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Flashcards · AP® Physics C: Electricity and Magnetism · Unit 8

Unit 8: Electric Charges, Fields, and Gauss’s Law flashcards

47 cards on topics 8.1 to 8.6. Pick a topic or study the whole unit.

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8.1 Electric Charge and Electric Force

What is Coulomb's law?
F = (1/(4πε₀)) |q₁q₂|/r², with 1/(4πε₀) = 8.99 × 10⁹ N·m²/C² and r the separation. Like charges repel, unlike attract, and the two forces of the pair are equal and opposite.
How do you find the net electric force from several point charges?
Find each force's size from Coulomb's law using magnitudes, decide each direction by "like repel, unlike attract", then add the vectors by components.
How is charge quantized?
Every object's charge is q = ne, with n a whole number and e = 1.60 × 10⁻¹⁹ C. A proton has +e, an electron −e, a neutron 0.
How does an object become charged?
By gaining electrons (negative) or losing them (positive). Protons do not move between objects.
How does the electric force compare with gravity?
Between particles the electric force is far larger: for a proton and electron F_e/F_g ≈ 2.3 × 10³⁹, independent of r. Gravity dominates at large scales because large bodies are nearly neutral.
What is polarization, and what does permittivity measure?
Polarization is a field shifting electrons relative to nuclei inside a material. Permittivity measures how much a material or medium polarizes; free space has ε₀ = 8.85 × 10⁻¹² C²/(N·m²).
Why does a charged rod attract a neutral piece of paper?
It polarizes the paper. The nearer, opposite charges are attracted more strongly than the farther, like charges are repelled, because F ∝ 1/r².
How does the Coulomb force change if the separation is halved?
F ∝ 1/r², so it becomes 4 times larger. Convert nC, μC and cm to SI units before calculating.

8.2 Conservation of Electric Charge and the Process of Charging

What is conservation of charge?
The net charge of a system is constant unless charge moves between the system and its surroundings. Charge is moved, never created or destroyed.
What happens when two objects are charged by friction?
Electrons move from one material to the other, so the two objects end with equal and opposite charges. If the rod gains −q, the cloth gains +q.
What happens when two identical conductors touch?
Charge flows until they share it equally: q_each = (q₁ + q₂)/2. Add the charges first, then share. Unequal sizes share unequally.
How do you charge a sphere by induction?
Bring the rod near to polarize the sphere, ground the sphere, remove the ground, then remove the rod. The sphere ends with the opposite sign to the rod.
What goes wrong if you remove the rod before the ground when charging by induction?
The electrons flow back through the ground connection and the sphere ends neutral.
What is grounding?
Connecting an object to a huge, nearly neutral system such as Earth, which can supply or absorb any number of electrons.
Why is attraction not proof that an object is charged?
Neutral objects are attracted to charged ones by polarization. Only repulsion proves an object is charged.
How many electrons are moved when a rod gains −3.2 nC?
N = |Δq|/e = 3.2 × 10⁻⁹ ÷ 1.60 × 10⁻¹⁹ = 2.0 × 10¹⁰ electrons.

8.3 Electric Fields

What is the electric field?
The force per unit charge on a small test charge: E = F/q₀, in N/C. It is a vector set up by the source charges and does not depend on q₀.
What is the force on a charge in an electric field?
F = qE. A positive charge is pushed along E; a negative charge is pushed opposite to E.
What is the field of a point charge?
E = kq/r² = q/(4πε₀r²), pointing away from a positive charge and toward a negative one. E ∝ q and E ∝ 1/r².
How do you find the field of several point charges?
Superposition: E_net = E₁ + E₂ + …. Find each field using the distance from that source to the point, then add x- and y-components.
What are the rules for drawing electric field lines?
Lines are tangent to E, start on positive charges and end on negative ones, are denser where E is stronger, and never cross. They show force direction, not paths.
What is the field inside a conductor in electrostatic equilibrium?
E = 0 inside the metal, with all excess charge on the surface, and E perpendicular to the surface just outside. Inside an insulator, E can be nonzero.
What is the field outside a spherically symmetric charge distribution?
E = kQ/r² for r ≥ R, as if all the charge Q were a point charge at the center.
What is the force on a −3.0 nC charge in a 2.0 × 10⁴ N/C field pointing east?
F = |q|E = 3.0 × 10⁻⁹ × 2.0 × 10⁴ = 6.0 × 10⁻⁵ N, directed west because the charge is negative.

