Two point charges of equal magnitude $q$ but opposite sign are fixed on the $x$-axis at $x = -d$ and $x = +d$. Points $A$, $B$, and $C$ lie on the positive $y$-axis at $y = 0$, $y = d$, and $y = 2d$, respectively. Which of the following correctly ranks the magnitude of the electric field $E$ at points $A$, $B$, and $C$? • Nerd Notes

AP Physics

Unit 1 - Vectors and Kinematics

MCQ

Mathematical

Beginner

Step	Reasoning
Identify the goal and principle needed to determine the electric field ranking.	The question asks for a ranking of electric field magnitudes at points along the perpendicular bisector of a dipole, which requires calculating or analyzing the net field vector via electric field superposition.
Evaluate the net electric field magnitude at point A (the origin, $y = 0$). \[ E_A = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{d^2} + \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{d^2} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2q}{d^2} \]	Point A lies directly between the two charges. The positive charge at $x = -d$ creates a field to the right, and the negative charge at $x = +d$ also creates a field to the right, leading to maximum constructive addition.
Derive the general expression for the electric field magnitude at any point along the perpendicular bisector $(0, y)$. \[ E(y) = 2 E_{\text{single}} \cos\theta = 2 \left(\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{d^2 + y^2}\right) \left(\dfrac{d}{\sqrt{d^2 + y^2}}\right) = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2qd}{(d^2 + y^2)^{3/2}} \]	To compare points B and C at non-zero y-coordinates, we must express $E(y)$ as a function of $y$.
Compare the resulting field magnitudes as $y$ increases from $y = 0$ to $y = d$ and $y = 2d$. \[ E_A = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2q}{d^2}, \quad E_B = \dfrac{1}{4\pi\varepsilon_0}\dfrac{\sqrt{2}q}{2d^2}, \quad E_C = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2q}{5\sqrt{5}d^2} \]	As $y$ increases, the denominator $(d^2 + y^2)^{3/2}$ increases monotonically, which means $E(y)$ strictly decreases as distance from the origin along the y-axis increases.

Why each choice is correct or incorrect:

(A) Incorrectly assumes electric field magnitude increases with distance from the dipole axis, reversing the inverse-distance dependence.

(B) Incorrectly assumes that the electric field along the perpendicular bisector is uniform regardless of distance.

(C) This is the correct answer. The net electric field magnitude is greatest at the midpoint between opposite charges and decreases monotonically with increasing distance along the perpendicular bisector.

(D) Confuses opposite charges with like charges, incorrectly assuming the fields cancel out at the midpoint to give a local minimum at point A.

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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 $t$?

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A flat rectangular loop of wire with length $L$, width $w$, and resistance $R$ moves in the $+x$-direction at constant speed $v_0$. At time $t = 0$, the front edge of the loop enters a magnetic field region extending from $x = 0$ to $x = 2L$. Within this region, the magnetic field is directed into the page with spatially varying magnitude $B(x) = B_0 \dfrac{x}{L}$, where $B_0$ is a positive constant, and $B = 0$ elsewhere. Defining counterclockwise current as positive, the induced current $I(t)$ as a function of time increases linearly from $0$ to $+I_0$ for $0 \le t \le T$, remains constant at $+I_0$ for $T \le t \le 2T$, and jumps to $-I_0$ at $t = 2T$ before rising linearly to $0$ at $t = 3T$, where $T = \dfrac{L}{v_0}$ and $I_0 = \dfrac{B_0 w v_0}{R}$. Which of the following claims correctly explains the physical origin of a feature in the $I(t)$ graph?

Beginner

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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

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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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Advanced

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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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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
Identify the goal and principle needed to determine the electric field ranking.	The question asks for a ranking of electric field magnitudes at points along the perpendicular bisector of a dipole, which requires calculating or analyzing the net field vector via electric field superposition.
Evaluate the net electric field magnitude at point A (the origin, \(y = 0\)). \[ E_A = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{d^2} + \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{d^2} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2q}{d^2} \]	Point A lies directly between the two charges. The positive charge at \(x = -d\) creates a field to the right, and the negative charge at \(x = +d\) also creates a field to the right, leading to maximum constructive addition.
Derive the general expression for the electric field magnitude at any point along the perpendicular bisector \((0, y)\). \[ E(y) = 2 E_{\text{single}} \cos\theta = 2 \left(\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{d^2 + y^2}\right) \left(\dfrac{d}{\sqrt{d^2 + y^2}}\right) = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2qd}{(d^2 + y^2)^{3/2}} \]	To compare points B and C at non-zero y-coordinates, we must express \(E(y)\) as a function of \(y\).
Compare the resulting field magnitudes as \(y\) increases from \(y = 0\) to \(y = d\) and \(y = 2d\). \[ E_A = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2q}{d^2}, \quad E_B = \dfrac{1}{4\pi\varepsilon_0}\dfrac{\sqrt{2}q}{2d^2}, \quad E_C = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2q}{5\sqrt{5}d^2} \]	As \(y\) increases, the denominator \((d^2 + y^2)^{3/2}\) increases monotonically, which means \(E(y)\) strictly decreases as distance from the origin along the y-axis increases.

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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AP Physics

Unit 1 - Vectors and Kinematics

MCQ

Mathematical

Beginner

Pro

Pro

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