This question tests AP Physics C understanding of charge oscillation equations in ideal LC circuits. In an ideal LC circuit with no resistance, the charge oscillates sinusoidally with angular frequency ω₀ = 1/√(LC). If the capacitor is initially charged to Q, the charge varies as q(t) = Q cos(ω₀t), assuming q = Q at t = 0. Choice B is correct because it shows the cosine function with the proper angular frequency ω₀ = 1/√(LC). Choice A describes exponential decay in an RC circuit, not oscillations in an LC circuit. Choice C has the wrong formula for ω₀ (should be 1/√(LC), not √(LC)). Choice D incorrectly suggests linear charge growth, which violates energy conservation. To help students: Recognize that LC circuits produce sinusoidal oscillations, not exponential decay. The initial conditions determine whether to use sine or cosine - if q(0) = Q, use cosine; if q(0) = 0, use sine.

This question tests AP Physics C understanding of energy storage mechanisms in LC circuits. In LC oscillations, energy alternates between electric field energy in the capacitor and magnetic field energy in the inductor. The inductor stores energy in its magnetic field according to U = ½LI², where the current through the inductor creates a magnetic field in and around the coil. Choice B is correct because inductors specifically store magnetic field energy through the magnetic flux created by current flow. Choice A is incorrect as capacitors store electric field energy (U = ½CV²), Choice C is incorrect because resistors dissipate rather than store energy, and Choice D is incorrect as wires ideally have negligible inductance and don't store significant magnetic energy. To help students: Visualize the energy oscillation - when current is maximum through the inductor, all energy is magnetic; when voltage is maximum across the capacitor, all energy is electric. Remember the fundamental property: inductors oppose current changes through magnetic fields.

This question tests AP Physics C understanding of calculating resonant frequency in LC circuits. The resonant frequency formula for an LC circuit is f₀ = 1/(2π√(LC)), which describes the natural oscillation frequency when energy alternates between electric and magnetic fields. Substituting the given values: f₀ = 1/(2π√(2.0×10⁻⁴ × 5.0×10⁻¹⁰)) = 1/(2π√(1.0×10⁻¹³)) = 1/(2π × 1.0×10⁻⁶·⁵) ≈ 1.59×10⁵ Hz. Choice C is correct because it gives 1.6×10⁵ Hz, which matches our calculation when rounded to two significant figures. The other choices represent common calculation errors: Choice A (1.6×10⁴ Hz) is off by a factor of 10, Choice B (5.0×10⁵ Hz) appears to use an incorrect formula, and Choice D (1.6×10⁶ Hz) is also off by a factor of 10 in the opposite direction. To help students: Always check unit consistency (H×F = s²), practice calculating square roots of scientific notation, and verify your answer makes physical sense for typical LC circuit frequencies.

This question tests AP Physics C understanding of how capacitance affects resonant frequency in LC circuits. The resonant frequency formula f₀ = 1/(2π√(LC)) shows that frequency is inversely proportional to the square root of capacitance. When capacitance C is doubled, the new frequency becomes f₀' = 1/(2π√(L×2C)) = 1/(2π√(2)√(LC)) = f₀/√2. Choice A is correct because doubling the capacitance decreases the frequency by a factor of √2, reflecting the inverse square root relationship. Choice B incorrectly suggests frequency increases, Choice C suggests a factor of 2 instead of √2, and Choice D incorrectly claims frequency is independent of capacitance. To help students: Remember that larger capacitance means more charge storage capacity, requiring more time to charge/discharge, thus lowering the oscillation frequency. Practice manipulating the resonant frequency formula algebraically before substituting numbers.

This question tests AP Physics C understanding of calculating required capacitance for a specific resonant frequency. Rearranging f₀ = 1/(2π√(LC)) to solve for C: C = 1/(4π²f₀²L). Substituting L = 1.0×10⁻⁶ H and f₀ = 1.0×10⁸ Hz: C = 1/(4π² × (1.0×10⁸)² × 1.0×10⁻⁶) = 1/(4π² × 10¹⁶ × 10⁻⁶) = 1/(4π² × 10¹⁰) = 1/(3.95×10¹¹) ≈ 2.5×10⁻¹² F. Choice A is correct with C = 2.5×10⁻¹² F, which matches our calculation. The other choices represent order-of-magnitude errors: B is 100 times too large, C is 10,000 times too large, and D is 1,000,000 times too large. To help students: When solving for C or L, first rearrange the formula algebraically, then substitute values carefully tracking units and exponents. Remember that high frequencies require small capacitances in LC circuits.

