $I(t)$ curve.
Analyze how the magnitude of the slope behaves as a function of time for exponential decay. \[ \left\| \dfrac{dI}{dt} \right\| = \left\| -\dfrac{I_0}{\tau} e^{-t/\tau} \right\| = \dfrac{I_0}{\tau} e^{-t/\tau} \]	Evaluating the slope of the exponential decay curve $I(t) = I_0 e^{-t/\tau}$ determines how $\|V_L\|$ evolves over time.
Synthesize the slope behavior to select the correct physical claim.	At $t = 0$, the curve is at its steepest point (maximum magnitude of slope), so $\|V_L\|$ is at its maximum value. As $t \to \infty$, the curve flattens out and the slope approaches zero, causing $\|V_L\|$ to decay exponentially toward zero.

Why each choice is correct or incorrect:

(A) Incorrectly assumes $|V_L|$ depends on the value of $I$ rather than its derivative $dI/dt$; at maximum current, the magnitude of the slope is actually at its peak, not zero.

(B) Incorrectly asserts that the magnitude of the slope increases; exponential decay becomes flatter over time, meaning the slope magnitude decreases.

(C) This is the correct answer. The potential difference across the inductor depends directly on the magnitude of the slope $|dI/dt|$, which is greatest at $t = 0$ and decays toward zero.

(D) Incorrectly connects voltage across an inductor to the area under the $I(t)$ curve (which gives total charge transferred, $Q = \int I\,dt$) rather than to the slope of the curve.

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Advanced

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Beginner

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An ideal circuit consists of an inductor of inductance $L$ connected in series with a capacitor of capacitance $C$. At time $t = 0$, the capacitor carries an initial charge $Q_0$ and the current in the circuit is zero. Which of the following graphs best represents the charge $q(t)$ on the capacitor as a function of time $t$?

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A horizontal circular ring of radius $r$, mass $m$, and electrical resistance $R$ falls vertically under the influence of gravity through a region with a non-uniform vertical magnetic field. The vertical component of the magnetic field varies linearly with height $z$ according to $B_z(z) = B_0 + bz$, where $B_0$ and $b$ are positive constants, and $z$ is measured upward. Air resistance is negligible. Which of the following expressions correctly represents the magnitude of the ring’s terminal velocity $v_T$?

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A long, straight cylindrical wire of radius $R$ carries a steady total current $I$ distributed uniformly across its circular cross section. The permeability of free space is $\mu_0$. Which of the following expressions represents the total magnetic energy stored per unit length inside the volume of the wire?

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A circular region of radius $R$ in the $xy$-plane contains a time-dependent, non-uniform magnetic field directed perpendicular to the plane. Inside the region ($r \le R$), the magnitude of the magnetic field is given by $B(r,t) = C t \left(\dfrac{r}{R}\right)^2$, where $C$ is a positive constant and $r$ is the radial distance from the central axis. Outside the region ($r > R$), the magnetic field is zero. Which of the following expressions gives the magnitude of the induced electric field $E(r)$ as a function of distance $r$ from the central axis inside the region ($r < R$)?

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$A schematic diagram of an electric circuit drawn with straight black lines forming two adjacent rectangular loops. On the far left vertical wire of the left loop is a DC battery labeled $V_0$ with its longer positive terminal on top. On the top horizontal wire of the left loop is a switch labeled $S$ shown in an open position and a resistor labeled $R_1 = R$ in series. A shared vertical wire separates the left and right loops, containing a resistor labeled $R_2 = 2R$. On the far right vertical wire of the right loop, a resistor labeled $R_3 = R$ is connected in series above an inductor coil labeled $L$. No other labels, lines, text, or axes appear.$

In the circuit shown, an ideal battery with potential difference $V_0$ is connected to three resistors ($R_1 = R$, $R_2 = 2R$, $R_3 = R$), an ideal inductor $L$, and a switch $S$. The switch $S$ has been closed for a long time. At time $t = 0$, switch $S$ is opened. Which of the following expressions represents the magnitude of the current $I(t)$ through resistor $R_2$ as a function of time $t$ for $t \ge 0$?

Step	Reasoning
Relate the magnitude of the potential difference across the inductor \(\|V_L\|\) to the current \(I(t)\). \[ \|V_L\| = L \left\| \dfrac{dI}{dt} \right\| \]	The question asks about the magnitude of the potential difference across the inductor, which is governed by Faraday’s law of induction applied to self-inductance.
Connect the derivative \(\dfrac{dI}{dt}\) to a visual feature of the given graph. \[ \text{Slope} = \dfrac{dI}{dt} \]	The rate of change of current with respect to time corresponds directly to the instantaneous slope of the \(I(t)\) curve.
Analyze how the magnitude of the slope behaves as a function of time for exponential decay. \[ \left\| \dfrac{dI}{dt} \right\| = \left\| -\dfrac{I_0}{\tau} e^{-t/\tau} \right\| = \dfrac{I_0}{\tau} e^{-t/\tau} \]	Evaluating the slope of the exponential decay curve \(I(t) = I_0 e^{-t/\tau}\) determines how \(\|V_L\|\) evolves over time.
Synthesize the slope behavior to select the correct physical claim.	At \(t = 0\), the curve is at its steepest point (maximum magnitude of slope), so \(\|V_L\|\) is at its maximum value. As \(t \to \infty\), the curve flattens out and the slope approaches zero, causing \(\|V_L\|\) to decay exponentially toward zero.

