AP Physics C: Electricity and Magnetism Unit 9 Review: Electric Potential

Q: What's on the AP Physics E&M Unit 9 progress check (MCQ and FRQ)?

The AP Physics E&M Unit 9 progress check includes both MCQ and FRQ parts drawn from all three unit topics: electric potential energy (9.1), electric potential (9.2), and conservation of electric energy (9.3). MCQ questions typically ask you to calculate or compare potential values and energy changes, while the FRQ part asks you to derive relationships and apply conservation principles to charged-particle scenarios. Check Unit 9 for practice that mirrors the progress check format.

Q: How do I practice AP Physics E&M Unit 9 FRQs?

The best way to practice AP Physics E&M Unit 9 FRQs is to work through problems that ask you to derive expressions for electric potential energy and electric potential, then apply conservation of electric energy to find final speeds or positions of charged particles. FRQs from this unit often combine a derivation step with a graphical interpretation or a multi-part scenario involving point charges or uniform fields. Start by writing out your energy conservation equation explicitly before solving, since College Board awards method points. Head to Unit 9 for FRQ practice sets tied to these topics.

Q: Where can I find AP Physics E&M Unit 9 practice questions?

You can find AP Physics E&M Unit 9 practice questions, including multiple-choice and practice test sets, at Unit 9 . The page organizes MCQ and FRQ practice by topic, covering electric potential energy, electric potential, and conservation of electric energy. Working through topic-by-topic MCQs first, then full practice test sections, is the most efficient way to build confidence across all three topics.

Q: How should I study AP Physics E&M Unit 9?

Start with electric potential energy (9.1) and make sure you can calculate work done by electric forces before moving to electric potential (9.2), since potential is just potential energy per unit charge. Once those two are solid, conservation of electric energy (9.3) clicks into place naturally. A concrete plan: sketch field diagrams alongside potential diagrams for the same charge configuration so you see how they relate, then practice writing energy conservation equations from scratch without a formula sheet. Prioritize problems where a charged particle moves between two points, since those appear most often on the exam. Use Unit 9 to find topic-specific practice after each section.

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AP exam review verified for 2027

Electric potential energy (U_E) is a property of a charge configuration; electric potential (V) is energy per unit charge at a point in space. When a charge q moves through a potential difference delta V, its potential energy changes by q delta V, and conservation of energy links that change to a change in kinetic energy.

Fiveable's AP Physics C: Electricity and Magnetism Unit 9 review covers 3 topics, with guided practice questions for the unit.

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AP Physics C: Electricity and Magnetism unit 9 topics

9.1

Electric Potential Energy

Defines U_E = kq1q2/r for two point charges, establishes the zero-reference at infinity, explains the sign convention, and extends to multi-charge systems via pairwise summation.

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9.2

Electric Potential

Introduces V as a scalar field (J/C), covers point-charge and continuous-distribution formulas, scalar superposition, the relationship E_x = -dV/dx and delta V = -integral E dot dr, and equipotential maps.

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9.3

Conservation of Electric Energy

Applies delta U_E = q delta V and conservation of energy to predict kinetic energy changes when charged particles move through potential differences, including sign-convention analysis and the electron-volt unit.

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Practice AP Physics C: Electricity and Magnetism unit 9 questions

Try AP-style questions on unit 9, then use the review notes below for anything you miss.

Example AP-style MCQs

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MCQ

AP-style practice question

Question

A student wants to determine the dielectric constant $\kappa$ of a plastic slab. The student charges an isolated parallel-plate capacitor to potential $V_0$ and then disconnects the battery. Which procedure yields $\kappa$ ?

Insert the slab while maintaining the connection to the battery and measure the new potential difference.

Insert the slab after disconnecting the battery and measure the new potential difference.

Insert the slab after disconnecting the battery and measure the new charge on the plates.

Insert the slab after disconnecting the battery and measure the new electric field between the plates.

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MCQ

AP-style practice question

Question

The electric potential along the x-axis is given by the function $V(x) = ax^2 - b$ , where $a$ and $b$ are positive constants. Which of the following correctly compares the magnitude of the electric field at $x = d$ to the magnitude of the electric field at $x = 2d$ ?

