Two identical spheres of mass are separated by . They carry equal and opposite charges and . What value of would make the magnitudes of the electric and gravitational forces equal? Use and (SI units).
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Practice Compare Gravitational And Electric Forces in Physics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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Two identical spheres of mass m=1.0kg are separated by r=1.0m. They carry equal and opposite charges +q and −q. What value of q would make the magnitudes of the electric and gravitational forces equal? Use G=6.67×10−11 and k=9.0×109 (SI units).
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Two identical spheres of mass m=1.0kg are separated by r=1.0m. They carry equal and opposite charges +q and −q. What value of q would make the magnitudes of the electric and gravitational forces equal? Use G=6.67×10−11 and k=9.0×109 (SI units).
Explanation: This question tests understanding of how to equate gravitational and electric force magnitudes for macroscopic objects. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = k|q₁q₂|/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values; additionally, gravity is always attractive while electric force can be attractive or repulsive, and gravity depends on mass while electric force depends on charge. Setting F_elec = F_grav gives k q² / r² = G m² / r², so q² = (G m²)/k, yielding q ≈ 2.7 × 10⁻¹⁰ C for the given parameters—this tiny charge equalizes the forces because k >> G. Choice C is correct because it properly calculates q from equating the force expressions. Choice B has an error in the power of 10, reporting q as 10^{-6} instead of the correct 10^{-10}, likely from incorrectly handling exponents when dividing powers of 10 in the calculation: G/k ≈ 10^{-11}/10^9 = 10^{-20}, sqrt(m² * 10^{-20}) = m * 10^{-10}. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav enormous while F_elec ≈ 0.
Two particles are separated by a distance r. Both the gravitational and electric forces between them follow an inverse-square law. If the separation distance is doubled (from r to 2r), how do the magnitudes of Fgrav and Felec change, and what happens to the ratio Felec/Fgrav?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For scale dependence: The ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of distance r (since both forces have r² in denominator which cancels in the ratio), meaning the relative strength depends only on the charge-to-mass ratios of the objects—this is why electric forces dominate at atomic scales (where charge-to-mass ratio is high for particles like electrons) while gravity dominates at cosmic scales (where objects are electrically neutral with charge-to-mass ratio near zero, but masses are enormous); when r doubles, both F_grav and F_elec decrease by 1/(2)^2=1/4. Choice A is correct because it correctly explains that both forces follow inverse square laws (F ∝ 1/r²) and decrease at the same rate with distance—the ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of r. Choice D suggests that the ratio decreases by a factor of 4, when actually the ratio stays the same since both forces scale identically with r. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav = G(M_planet)(M_star)/r² enormous while F_elec ≈ k(~0)(~0)/r² ≈ 0.
A proton and an electron in a hydrogen atom are separated by r=5.3×10−11m. Use mp=1.67×10−27kg, me=9.11×10−31kg, qp=+1.6×10−19C, qe=−1.6×10−19C, G=6.67×10−11, and k=9.0×109. What is the ratio Felec/Fgrav between the electric and gravitational forces for this pair?\n\n(Recall: Fgrav=Gr2m1m2 and Felec=kr2∣q1q2∣.)
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F=Gr2m1m2) and electric force (F=kr2q1q2) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k=9.0×109N\cdotpm2/C2 is about 1020 times larger than the gravitational constant G=6.67×10−11N\cdotpm2/kg2, making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For a proton and electron separated by the Bohr radius r≈5.3×10−11m, the gravitational force is Fgrav=G(mp)(me)/r2=(6.67×10−11)(1.67×10−27)(9.11×10−31)/(5.3×10−11)2≈3.6×10−47N, while the electric force is Felec=k(e)(e)/r2=(9.0×109)(1.6×10−19)2/(5.3×10−11)2≈8.2×10−8N. The ratio Felec/Fgrav≈(8.2×10−8)/(3.6×10−47)≈2.3×1039, meaning the electric force is about 1039 times stronger—this enormous difference explains why gravity is completely negligible at atomic scales and electron orbits are determined entirely by electric attraction. Choice C is correct because it properly calculates the ratio (kq1q2)/(Gm1m2) with correct scientific notation. Choice A inverts the ratio, calculating Fgrav/Felec instead of Felec/Fgrav, which gives a tiny number like 10−39 instead of the enormous ratio 1039, incorrectly suggesting gravity is stronger when actually electric force is vastly stronger at this scale. When comparing gravitational and electric forces: (1) both follow inverse square laws F∝1/r2, so distance affects them equally, (2) the ratio of strengths Felec/Fgrav=(kq1q2)/(Gm1m2) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G≈1020 when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈10−47N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net Felec≈0). The key insight is that electric force is far stronger intrinsically (k≫G by 1020), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making Fgrav=G(Mplanet)(Mstar)/r2 enormous while Felec≈k(∼0)(∼0)/r2≈0.