Kinematics	Forces
\(\Delta x = v_i t + \frac{1}{2} at^2\)	\(F = ma\)
\(v = v_i + at\)	\(F_g = \frac{G m_1 m_2}{r^2}\)
\(v^2 = v_i^2 + 2a \Delta x\)	\(f = \mu N\)
\(\Delta x = \frac{v_i + v}{2} t\)	\(F_s =-kx\)
\(v^2 = v_f^2 \,-\, 2a \Delta x\)

Circular Motion	Energy
\(F_c = \frac{mv^2}{r}\)	\(KE = \frac{1}{2} mv^2\)
\(a_c = \frac{v^2}{r}\)	\(PE = mgh\)
\(T = 2\pi \sqrt{\frac{r}{g}}\)	\(KE_i + PE_i = KE_f + PE_f\)
	\(W = Fd \cos\theta\)

Momentum	Torque and Rotations
\(p = mv\)	\(\tau = r \cdot F \cdot \sin(\theta)\)
\(J = \Delta p\)	\(I = \sum mr^2\)
\(p_i = p_f\)	\(L = I \cdot \omega\)

Simple Harmonic Motion	Fluids
\(F = -kx\)	\(P = \frac{F}{A}\)
\(T = 2\pi \sqrt{\frac{l}{g}}\)	\(P_{\text{total}} = P_{\text{atm}} + \rho gh\)
\(T = 2\pi \sqrt{\frac{m}{k}}\)	\(Q = Av\)
\(x(t) = A \cos(\omega t + \phi)\)	\(F_b = \rho V g\)
\(a = -\omega^2 x\)	\(A_1v_1 = A_2v_2\)

Constant	Description
[katex]g[/katex]	Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface
[katex]G[/katex]	Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex]
[katex]\mu_k[/katex] and [katex]\mu_s[/katex]	Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion.
[katex]k[/katex]	Spring constant, in [katex]\text{N/m}[/katex]
[katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex]	Mass of the Earth
[katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex]	Mass of the Moon
[katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex]	Mass of the Sun

Variable	SI Unit
[katex]s[/katex] (Displacement)	[katex]\text{meters (m)}[/katex]
[katex]v[/katex] (Velocity)	[katex]\text{meters per second (m/s)}[/katex]
[katex]a[/katex] (Acceleration)	[katex]\text{meters per second squared (m/s}^2\text{)}[/katex]
[katex]t[/katex] (Time)	[katex]\text{seconds (s)}[/katex]
[katex]m[/katex] (Mass)	[katex]\text{kilograms (kg)}[/katex]

Variable	Derived SI Unit
[katex]F[/katex] (Force)	[katex]\text{newtons (N)}[/katex]
[katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy)	[katex]\text{joules (J)}[/katex]
[katex]P[/katex] (Power)	[katex]\text{watts (W)}[/katex]
[katex]p[/katex] (Momentum)	[katex]\text{kilogram meters per second (kgm/s)}[/katex]
[katex]\omega[/katex] (Angular Velocity)	[katex]\text{radians per second (rad/s)}[/katex]
[katex]\tau[/katex] (Torque)	[katex]\text{newton meters (Nm)}[/katex]
[katex]I[/katex] (Moment of Inertia)	[katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex]
[katex]f[/katex] (Frequency)	[katex]\text{hertz (Hz)}[/katex]

Prefix	Symbol	Power of Ten
Pico-	p	[katex]10^{-12}[/katex]
Nano-	n	[katex]10^{-9}[/katex]
Micro-	µ	[katex]10^{-6}[/katex]
Milli-	m	[katex]10^{-3}[/katex]
Centi-	c	[katex]10^{-2}[/katex]
Deci-	d	[katex]10^{-1}[/katex]
(Base unit)	–	[katex]10^{0}[/katex]
Deca- or Deka-	da	[katex]10^{1}[/katex]
Hecto-	h	[katex]10^{2}[/katex]
Kilo-	k	[katex]10^{3}[/katex]
Mega-	M	[katex]10^{6}[/katex]
Giga-	G	[katex]10^{9}[/katex]
Tera-	T	[katex]10^{12}[/katex]

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