The field at $x = 2d$ is half the magnitude of the field at $x = d$

The field at $x = 2d$ is equal in magnitude to the field at $x = d$

The field at $x = 2d$ is twice the magnitude of the field at $x = d$

The field at $x = 2d$ is four times the magnitude of the field at $x = d$

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Example FRQs

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FRQ

Electric potential and field from opposite charges

4. Two point charges are fixed on the x-axis, as shown in Figure 1. Charge 1 has magnitude +3.0 nC and is located at x = -0.20 m. Charge 2 has magnitude -3.0 nC and is located at x = +0.20 m. Point A is located at x = 0.00 m. Point B is located at x = +0.60 m. A particle of charge +2.0 nC and mass 1.0 g is released from rest at point B and is constrained to move only along the x-axis. Gravitational effects are negligible.

Figure 1. Two fixed point charges on the x-axis and a +2.0 nC particle released from rest at point B, constrained to move only along the x-axis.

$V_A$ is the electric potential at point A due to Charges 1 and 2. $V_B$ is the electric potential at point B due to Charges 1 and 2.

Indicate whether $V_B$ is greater than, less than, or equal to $V_A$ by writing one of the following.

$V_B > V_A$
$V_B < V_A$
$V_B = V_A$

Justify your answer.

Derive an expression for the x-component $E_x(B)$ of the electric field at point B due to Charges 1 and 2 in terms of $k$ , the charge magnitude $q = 3.0\ \text{nC}$ , and the given distances. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 2. Same geometry as Figure 1, but Charge 2 has magnitude −6.0 nC at x = +0.20 m; all positions remain identical.

Indicate whether $\left|E_{x,\text{new}}(B)\right|$ is greater than, less than, or equal to $\left|E_x(B)\right|$ by writing one of the following. Later, Charge 2 is changed to $-6.0\ \text{nC}$ , but it remains at $x=+0.20\ \text{m}$ , as shown in Figure 2. Charge 1 remains $+3.0\ \text{nC}$ at $x=-0.20\ \text{m}$ . Let $E_{x,\text{new}}(B)$ be the new x-component of the electric field at point B.

$\left|E_{x,\text{new}}(B)\right| > \left|E_x(B)\right|$
$\left|E_{x,\text{new}}(B)\right| < \left|E_x(B)\right|$
$\left|E_{x,\text{new}}(B)\right| = \left|E_x(B)\right|$

Briefly justify your answer by referencing your derivation in part B.

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FRQ

Electric field from two point charges on x-axis

1. Two fixed point charges lie on the x-axis in air, as shown in Figure 1. Charge $q_1 = +3.0\ \mu\text{C}$ is located at $x = 0.00\ \text{m}$ and charge $q_2 = -6.0\ \mu\text{C}$ is located at $x = 0.40\ \text{m}$ . Consider points on the x-axis only. Ignore gravitational effects.

Figure 1. Two fixed point charges on the x-axis: q₁ = +3.0 μC at x = 0.00 m and q₂ = −6.0 μC at x = 0.40 m; a field point P lies on the x-axis at coordinate x.

Figure 2. Axes for graphing electric potential V as a function of position x (from −0.20 m to 0.80 m), with vertical reference lines at x = 0.00 m and x = 0.40 m.

Using the superposition principle, derive an expression for the electric potential $V(x)$ at a point on the x-axis due to the two charges for a location with $x ≠ 0.00\ \text{m}$ and $x ≠ 0.40\ \text{m}$ . Express your answer in terms of $x$ , $q_1$ , $q_2$ , and physical constants, as appropriate.

ii.

Derive an equation that could be solved to determine all positions $x$ on the x-axis where the electric potential is zero, $V(x)=0$ , for the region $x > 0.40\ \text{m}$ . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

iii.

On the axes shown in Figure 2, sketch a graph of the electric potential $V$ as a function of position $x$ from $x=-0.20\ \text{m}$ to $x=0.80\ \text{m}$ . Clearly indicate any vertical asymptotes and where $V$ is positive, negative, or zero.

Figure 3. Axes for graphing the x-component of the electric field Eₓ as a function of position x, with a vertical reference line at x = 0.40 m (boundary for the region x > 0.40 m).

Derive an expression for the x-component of the electric field $E_x(x)$ in the region $x > 0.40\ \text{m}$ by differentiating your expression for $V(x)$ from part A.i. Then, on the axes shown in Figure 3, sketch the qualitative behavior of $E_x$ versus $x$ in that region, including how $E_x$ behaves as $x \to 0.40^+$ and as $x \to \infty$ . The relationship between electric field and electric potential along the x-axis is given by $E_x = -\frac{dV}{dx}$ . Consider only the region $x > 0.40\ \text{m}$ .