Two identical objects each have mass m=5.0kg and are separated by r=1.0m. Approximately what net charge magnitude q must each object carry (same sign) so that the electric repulsion equals the gravitational attraction in magnitude? Use G=6.67×10−11 and k=9.0×109.
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. To find q such that F_elec = F_grav for two 5kg objects at 1m (same sign, repulsion), set k q^2 / r^2 = G m^2 / r^2, so q^2 = G m^2 / k, q = m sqrt(G/k) =5 sqrt(6.67e-11 /9e9) =5 sqrt(7.41e-21) ≈5 * 2.72e-10.5 wait, sqrt(7.41e-21)=sqrt(7.41)10^{-10.5}≈2.7210^{-10.5} but better: sqrt(7.41e-21)=sqrt(7.41)10^{-10.5}≈2.723.162e-11≈8.6e-11, then 5*8.6e-11≈4.3e-10 C, yes. Choice A is correct because it properly calculates q = sqrt( G m^2 / k ), yielding ≈4.3×10^{-10} C. Choice B has an error in the power of 10, reporting q as 10^{-4} instead of 10^{-10}, likely from incorrectly handling exponents when taking square root: sqrt(G/k)=sqrt(10^{-11}/10^9)=sqrt(10^{-20})=10^{-10}, combined with sqrt(m^2)=m=5. Choice D is much larger, perhaps inverting k/G. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). Practical implications: chemistry, molecular biology, material strength, friction, and essentially all everyday phenomena (except falling) are determined by electric forces between atoms and molecules, while planetary orbits, tides, satellite motion, and the large-scale structure of the universe are determined by gravitational forces—the electric force's dominance at small scales is why a charged balloon can lift paper against Earth's entire gravitational pull, yet gravity's dominance at large scales is why planets orbit stars despite any residual electric charges they might have.
In ordinary planetary systems, gravity dominates the motion of planets even though electric forces can be much stronger between charged objects. Which statement best explains why electric forces are usually negligible between planets and stars?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. At cosmic scales, gravity dominates not because electric force is weak, but because planets and stars have nearly equal amounts of positive and negative charge (electrically neutral), so net charge ≈ 0 makes F_elec ≈ 0 even though the intrinsic strength of electric force is much greater than gravity. Choice B is correct because it properly explains that planets and stars have enormous mass but near-zero net charge due to charge cancellation—matter contains equal numbers of protons and electrons, so large objects are electrically neutral despite containing vast amounts of charge. Choice A incorrectly suggests that electric forces decrease faster than gravity with distance, when actually both follow inverse square laws (F ∝ 1/r²) and decrease at the same rate with distance—the ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of r. The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav = G(M_planet)(M_star)/r² enormous while F_elec ≈ k(~0)(~0)/r² ≈ 0.
A proton and an electron are separated by some distance r. The distance is then doubled to 2r. Which statement correctly describes how the magnitudes of the gravitational and electric forces change?
Use Fgrav=Gr2m1m2 and Felec=kr2∣q1q2∣.
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For scale dependence: The ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of distance r (since both forces have r² in denominator which cancels in the ratio), meaning the relative strength depends only on the charge-to-mass ratios of the objects—this is why electric forces dominate at atomic scales (where charge-to-mass ratio is high for particles like electrons) while gravity dominates at cosmic scales (where objects are electrically neutral with charge-to-mass ratio near zero, but masses are enormous). Choice B is correct because it correctly explains that gravity dominates at cosmic scale because massive objects are electrically neutral. Choice C suggests that the distance dependence is different for the two forces (claiming one decreases faster with distance than the other), when actually both follow inverse square laws (F ∝ 1/r²) and decrease at the same rate with distance—the ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of r. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav = G(M_planet)(M_star)/r² enormous while F_elec ≈ k(~0)(~0)/r² ≈ 0.
