A positively charged particle moves with velocity $\vec{v}$ in a region containing a non-uniform magnetic field $\vec{B}$ aligned generally along the $z$-axis (a magnetic bottle configuration). Near the “neck” of the bottle at $z > 0$, the magnetic field lines converge toward the $z$-axis, introducing a radial magnetic field component $B_r < 0$ directed toward the axis. The particle spirals around the $z$-axis with an azimuthal velocity component $v_\theta$ while traveling in the $+z$-direction with axial speed $v_z$. As the particle approaches the neck, its axial speed $v_z$ decreases to zero and the particle reflects back toward $z = 0$. Which of the following explanations correctly accounts for the axial deceleration of the particle and the work done by the magnetic field during this reflection?(a) The converging magnetic field lines create an axial gradient in $B_z$ that exerts a direct magnetic force along the $-z$-direction. Because this force opposes the particle’s displacement, the magnetic field performs negative work on the particle, reducing its total kinetic energy until it momentarily stops and reflects. , (b) As the particle moves toward the neck, the spatial variation in magnetic field induces a retarding axial electric field. This induced electric field performs negative work on the particle, decreasing its kinetic energy until its axial velocity reaches zero. , (c) The inward radial field component $B_r$ exerts a retarding axial force $F_z$. The magnetic field does negative work on the axial component of motion while doing an equal amount of positive work on the azimuthal component of motion, resulting in a net work of zero. , (d) The inward radial field component $B_r$ interacts with the particle’s azimuthal velocity $v_\theta$ via $q(\vec{v}_\theta \times \vec{B}_r)$ to produce a retarding axial force $F_z$. The magnetic force performs zero work on the particle because $\vec{F}_B$ is perpendicular to $\vec{v}$ at every instant, continuously converting axial kinetic energy into transverse kinetic energy until $v_z = 0$. :

A positively charged particle moves with velocity $\vec{v}$ in a region containing a non-uniform magnetic field $\vec{B}$ aligned generally along the $z$-axis (a magnetic bottle configuration). Near the "neck" of the bottle at $z > 0$, the magnetic field lines converge toward the $z$-axis, introducing a radial magnetic field component $B_r < 0$ directed toward the axis. The particle spirals around the $z$-axis with an azimuthal velocity component $v_\theta$ while traveling in the $+z$-direction with axial speed $v_z$. As the particle approaches the neck, its axial speed $v_z$ decreases to zero and the particle reflects back toward $z = 0$. Which of the following explanations correctly accounts for the axial deceleration of the particle and the work done by the magnetic field during this reflection? • Nerd Notes

AP Physics

Unit 1 - Vectors and Kinematics

MCQ

Mathematical

Advanced

Step	Reasoning
Identify the origin of the retarding axial force from the magnetic force expression. \[ \vec{F}_B = q(\vec{v} \times \vec{B}) \]	The question asks for the mechanism responsible for decelerating the particle’s axial motion $v_z$.
Evaluate the axial component of the cross product in cylindrical coordinates. \[ F_z = \left[ q(\vec{v} \times \vec{B}) \right]_z = q(v_\theta B_r – v_r B_\theta) = q v_\theta B_r \]	The converging geometry introduces a radial magnetic field component $B_r$ directed toward the axis, which interacts with the spiraling particle’s azimuthal velocity $v_\theta$.
Evaluate the instantaneous power and total work done by the magnetic field. \[ P = \vec{F}_B \cdot \vec{v} = q(\vec{v} \times \vec{B}) \cdot \vec{v} = 0 \implies W = \int P\,dt = 0 \]	The question asks to account for the work done by the magnetic force on the particle.
Relate zero total work to the partitioning of kinetic energy. \[ \Delta K = \Delta K_z + \Delta K_\perp = 0 \implies \Delta K_\perp = -\Delta K_z \]	Since the magnetic force does zero work on the particle, its total kinetic energy $K = K_z + K_\perp$ remains constant throughout the motion, transferring axial kinetic energy into transverse gyrational energy.

Why each choice is correct or incorrect:

(A) Incorrectly claims that the magnetic force performs negative work and reduces the particle’s total kinetic energy. Magnetic forces are always perpendicular to velocity and perform zero work.

(B) Fails to recognize that the static magnetic field configuration does not produce an induced electric field, and incorrectly claims that the particle loses total kinetic energy.

(C) Misinterprets work as a quantity that can be done separately on orthogonal velocity components. The magnetic force is perpendicular to the total velocity vector $\vec{v}$ at every instant, so the instantaneous power $\vec{F}_B \cdot \vec{v}$ is strictly zero, not a combination of positive and negative work components.

(D) This is the correct answer. The cross product of the azimuthal velocity $v_\theta$ with the inward radial field $B_r$ produces a retarding axial force, while $\vec{F}_B \perp \vec{v}$ guarantees zero net work, conserving total kinetic energy by converting axial kinetic energy to transverse kinetic energy.

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Intermediate

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A conducting loop is fixed in a region with a magnetic field directed perpendicular to the plane of the loop. The induced electromotive force (EMF) $\mathcal{E}$ in the loop is measured as a function of time $t$ and is shown in the graph. At time $t = 0$, the magnetic flux through the loop is zero ($\Phi_B = 0$). Which of the following graphs best represents the magnetic flux $\Phi_B$ through the loop as a function of time $t$?

Intermediate

Mathematical

MCQ

A rigid circular wire loop of radius $a$ and total resistance $R$ lies fixed in a plane perpendicular to a spatially uniform magnetic field. The magnitude of the magnetic field changes with time $t$ according to $B(t) = B_0 \left(1 + \dfrac{t}{\tau}\right)^3$, where $B_0$ and $\tau$ are positive constants. Which of the following expressions represents the total electric charge $Q$ that flows past a point in the loop between $t = 0$ and $t = \tau$?

Advanced

Mathematical

MCQ

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Beginner

Mathematical

MCQ

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 $t$?

Intermediate

Mathematical

MCQ

$A 3D Cartesian coordinate system with x, y, and z axes labeled at the ends of thin black lines with arrowheads. A flat square loop of side length L is situated in the xy-plane with one edge lying along the y-axis from y = 0 to y = L. A second identical square loop shares the same edge along the y-axis but is tilted upward into the xz-plane by an angle \theta relative to the xy-plane. An arc labeled \theta indicates the tilt angle between the two loop planes. Vertical upward arrows parallel to the z-axis represent a magnetic field \vec{B}; the arrows increase in length at larger values of x. No other labels, lines, text, or axes appear.$

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Advanced

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Mathematical

MCQ

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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Mathematical

MCQ

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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Mathematical

MCQ

$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 $t \ge 0$?

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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Unit 1 - Vectors and Kinematics

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