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FRQ

Electric potential and field from fixed point charges

2. Two point charges are fixed on the x-axis, as shown in Figure 1. Charge $q_1 = +4.0\,\text{nC}$ is located at $x = -0.30\,\text{m}$ and charge $q_2 = -2.0\,\text{nC}$ is located at $x = +0.30\,\text{m}$ . A point P is on the x-axis at $x = 0.00\,\text{m}$ . Consider the electric potential $V(x)$ on the x-axis due only to these two charges. Take the electric potential to be zero at infinity.

Figure 1. Two point charges on the x-axis and four labeled observation points (A, P, B, C) with exact x-positions.

Figure 2. Bar-chart template for electric potential V at A, P, B, and C, plotted in volts, with VP shown as a fixed positive reference level.

In Figure 2, draw bars to represent the electric potential $V$ at points A, B, and C relative to $V_P$ shown at point P. The sign of the potential must be indicated by drawing the bar above (positive) or below (negative) the horizontal axis. If $V = 0$ at a point, write a "0" in that column. The electric potential at point P is $V_P$ . The partially completed bar chart in Figure 2 shows a bar that represents $V_P$ as a positive value.

Derive an expression for the x-component of the electric field, $E_x(x)$ , at a general point on the x-axis in the region $-0.30\,\text{m} < x < +0.30\,\text{m}$ in terms of $q_1$ , $q_2$ , $x$ , and physical constants, as appropriate. Begin your derivation by writing an expression for the electric potential $V(x)$ due to the two charges and then relate $E_x$ to the spatial rate of change of $V$ .

Figure 3. Axes for sketching electric potential V(x) from x = -0.80 m to +0.80 m, with vertical asymptotes at the charge locations.

On the axes shown in Figure 3, sketch a graph of the electric potential $V(x)$ as a function of position x for $-0.80\,\text{m} ≤ x ≤ +0.80\,\text{m}$ due to the two charges. The sketch must show the correct qualitative behavior near $x=-0.30\,\text{m}$ and $x=+0.30\,\text{m}$ and must be consistent with the relative values of $V$ at points A, P, B, and C from part A.

Indicate whether the electric potential energy of the test charge increases, decreases, or remains the same as it moves from P to C. Then calculate the change in electric potential energy $\Delta U$ of the test charge for the move from P to C. Briefly justify your answer by referencing the relationship between electric potential difference and potential energy. A test charge of magnitude $q_t = +1.50\,\text{nC}$ moves along the x-axis from point P at $x=0.00\,\text{m}$ to point C at $x=+0.60\,\text{m}$ . Use $k = 8.99\times10^9\,\text{N}\,\text{m}^2/\text{C}^2$ .

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AP Physics C: Electricity and Magnetism unit 9 practice test

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Unit 9 review notes

9.1

Electric Potential Energy

The electric potential energy of two point charges equals the work an external agent must do to assemble the configuration from infinite separation. The reference point is U_E = 0 at r = infinity. For any pair, U_E = kq1q2/r. For a system of three charges, add the three unique pairwise terms: U_12 + U_13 + U_23. The sign of U_E tells you whether the configuration is bound (negative, opposite charges) or requires stored energy to maintain (positive, like charges).

U_E = kq1q2/r: Potential energy of two point charges; k = 1/(4piepsilon_0) = 9x10^9 Nm^2/C^2; r is the separation distance.
Sign of U_E: Positive when q1q2 > 0 (like charges repel, energy stored); negative when q1q2 < 0 (opposite charges attract, energy released on assembly).
Pairwise superposition: Total U_E for N charges = sum of U_E for every unique pair; for three charges that is three terms.
Work-energy connection: Work done by the electric field on a charge equals -delta U_E; work done by an external force to move a charge quasi-statically equals +delta U_E.

A +2 micro-C charge and a -3 micro-C charge are 0.10 m apart. Is U_E positive or negative, and what does that sign mean physically?

Charge product q1*q2	Sign of U_E	Physical meaning
Positive (like charges)	Positive	Energy must be added to hold them at distance r
Negative (opposite charges)	Negative	System releases energy as charges approach from infinity

9.2

Electric Potential and Its Relationship to the Electric Field

Electric potential V is a scalar field: it assigns a single number (in volts = J/C) to every point in space. For a point charge, V = kq/r. For multiple point charges, add the individual potentials algebraically (scalar superposition). For a continuous distribution, integrate: V = (1/4pi*epsilon_0) integral dq/r. The potential difference between two points is delta V = -integral_a^b E dot dr. Conversely, the electric field component in any direction is E_x = -dV/dx. Equipotential lines are perpendicular to field lines everywhere; moving along an equipotential requires no work.

