Two isolated conducting spheres of radii R and 4R are connected by a thin conducting wire and charged to a total potential V. Which expression correctly compares the surface charge density σ_small of the smaller sphere to the surface charge density σ_large of the larger sphere?

The charge density σ_small is four times the density σ_large.

The charge density σ_small is one-fourth the density σ_large.

The charge density σ_small is equal to the density σ_large.

The charge density σ_small is two times the density σ_large.

A cylindrical wire of radius r delivers a constant current density J to charge an initially neutral object for time t. If the wire is replaced by one with radius 2r carrying the same current density J, what is the total charge delivered over the same time t?

The total charge is four times the original charge.

The total charge is two times the original charge.

The total charge is eight times the original charge.

The total charge is sixteen times the original charge.

AP Physics C: E&M Unit 8 Practice Test

Pick a unit and take a timed, AP^®-style test built only from that unit's questions — with AI feedback on every free-response answer.

unit 8 unit 9 unit 10 unit 11 unit 12 unit 13

Electric Charges & Fields: Gauss's Law

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Two ways to practice unit 8

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The exam experience, sized to one unit — timed like the real thing

20 Multiple Choice Questions

40 min

1 FRQ – FRQ 1 – Mathematical Routines

24 min

1 FRQ – FRQ 2 – Translation Between Representations

24 min

Sections run back-to-back with the same directions, timers, and scoring as the real exam

Total: 1 hr 28 min

customizable

Pick your own mix of question types

multiple choice questions

free response questions (choose as many as you want)

FRQ 1 – Mathematical Routines · 24 minFRQ 2 – Translation Between Representations · 24 minFRQ 3 – Experimental Design · 24 minFRQ 4 – Qualitative/Quantitative Translation · 24 min

Total: 40 min

How scoring works: multiple choice is scored instantly, free-response answers get point-by-point AI feedback, and your result is a percentage for this unit — not a projected AP^® score.

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AP Physics C: E&M Unit 8 practice questions

Try 10 multiple-choice questions from Unit 8 here. Pick an answer to see why it's right or wrong, then take the timed test above for a full AP-style set.

