Praxis 5266

Explanation

Coulomb's law states F = k|q₁||q₂|/r². With charges +Q and −Q separated by distance d, the magnitude of the force is F = kQ²/d². When the separation is changed to r' = d/3, the new force is F' = kQ²/(d/3)² = kQ²·9/d² = 9F. Thus the force increases by a factor of 9. Option B incorrectly states the force increases by a factor of 3, confusing an inverse-square relationship with an inverse-linear relationship. Option C incorrectly writes the original force as kQ/d² rather than kQ²/d², omitting one factor of Q; this would be dimensionally incorrect. Option D incorrectly states the force decreases; reducing separation always increases the Coulomb force between two charges.

Explanation

The large-angle scattering of a small fraction of alpha particles could only be explained if most of the atom's mass and all positive charge were concentrated in an extremely small, dense nucleus. This ruled out the Thomson 'plum pudding' model. Choice A describes the Thomson model, which Rutherford's experiment disproved — a diffuse positive charge would not produce large-angle deflections. Choice C describes Bohr's model of quantized electron shells, which was developed after and inspired by Rutherford's nuclear model, but is not what Rutherford directly concluded from his scattering data. Choice D is incorrect because alpha particles are deflected by the nucleus via Coulomb repulsion, not by interactions with electrons, which are far too light to cause significant deflection.

Explanation

The maximum theoretical efficiency of a heat engine is given by the Carnot efficiency: η = 1 − (T_C / T_H), where T_C is the cold reservoir temperature and T_H is the hot reservoir temperature, both in Kelvin. Here, η = 1 − (320/800) = 1 − 0.40 = 0.60, or 60%. Option A (40%) represents the ratio T_C/T_H, which is the fraction of heat that is rejected to the cold reservoir, not the efficiency. Option C (75%) is incorrect and does not follow from the Carnot formula with these temperatures. Option D (25%) is incorrect and likely results from arithmetic errors or an incorrect formula such as using temperature differences divided by the sum of temperatures.

Explanation

In alpha decay, the parent nucleus emits an alpha particle (⁴He, Z = 2, A = 4). The daughter nucleus therefore has atomic number Z_daughter = 92 − 2 = 90 (thorium) and mass number A_daughter = 238 − 4 = 234. This gives thorium-234 (²³⁴Th). Option B incorrectly subtracts only 2 from the mass number instead of 4. Option C incorrectly subtracts only 1 from Z (as in beta-minus decay) while correctly subtracting 4 from A. Option D leaves Z unchanged and subtracts 4 from A, which would correspond to a decay mode that emits only neutrons, not alpha decay.

Explanation

Einstein's photoelectric equation states that the maximum kinetic energy of ejected electrons equals the energy of the incident photon (hf) minus the work function (φ), which is the minimum energy required to remove an electron from the metal surface: KE_max = hf − φ. Option B is incorrect because adding φ would imply the metal provides extra energy to the electron, which violates energy conservation. Option C is incorrect because it would yield a negative KE_max when hf > φ, which is the normal condition for photoemission. Option D is incorrect because it multiplies rather than subtracts the two energy terms, which has no physical basis in the photoelectric model.

Explanation

Coulomb's law gives F = k|q₁||q₂|/r² = (9.0 × 10⁹)(3 × 10⁻⁶)(3 × 10⁻⁶)/(0.10)² = (9.0 × 10⁹)(9 × 10⁻¹²)/(0.01) = (8.1 × 10⁻²)/(0.01) = 8.1 N. The force is attractive because the charges are opposite in sign, but the magnitude is 8.1 N. Choice B (2.7 N) results from incorrectly using r instead of r² in the denominator, giving F = kq²/r = (9×10⁹)(9×10⁻¹²)/0.10 = 0.81/0.10 = 8.1—actually this also gives 8.1 if done correctly; 2.7 might arise from arithmetic errors or using a single charge value. Choice C (81 N) results from forgetting to square the distance (using d = 0.01 m² rather than d² = 0.01 m²) or misplacing a decimal. Choice D (0.81 N) results from using r² = (0.10)² = 0.01 but then making an additional factor-of-10 error, perhaps dividing by 0.10 m instead of 0.01 m².

Explanation

According to the first law of thermodynamics, ΔU = Q - W, where Q is the heat added to the system and W is the work done BY the system. Here, ΔU = 500 J - 200 J = 300 J. Option B (700 J) incorrectly adds Q and W instead of subtracting. Option C (-300 J) would result if the signs of Q and W were reversed or if W were subtracted from a negative Q. Option D (-700 J) applies the wrong sign convention throughout. The correct application of the first law yields a positive 300 J increase in internal energy.

Explanation

In a series circuit, total resistance is the sum of individual resistances: R_total = 4 + 8 = 12 ohms. Using Ohm's Law, I = V/R = 12V / 12Ω = 1.0 A. This is correct. Option A (0.75 A) would result from incorrectly using only one resistor or a calculation error. Option C (1.5 A) might result from dividing 12V by only 8Ω, ignoring the 4Ω resistor. Option D (3.0 A) would result from dividing 12V by only 4Ω, ignoring the 8Ω resistor.

Explanation

The Carnot efficiency is given by η = 1 - (T_cold/T_hot) = 1 - (300/600) = 1 - 0.5 = 0.50, or 50%. This is the maximum efficiency any heat engine can achieve operating between these two temperatures. Option A (25%) would result from an incorrect calculation such as using (T_hot - T_cold)/T_hot² or other arithmetic errors. Option C (75%) might result from incorrectly computing 1 - (T_cold/T_hot) as (T_hot - T_cold)/T_cold = 300/300 = 1, or misapplying the formula. Option D (100%) is impossible by the second law of thermodynamics; no real or ideal engine can convert heat entirely into work when operating between two finite-temperature reservoirs.

Explanation

Gauss's Law states that the total electric flux Φ through any closed (Gaussian) surface equals the net enclosed charge divided by the permittivity of free space: Φ = Q_enc/ε₀. This is exactly described by option A. Option B describes the inverse-square dependence of Coulomb's Law, not Gauss's Law directly. Option C is a statement related to the conservative nature of electrostatic fields (derived from Faraday's Law in the static limit), which is a different Maxwell equation (∮E·dl = 0 for static fields). Option D describes the definition of electric potential difference (work per unit charge), which is a separate concept from Gauss's Law.