8.4 Electric Fields of Charge Distributions

How do you find the field of a continuous charge distribution?
Split it into point charges dq, each giving dE = k dq/r², and integrate: E = ∫ k dq/r² r̂. Integrate components, not magnitudes, unless every dE points the same way.
How do you write dq for a line or an arc?
dq = λ dx for a line or dq = λR dθ for an arc. Uniform rod: λ = Q/L. Uniform semicircle: λ = Q/(πR), since its length is πR.
How do you set up a field integral correctly?
Write r and every sin θ or cos θ in terms of the single variable you integrate over, use symmetry to decide which components cancel, then check the far-away limit and direction.
What is the field of a long line of charge?
E = λ/(2πε₀r) = 2kλ/r, radially outward for positive λ. It falls as 1/r, not 1/r².
What is the field on the axis of a charged ring?
E = kQz/(z² + a²)^(3/2) along the axis. It is zero at the center and greatest at z = a/√2. Far away it approaches kQ/z².
What is the field at the center of a uniformly charged semicircle?
E = 2kλ/R = 2kQ/(πR²), along the symmetry line, away from the arc for positive charge. The components parallel to the diameter cancel.
What is the field of a finite rod at a point on its own line?
E = kQ/[d(d + L)], along the line away from the rod (positive charge), where d is the distance from the near end. For d much larger than L this tends to kQ/d².
What is the field of a finite rod on its perpendicular bisector?
E = kQ/[x√(x² + L²/4)], perpendicular to the rod. When L is much larger than x, it becomes 2kλ/x, the infinite-line field.

8.5 Electric Flux

What is electric flux?
A measure of the field passing through a surface, proportional to the number of field lines crossing it. For a uniform field and flat surface, Φ_E = E·A = EA cos θ. Flux is a signed scalar in N·m²/C.
What is the area vector?
A vector with the size of the area, perpendicular to the surface. For an open surface you choose a side; for a closed surface it always points outward.
Which angle goes into Φ_E = EA cos θ?
θ is between E and the area vector (the normal), not the surface. If α is given between E and the surface, Φ_E = EA sin α.
How do you find flux when the field varies over a surface?
Integrate: Φ_E = ∫E·dA. Choose strips along which E is constant, perpendicular to the direction in which E changes. For a closed surface, Φ_E = ∮E·dA with outward dA.
What is the net flux of a uniform field through a closed surface?
Zero: every line that enters also leaves. Entering lines give negative flux and leaving lines positive. Zero net flux does not mean zero field.
What is the flux through a curved open surface in a uniform field?
The same as through the flat area across its rim. For a dome of radius R, use πR², not the curved area.
What is the flux through a surface parallel to the field?
Zero, since the area vector is perpendicular to E and cos 90° = 0.

8.6 Gauss’s Law

What does Gauss's law state?
The net flux through any closed surface equals the enclosed charge divided by ε₀: ∮E·dA = q_enc/ε₀, with outward area vectors. Resizing the surface does not change the flux if q_enc is the same.
When can Gauss's law give the field directly?
Always true, but useful only when symmetry makes E constant and perpendicular on parts of the Gaussian surface and parallel (zero flux) on the rest: spherical, cylindrical or planar symmetry.
What is the field inside and outside a uniformly charged solid sphere?
Outside (r ≥ R): E = Q/(4πε₀r²). Inside: E = Qr/(4πε₀R³) = ρr/(3ε₀), rising linearly with r. The two match at r = R.
What is the field of an infinite thin sheet of charge?
E = σ/(2ε₀), independent of distance. Two faces of the pillbox carry flux, which is why it is not σ/ε₀.
What is the field of a line charge from Gauss's law?
Using a coaxial cylinder of radius r and length ℓ: E(2πrℓ) = λℓ/ε₀, so E = λ/(2πε₀r). The end caps carry no flux.
What is the field inside a uniformly charged long cylinder?
E = ρr/(2ε₀) for points inside the cylinder, using a coaxial Gaussian cylinder that encloses ρπr²ℓ.
How do you use Gauss's law for a sphere with non-uniform density ρ(r)?
Integrate to find the enclosed charge: q_enc = ∫₀ʳ ρ(r′)4πr′² dr′, then E = q_enc/(4πε₀r²). For ρ = ρ₀r/R, q_enc = πρ₀r⁴/R.
Does zero flux through a closed surface mean zero field on it?
No. Zero net flux means zero enclosed charge. The field on the surface can still be nonzero.

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