Question 1
A conducting sphere of radius R carries an initial net charge +Q, while a second conducting sphere of radius 3R is initially neutral. The two spheres are connected by a long, thin conducting wire until equilibrium is reached. What is the final charge on the sphere of radius 3R?
A final charge of +Q/4
A final charge of +Q/3
A final charge of +Q/2
A final charge of +3Q/4
Show answer
Answer: (D)
Equating the electric potentials (V = kQ/r) of the connected spheres reveals that charge distributes proportional to radius. The sphere with radius 3R holds three times the charge of the sphere with radius R, accounting for 3/4 of the total charge +Q.
Question 2
Two isolated conducting spheres of radii R and 4R are connected by a thin conducting wire and charged to a total potential V. Which expression correctly compares the surface charge density σ_small of the smaller sphere to the surface charge density σ_large of the larger sphere?
The charge density σ_small is one-fourth the density σ_large.
The charge density σ_small is equal to the density σ_large.
The charge density σ_small is two times the density σ_large.
The charge density σ_small is four times the density σ_large.
Show answer
Answer: (D)
Combining the potential formula (V = kQ/R) with the density formula (σ = Q/4πR²) shows that σ is proportional to V/R. Since V is constant for both connected spheres, the sphere with 1/4 the radius must have 4 times the surface charge density.
Question 3
A cylindrical wire of radius r delivers a constant current density J to charge an initially neutral object for time t. If the wire is replaced by one with radius 2r carrying the same current density J, what is the total charge delivered over the same time t?
The total charge is two times the original charge.
The total charge is four times the original charge.
The total charge is eight times the original charge.
The total charge is sixteen times the original charge.
Show answer
Answer: (B)
Current I is the product of current density J and cross-sectional area A (I = Jπr²). Doubling the radius increases the area by a factor of 4. Since Q = It, the total charge delivered also increases by a factor of 4.
Question 4
Two small spheres with charges $q_1$ and $q_2$ are separated by a distance $r$ and exert an electrostatic force of magnitude $F$ . If charge $q_1$ is doubled and the distance $r$ is tripled, what is the new magnitude of the force?
$\frac{2}{9}F$
$\frac{2}{3}F$
$\frac{4}{9}F$
$6F$
Show answer
Answer: (A)
Applying Coulomb's Law, the force is proportional to the product of charges and inversely proportional to the square of the distance. Doubling one charge multiplies the force by 2, and tripling the distance divides the force by $3^2=9$ , resulting in a factor of $2/9$ .
Question 5
Two identical conducting spheres carry charges $+Q$ and $-3Q$ respectively and are separated by a distance $d$ . They attract each other with a force of magnitude $F_0$ . The spheres are brought into contact, allowing charge to equilibrate, and then returned to the same distance $d$ . What is the new magnitude of the force between them?
$F_0/3$
$F_0/9$
$3F_0$
$9F_0$
Show answer
Answer: (A)
Calculating the initial force proportional to $|(+Q)(-3Q)| = 3Q^2$ . Upon contact, the net charge $-2Q$ is shared equally, leaving each with $-Q$ . The final force is proportional to $|(-Q)(-Q)| = Q^2$ , which is one-third the initial magnitude.
Question 6
A proton and an alpha particle (charge $+2e$ , mass $4m_p$ ) are held at rest a distance $r$ apart. When released, the proton experiences an initial acceleration of magnitude $a_p$ . What is the magnitude of the initial acceleration of the alpha particle?
$a_p/4$
$a_p/2$
$2a_p$
$4a_p$
Show answer
Answer: (A)
Using Newton's Third Law, the electrostatic force $F$ on both particles is equal in magnitude. Since $a = F/m$ and the alpha particle has 4 times the mass of the proton, its acceleration is $F/(4m_p) = a_p/4$ .
Question 7
Two point charges, $+q$ and $+9q$ , are fixed at positions $x=0$ and $x=L$ , respectively. At what position $x$ along the line connecting them is the net electric force on a third positive test charge equal to zero?
$L/4$
$L/3$
$L/2$
$3L/4$
Show answer
Answer: (A)
Setting the forces equal, $k\frac{qQ}{x^2} = k\frac{9qQ}{(L-x)^2}$ , leads to $\frac{1}{x} = \frac{3}{L-x}$ . Solving for $x$ yields $L-x = 3x$ , so $4x = L$ and $x = L/4$ .
Question 8
Three identical point charges $+q$ are placed at the vertices of an equilateral triangle with side length $a$ . The magnitude of the net electric force on any one charge is $F$ . If the side length of the triangle is increased to $2a$ , what is the new magnitude of the net force on one charge?
$F/4$
$F/2$
$2F$
$4F$
Show answer
Answer: (A)
Recognizing that the geometry (angles) remains constant, the net force depends only on the distance scaling. Since Coulomb's force scales as $1/r^2$ , doubling the distance reduces the force by a factor of $2^2=4$ .
Question 9
Two point charges exert a repulsive force of magnitude $F$ on each other. If the charge of one particle is tripled and the distance between them is reduced to one-third of its original value, what is the new force magnitude?
$27F$
$9F$
$3F$
$F/3$
Show answer
Answer: (A)
Applying Coulomb's Law, the new force is proportional to $\frac{(3)(1)}{(1/3)^2}$ . The denominator becomes $1/9$ , so the total factor is $3 \times 9 = 27$ .
Question 10
An electron travels in a circular orbit of radius $r$ around a proton. If the radius of the orbit is increased to $4r$ , by what factor does the orbital speed of the electron change?
It decreases by a factor of 2
It decreases by a factor of 4
It increases by a factor of 2
It increases by a factor of 4
Show answer
Answer: (A)
Equating electric force to centripetal force ( $k e^2/r^2 = mv^2/r$ ) shows that $v^2 \propto 1/r$ , so $v \propto 1/\sqrt{r}$ . Increasing $r$ by 4 decreases $v$ by $\sqrt{4}=2$ .

What's on the AP Physics C: E&M Unit 8 practice test

The standard version takes about 88 minutes and follows the timing and directions of the AP exam, sized to one unit:

20 Multiple Choice Questions (40 minutes)
1 FRQ – FRQ 1 – Mathematical Routines (24 minutes)
1 FRQ – FRQ 2 – Translation Between Representations (24 minutes)

Questions are drawn from 1062 multiple-choice questions and 8 FRQs written for Unit 8, Electric Charges & Fields: Gauss's Law.

Practice AP Physics C: E&M Unit 8 FRQs

Answer the unit's free-response questions one at a time and get a score on every part.

FRQ 1 – Mathematical Routines (1 question)
FRQ 2 – Translation Between Representations (1 question)
FRQ 3 – Experimental Design (4 questions)
FRQ 4 – Qualitative/Quantitative Translation (2 questions)

AP Physics C: E&M Unit 8 Practice Test

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AP Physics C: E&M Unit 8 practice questions

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