A proton and an electron are separated by a distance of r=5.3×10−11m (about the Bohr radius). Use mp=1.67×10−27kg, me=9.11×10−31kg, qp=+1.6×10−19C, qe=−1.6×10−19C, G=6.67×10−11N\cdotpm2/kg2, and k=9.0×109N\cdotpm2/C2. What is the ratio Felec/Fgrav between them?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales, specifically for subatomic particles. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = k|q₁q₂|/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values; additionally, gravity is always attractive while electric force can be attractive or repulsive, and gravity depends on mass while electric force depends on charge. For a proton and electron separated by the Bohr radius r ≈ 5.3 × 10⁻¹¹ m, the gravitational force is F_grav = G(m_p)(m_e)/r² ≈ 3.6 × 10⁻⁴⁷ N, while the electric force is F_elec = k(e)(e)/r² ≈ 8.2 × 10⁻⁸ N, yielding a ratio F_elec/F_grav ≈ 2.3 × 10³⁹—this enormous difference explains why gravity is negligible at atomic scales. Choice B is correct because it properly calculates the ratio (kq₁q₂)/(Gm₁m₂) with correct scientific notation, reflecting the vast dominance of electric force. Choice A inverts the ratio, calculating F_grav/F_elec instead of F_elec/F_grav, which gives a tiny number like 10⁻³⁹ instead of the enormous ratio 10³⁹, incorrectly suggesting gravity is stronger when actually electric force is vastly stronger at this scale. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge.
Two objects are separated by a fixed distance r. Object 1 has mass m1 and charge q1; object 2 has mass m2 and charge q2. Which expression correctly gives the ratio of the magnitudes of the electric to gravitational forces, Felec/Fgrav, and what does it imply about dependence on distance?
Explanation: This question tests understanding of how gravitational and electric forces compare in their mathematical expressions and distance dependence. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = k|q₁q₂|/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values; additionally, gravity is always attractive while electric force can be attractive or repulsive, and gravity depends on mass while electric force depends on charge. The ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of distance r (since both forces have r² in denominator which cancels in the ratio), meaning the relative strength depends only on the charge-to-mass ratios of the objects—this is why electric forces dominate at atomic scales while gravity dominates at cosmic scales. Choice A is correct because it accurately expresses the ratio and notes its independence from r. Choice D incorrectly includes a 1/r² term, suggesting the ratio decreases with r², when actually both forces decrease equally with r² and the ratio remains constant. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav enormous while F_elec ≈ 0.
Two protons are separated by r=1.0×10−15m. Use mp=1.67×10−27kg, qp=+1.6×10−19C, G=6.67×10−11, and k=9.0×109 (SI units). Which statement best compares the magnitudes and directions of the gravitational and electric forces between the protons?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and direction for identical charged particles at nuclear scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = k|q₁q₂|/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values; additionally, gravity is always attractive while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For two protons separated by 1.0 × 10⁻¹⁵ m, the ratio F_elec/F_grav ≈ 1.2 × 10³⁶, meaning the electric force is vastly larger, and since both have positive charge, electric force is repulsive while gravity is attractive. Choice C is correct because it accurately identifies that gravity is attractive and electric is repulsive, and the electric force is vastly larger. Choice A incorrectly claims gravitational force dominates at atomic scales, when actually gravity is negligible compared to electric force—the ratio of ~10³⁶ means you would need 10³⁶ times more mass to get gravitational force equal to the electric force from elementary charges, which is why nuclear stability involves electric repulsion countered by strong force, not gravity. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav enormous while F_elec ≈ 0.
Two objects are separated by the same distance r. The gravitational force is Fgrav=Gm1m2/r2 and the electric force is Felec=kq1q2/r2. If the distance between the objects is doubled to 2r, how do Fgrav and Felec change, and what happens to the ratio Felec/Fgrav?