Scalar superposition: V_total = sum of kqi/ri for all point charges; no vector addition needed, just algebraic sum with signs.
Integral form: V = (1/4pi*epsilon_0) integral dq/r; used for rings, rods, disks, and other continuous distributions.
delta V = -integral E dot dr: Potential difference found by integrating the dot product of E and the path element; path independence means any path gives the same result.
E_x = -dV/dx: The electric field component in a direction equals the negative rate of change of V in that direction; steeper potential gradient means stronger field.
Equipotential lines: Curves of constant V; always perpendicular to E field vectors; no work is done moving a charge along an equipotential.

A thin ring of charge Q and radius R: write the expression for V at a point on the axis a distance x from the center, and explain why no integration over direction is needed.

Quantity	Type	Superposition rule	Key formula
Electric field E	Vector	Vector sum of components	E = kq/r^2 (direction matters)
Electric potential V	Scalar	Algebraic sum with signs	V = kq/r (no direction)

9.3

Conservation of Electric Energy

When a charge q moves between two points with a potential difference delta V, the change in electric potential energy is delta U_E = q delta V. Because the electrostatic force is conservative, the total mechanical energy is conserved when no other forces act: delta K + delta U_E = 0, so delta K = -q delta V. A positive charge moving from high V to low V loses potential energy and gains kinetic energy. A negative charge moving from high V to low V gains potential energy and slows down. The electron-volt (eV) is a convenient energy unit: 1 eV = 1.6x10^-19 J, the energy gained by one elementary charge moving through 1 V.

delta U_E = q delta V: Core equation for this topic; delta V = V_final - V_initial; sign of q and sign of delta V both matter.
delta K = -delta U_E: Conservation of energy with no non-conservative forces; kinetic energy gained equals potential energy lost.
Electron-volt (eV): Energy unit equal to 1.6x10^-19 J; useful when an elementary charge moves through a potential difference of 1 V.
Path independence: The work done by the electrostatic force depends only on the endpoints, not the path; this is what makes delta U_E = q delta V universally applicable.

An electron (q = -1.6x10^-19 C) is released from rest near a negative plate and moves toward a positive plate where the potential is 50 V higher. Does the electron speed up or slow down? Find its final kinetic energy.

Charge sign	Moves toward higher V	delta U_E	delta K
Positive (+q)	Yes	Positive (increases)	Negative (slows down)
Positive (+q)	No (lower V)	Negative (decreases)	Positive (speeds up)
Negative (-q)	Yes	Negative (decreases)	Positive (speeds up)
Negative (-q)	No (lower V)	Positive (increases)	Negative (slows down)

Key terms

Term	Definition
ΔU_{E}=q ΔV	The change in electric potential energy of a charged object equals its charge times the potential difference between the two locations; both the sign of q and the sign of delta V determine whether energy increases or decreases.
scalar field	A field that assigns a single scalar value to every point in space; electric potential V is a scalar field, which is why contributions from multiple charges are added algebraically rather than as vectors.
superposition principle	For electric potential, the total V at a point equals the algebraic sum of kqi/ri from each charge; no vector components are needed because V is a scalar.
equipotential line	A curve along which V is constant; the electric field is always perpendicular to equipotential lines, and no work is done moving a charge along one.
line integral	Used in delta V = -integral_a^b E dot dr to find the potential difference between two points by integrating the electric field along any path connecting them; path independence means the result is the same for all paths.
Potential from continuous charge distribution	V = (1/4pi*epsilon_0) integral dq/r; each infinitesimal charge element dq contributes kq/r to the total potential, and the contributions are summed by integration.
integration	The calculus technique used to sum infinitesimal dq contributions when finding V for a ring, rod, disk, or other continuous charge distribution.
dot product	In the relation delta V = -integral E dot dr, the dot product selects the component of E along the path direction; only the component of E parallel to displacement contributes to the potential difference.
voltage	The electric potential difference between two points, measured in volts (V = J/C); in Unit 9 it appears as delta V in both the field-integral relation and the energy equation delta U_E = q delta V.

How to review unit 9

Topics students miss most

Miss rates come from Fiveable multiple-choice practice.