This question tests AP Physics C understanding of electromagnetic induction in LC circuit oscillations. Faraday's law states that a changing magnetic flux induces an emf, and Lenz's law specifies that this induced emf opposes the change causing it. In an inductor, the self-induced emf is ε = -L(dI/dt), where the negative sign indicates opposition to current changes. Choice A is correct because it accurately describes how changing current in the inductor creates an opposing emf, which is the fundamental mechanism enabling LC oscillations. Choice B is incorrect because constant current produces no induced emf (dI/dt = 0). Choice C incorrectly mentions magnetic monopoles, which don't exist. Choice D incorrectly relates charge changes directly to emf without considering the role of current. To help students: Focus on the rate of change - inductors respond to dI/dt, not I itself. This opposition to change is what creates the oscillatory behavior when combined with a capacitor.

This question tests AP Physics C understanding of calculating required inductance for a target resonant frequency. Rearranging f₀ = 1/(2π√(LC)) to solve for L: L = 1/(4π²f₀²C). Substituting C = 1.0×10⁻¹⁰ F and f₀ = 5.0×10⁶ Hz: L = 1/(4π² × (5.0×10⁶)² × 1.0×10⁻¹⁰) = 1/(4π² × 25×10¹² × 10⁻¹⁰) = 1/(4π² × 25×10²) = 1/(9.87×10³) ≈ 1.0×10⁻⁴ H. Choice B is correct with L = 1.0×10⁻⁴ H, matching our calculation. Choices A (10⁻² H) and C (10⁻³ H) are too large by factors of 100 and 10 respectively, while Choice D (10⁻⁵ H) is too small by a factor of 10. To help students: Practice algebraic manipulation of the resonant frequency formula before substituting numbers. Verify your answer produces the desired frequency when substituted back into the original formula.

This question tests AP Physics C understanding of how inductance changes affect resonant frequency. From f₀ = 1/(2π√(LC)), frequency is inversely proportional to the square root of inductance. When L increases by a factor of 9, the new frequency becomes f₀' = 1/(2π√(9L×C)) = 1/(2π×3√(LC)) = f₀/3. Choice B is correct because increasing inductance by a factor of 9 decreases the frequency by a factor of 3 (the square root of 9). Choice A incorrectly suggests frequency increases, Choice C incorrectly uses the factor 9 directly instead of its square root, and Choice D incorrectly claims frequency is independent of inductance. To help students: Remember that larger inductance means stronger opposition to current changes, slowing down the oscillations. The square root relationship is crucial - if L changes by a factor of n², frequency changes by a factor of n.

This question tests AP Physics C understanding of how circuit parameters affect resonant frequency in LC oscillations. LC circuits have a resonant frequency that depends inversely on the square root of both inductance and capacitance. Based on the passage, the relationship f₀ = 1/(2π√(LC)) shows that frequency is inversely proportional to √C. Choice B is correct because when C increases by a factor of 4, the square root of C increases by a factor of 2, causing f₀ to decrease by a factor of 2. Choice A incorrectly assumes a direct relationship, choice C confuses the square root relationship, and choice D reverses the inverse relationship. To help students: Remember that f₀ ∝ 1/√C, not 1/C. Draw graphs showing how frequency changes with capacitance. Practice problems involving proportional relationships with square roots to build intuition.

This question tests AP Physics C understanding of electromagnetic induction in inductors within LC circuits. LC circuits rely on the inductor's ability to oppose changes in current through electromagnetic induction, creating oscillations. Based on the passage and Faraday's law, an inductor generates an induced emf that opposes changes in current, given by ε = -L(dI/dt). Choice A is correct because this equation represents the fundamental relationship for inductors, where the negative sign indicates opposition to current changes (Lenz's law), L is the inductance, and dI/dt is the rate of current change. Choice B incorrectly applies a capacitor-like formula, choice C gives the voltage across a capacitor (V = Q/C), and choice D is Ohm's law for resistors. To help students: Remember that inductors oppose current changes, not current itself. The derivative dI/dt is crucial - it's the rate of change that matters. Connect this to energy storage: changing current means changing magnetic field means induced emf.