Explanation: This question tests understanding of how gravitational and electric forces compare in their dependence on distance. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = k|q₁q₂|/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values; additionally, gravity is always attractive while electric force can be attractive or repulsive, and gravity depends on mass while electric force depends on charge. The ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of distance r (since both forces have r² in denominator which cancels in the ratio), meaning the relative strength depends only on the charge-to-mass ratios of the objects—this is why electric forces dominate at atomic scales while gravity dominates at cosmic scales. Choice A is correct because it properly applies both force formulas and compares the resulting magnitudes after doubling distance, showing both become 1/4 as large and ratio unchanged. Choice B suggests that the distance dependence is different for the two forces (claiming one decreases faster with distance than the other), when actually both follow inverse square laws (F ∝ 1/r²) and decrease at the same rate with distance—the ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of r. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). Practical implications: chemistry, molecular biology, material strength, friction, and essentially all everyday phenomena (except falling) are determined by electric forces between atoms and molecules, while planetary orbits, tides, satellite motion, and the large-scale structure of the universe are determined by gravitational forces—the electric force's dominance at small scales is why a charged balloon can lift paper against Earth's entire gravitational pull, yet gravity's dominance at large scales is why planets orbit stars despite any residual electric charges they might have.
Two small spheres are r=0.20m apart. Sphere 1 has m1=0.50kg and q1=+2.0μC. Sphere 2 has m2=1.5kg and q2=−2.0μC. Using G=6.67×10−11 and k=9.0×109, which pair of magnitudes is closest to Fgrav and Felec?
(Report magnitudes; the electric force is attractive because charges are opposite.)
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For macroscopic charged objects: For two spheres with m1=0.5 kg, m2=1.5 kg, q1=+2μC, q2=-2μC separated by 0.2 m, the gravitational force is F_grav = (6.67×10⁻¹¹)(0.5)(1.5)/(0.2)² ≈ 1.25 × 10^{-9} N (tiny), while the electric force is F_elec = (9.0×10⁹)(2×10^{-6})(2×10^{-6})/(0.2)² ≈ 0.9 N (noticeable), a ratio of about 10^9—even this modest laboratory charge produces electric forces that are a billion times stronger than gravity between the same objects. Choice A is correct because it properly applies both force formulas and compares the resulting magnitudes. Choice D has an error in the power of 10, reporting F_elec as 9.0 × 10^{-7} instead of 0.9, likely from incorrectly handling exponents when dividing powers of 10 in the calculation: (10⁹)(10^{-12}) / (4×10^{-2}) = 10^9 × 10^{-12} / 4×10^{-2} = (10^{-3}) / 4×10^{-2} wait, actually miscalculating the charge product. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). Practical implications: chemistry, molecular biology, material strength, friction, and essentially all everyday phenomena (except falling) are determined by electric forces between atoms and molecules, while planetary orbits, tides, satellite motion, and the large-scale structure of the universe are determined by gravitational forces—the electric force's dominance at small scales is why a charged balloon can lift paper against Earth's entire gravitational pull, yet gravity's dominance at large scales is why planets orbit stars despite any residual electric charges they might have.
A proton and an electron in a hydrogen atom are separated by r=5.3×10−11m. Use mp=1.67×10−27kg, me=9.11×10−31kg, qp=+1.6×10−19C, qe=−1.6×10−19C, G=6.67×10−11, and k=9.0×109. What is the ratio Felec/Fgrav between the electric and gravitational forces for this pair?
(Recall: Fgrav=Gr2m1m2 and Felec=kr2∣q1q2∣.)
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For a proton and electron separated by the Bohr radius r ≈ 5.3 × 10⁻¹¹ m, the gravitational force is F_grav = G(m_p)(m_e)/r² = (6.67×10⁻¹¹)(1.67×10⁻²⁷)(9.11×10⁻³¹)/(5.3×10⁻¹¹)² ≈ 3.6 × 10⁻⁴⁷ N, while the electric force is F_elec = k(e)(e)/r² = (9.0×10⁹)(1.6×10⁻¹⁹)²/(5.3×10⁻¹¹)² ≈ 8.2 × 10⁻⁸ N. The ratio F_elec/F_grav ≈ (8.2×10⁻⁸)/(3.6×10⁻⁴⁷) ≈ 2.3 × 10³⁹, meaning the electric force is about 10³⁹ times stronger—this enormous difference explains why gravity is completely negligible at atomic scales and electron orbits are determined entirely by electric attraction. Choice C is correct because it properly calculates the ratio (kq₁q₂)/(Gm₁m₂) with correct scientific notation. Choice A inverts the ratio, calculating F_grav/F_elec instead of F_elec/F_grav, which gives a tiny number like 10⁻³⁹ instead of the enormous ratio 10³⁹, incorrectly suggesting gravity is stronger when actually electric force is vastly stronger at this scale. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav = G(M_planet)(M_star)/r² enormous while F_elec ≈ k(~0)(~0)/r² ≈ 0.