9.1

Electric Potential Energy

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9.3

Conservation of Electric Energy

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Common unit 9 mistakes

Confusing U_E (energy) with V (potential)

U_E is the energy of a charge configuration in joules; V is energy per unit charge in volts. The relation is U_E = qV only when V is the potential at the location of charge q due to all other charges. Mixing up the two leads to incorrect setup of energy-conservation equations.

Getting the sign of delta U_E wrong

delta U_E = q delta V requires both the sign of q and the sign of delta V = V_f - V_i. A negative charge moving to higher potential has a negative delta U_E (it gains kinetic energy), which is the opposite of what many students expect.

Treating V like a vector when superposing

Electric potential is a scalar. When finding V due to multiple charges, add the individual kqi/ri values algebraically, including their signs. Do not add them as vectors the way you would for electric field components.

Forgetting to count all unique pairs in multi-charge systems

For three charges, there are three unique pairs (1-2, 1-3, 2-3). Students often miss one pair or double-count. Systematically list every pair before summing.

Misapplying E_x = -dV/dx direction

The electric field points in the direction of decreasing potential, not increasing potential. If V increases in the +x direction, E_x is negative (field points in -x). Sketch V vs. x and take the negative slope to avoid sign errors.

How this unit shows up on the AP exam

Deriving V from a charge distribution using integration

Free-response questions in AP Physics C: E&M frequently ask you to set up and evaluate the integral V = (1/4pi*epsilon_0) integral dq/r for a specific geometry such as a ring, arc, or rod. You are expected to identify the correct expression for dq, write r in terms of the geometry, and carry out or simplify the integral. Showing the setup clearly earns method credit even if the algebra is complex.

Connecting V and E through derivatives and integrals

Exam tasks often give you a potential function V(x) and ask for the electric field, or give you E and ask for delta V. You need to apply E_x = -dV/dx in one direction and delta V = -integral E dot dr in the other. Equipotential diagrams may also appear, requiring you to sketch field lines or determine the direction and relative magnitude of E from the spacing of isolines.

Energy conservation with sign-convention reasoning

Multi-part problems commonly combine delta U_E = q delta V with the work-energy theorem to find the speed of a charged particle after it moves through a potential difference. You must handle the sign of q (positive or negative charge) and the sign of delta V correctly to determine whether the particle speeds up or slows down. Justifying your sign reasoning in words is a common written-response expectation.

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Topic study guides

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Frequently Asked Questions

What's on the AP Physics E&M Unit 9 progress check (MCQ and FRQ)?

The AP Physics E&M Unit 9 progress check includes both MCQ and FRQ parts drawn from all three unit topics: electric potential energy (9.1), electric potential (9.2), and conservation of electric energy (9.3). MCQ questions typically ask you to calculate or compare potential values and energy changes, while the FRQ part asks you to derive relationships and apply conservation principles to charged-particle scenarios. Check Unit 9 for practice that mirrors the progress check format.

How do I practice AP Physics E&M Unit 9 FRQs?

The best way to practice AP Physics E&M Unit 9 FRQs is to work through problems that ask you to derive expressions for electric potential energy and electric potential, then apply conservation of electric energy to find final speeds or positions of charged particles. FRQs from this unit often combine a derivation step with a graphical interpretation or a multi-part scenario involving point charges or uniform fields. Start by writing out your energy conservation equation explicitly before solving, since College Board awards method points. Head to Unit 9 for FRQ practice sets tied to these topics.

Where can I find AP Physics E&M Unit 9 practice questions?

You can find AP Physics E&M Unit 9 practice questions, including multiple-choice and practice test sets, at Unit 9. The page organizes MCQ and FRQ practice by topic, covering electric potential energy, electric potential, and conservation of electric energy. Working through topic-by-topic MCQs first, then full practice test sections, is the most efficient way to build confidence across all three topics.

How should I study AP Physics E&M Unit 9?

Start with electric potential energy (9.1) and make sure you can calculate work done by electric forces before moving to electric potential (9.2), since potential is just potential energy per unit charge. Once those two are solid, conservation of electric energy (9.3) clicks into place naturally. A concrete plan: sketch field diagrams alongside potential diagrams for the same charge configuration so you see how they relate, then practice writing energy conservation equations from scratch without a formula sheet. Prioritize problems where a charged particle moves between two points, since those appear most often on the exam. Use Unit 9 to find topic-specific practice after each section.

Ready to review Unit 9?Start with the notes, check the topic cards, and use the practice or resource links when they are available for this course.

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