Two protons are separated by r=1.0×10−15m. Use mp=1.67×10−27kg, qp=+1.6×10−19C, G=6.67×10−11, and k=9.0×109. Which pair of magnitudes is closest to the gravitational attraction and electric repulsion between them?
Compute Fgrav=Gr2mp2 and Felec=kr2qp2.
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For two protons separated by r = 1.0 × 10^{-15} m, the gravitational force is F_grav = G(m_p)^2 / r² = (6.67×10⁻¹¹)(1.67×10⁻²⁷)^2 / (10^{-15})^2 ≈ 1.9 × 10^{-34} N, while the electric force is F_elec = k(q_p)^2 / r² = (9.0×10⁹)(1.6×10^{-19})^2 / (10^{-15})^2 ≈ 2.3 × 10^{2} N, showing electric repulsion is vastly stronger at nuclear scales. Choice A is correct because it properly calculates the ratio (kq₁q₂)/(Gm₁m₂) with correct scientific notation. Choice C has an error in the power of 10, reporting F_elec as 2.3 × 10^{-18} instead of 2.3 × 10^{2}, likely from incorrectly handling exponents when dividing powers of 10 in the calculation: (10⁹)/(10^{-30}) for r² gives large positive exponent. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav = G(M_planet)(M_star)/r² enormous while F_elec ≈ k(~0)(~0)/r² ≈ 0.
Two objects are separated by the same distance r. Object 1 has mass m1 and charge q1, and object 2 has mass m2 and charge q2. Which expression correctly gives the ratio of the magnitude of the electric force to the magnitude of the gravitational force between them?
Use Fgrav=Gr2m1m2 and Felec=kr2∣q1q2∣.
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For scale dependence: The ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of distance r (since both forces have r² in denominator which cancels in the ratio), meaning the relative strength depends only on the charge-to-mass ratios of the objects—this is why electric forces dominate at atomic scales (where charge-to-mass ratio is high for particles like electrons) while gravity dominates at cosmic scales (where objects are electrically neutral with charge-to-mass ratio near zero, but masses are enormous). Choice A is correct because it properly applies both force formulas and compares the resulting magnitudes. Choice D incorrectly claims that the distance dependence is different for the two forces (claiming one decreases faster with distance than the other), when actually both follow inverse square laws (F ∝ 1/r²) and decrease at the same rate with distance—the ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of r. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). Practical implications: chemistry, molecular biology, material strength, friction, and essentially all everyday phenomena (except falling) are determined by electric forces between atoms and molecules, while planetary orbits, tides, satellite motion, and the large-scale structure of the universe are determined by gravitational forces—the electric force's dominance at small scales is why a charged balloon can lift paper against Earth's entire gravitational pull, yet gravity's dominance at large scales is why planets orbit stars despite any residual electric charges they might have.
Two identical metal spheres each have mass m=1.0kg and net charge q=+1.0μC=1.0×10−6C. Their centers are r=0.50m apart. Using G=6.67×10−11 and k=9.0×109, which statement best describes which force dominates and by approximately what factor (magnitude ratio)?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For macroscopic charged objects: For two 1 kg objects each with charge 1 μC = 10⁻⁶ C separated by 0.5 m, the gravitational force is F_grav = (6.67×10⁻¹¹)(1)(1)/(0.5)² ≈ 2.7 × 10⁻¹⁰ N (tiny), while the electric force is F_elec = (9.0×10⁹)(10⁻⁶)(10⁻⁶)/(0.5)² = 3.6 × 10^{-2} N (noticeable), a ratio of about 10^{11}—even this modest laboratory charge produces electric forces that are a hundred billion times stronger than gravity between the same objects. Choice B is correct because it accurately identifies that electric force dominates at atomic scale due to the enormous force ratio of ~10¹¹ and properly applies both force formulas and compares the resulting magnitudes. Choice D has an error in the power of 10, reporting the ratio as 10^{-11} instead of the correct 10^{11}, likely from incorrectly handling exponents when dividing powers of 10 in the calculation: (10⁹)/(10⁻¹¹) = 10²⁰ for the constants, combined with exponents from masses and charges. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). Practical implications: chemistry, molecular biology, material strength, friction, and essentially all everyday phenomena (except falling) are determined by electric forces between atoms and molecules, while planetary orbits, tides, satellite motion, and the large-scale structure of the universe are determined by gravitational forces—the electric force's dominance at small scales is why a charged balloon can lift paper against Earth's entire gravitational pull, yet gravity's dominance at large scales is why planets orbit stars despite any residual electric charges they might have.
Two protons are separated by r=1.0×10−15m. Use mp=1.67×10−27kg, qp=+1.6×10−19C, G=6.67×10−11, and k=9.0×109. Which pair of magnitudes is closest to the gravitational attraction and electric repulsion between them?
Compute Fgrav=Gr2mp2 and Felec=kr2qp2.
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For two protons separated by r = 1.0 × 10^{-15} m, the gravitational force is F_grav = G(m_p)^2 / r² = (6.67×10⁻¹¹)(1.67×10⁻²⁷)^2 / (10^{-15})^2 ≈ 1.9 × 10^{-34} N, while the electric force is F_elec = k(q_p)^2 / r² = (9.0×10⁹)(1.6×10^{-19})^2 / (10^{-15})^2 ≈ 2.3 × 10^{2} N, showing electric repulsion is vastly stronger at nuclear scales. Choice A is correct because it properly calculates the ratio (kq₁q₂)/(Gm₁m₂) with correct scientific notation. Choice C has an error in the power of 10, reporting F_elec as 2.3 × 10^{-18} instead of 2.3 × 10^{2}, likely from incorrectly handling exponents when dividing powers of 10 in the calculation: (10⁹)/(10^{-30}) for r² gives large positive exponent. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav = G(M_planet)(M_star)/r² enormous while F_elec ≈ k(~0)(~0)/r² ≈ 0.
Two small spheres are r=0.20m apart. Sphere 1 has m1=0.50kg and q1=+2.0μC. Sphere 2 has m2=1.5kg and q2=−2.0μC. Using G=6.67×10−11 and k=9.0×109, which pair of magnitudes is closest to Fgrav and Felec?
(Report magnitudes; the electric force is attractive because charges are opposite.)
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For macroscopic charged objects: For two spheres with m1=0.5 kg, m2=1.5 kg, q1=+2μC, q2=-2μC separated by 0.2 m, the gravitational force is F_grav = (6.67×10⁻¹¹)(0.5)(1.5)/(0.2)² ≈ 1.25 × 10^{-9} N (tiny), while the electric force is F_elec = (9.0×10⁹)(2×10^{-6})(2×10^{-6})/(0.2)² ≈ 0.9 N (noticeable), a ratio of about 10^9—even this modest laboratory charge produces electric forces that are a billion times stronger than gravity between the same objects. Choice A is correct because it properly applies both force formulas and compares the resulting magnitudes. Choice D has an error in the power of 10, reporting F_elec as 9.0 × 10^{-7} instead of 0.9, likely from incorrectly handling exponents when dividing powers of 10 in the calculation: (10⁹)(10^{-12}) / (4×10^{-2}) = 10^9 × 10^{-12} / 4×10^{-2} = (10^{-3}) / 4×10^{-2} wait, actually miscalculating the charge product. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). Practical implications: chemistry, molecular biology, material strength, friction, and essentially all everyday phenomena (except falling) are determined by electric forces between atoms and molecules, while planetary orbits, tides, satellite motion, and the large-scale structure of the universe are determined by gravitational forces—the electric force's dominance at small scales is why a charged balloon can lift paper against Earth's entire gravitational pull, yet gravity's dominance at large scales is why planets orbit stars despite any residual electric charges they might have.
Consider the Earth and the Moon separated by about r=3.8×108m. Suppose (hypothetically) both bodies have extremely small net charges compared with their total number of protons/electrons, so their net charges are effectively q⊕≈0 and qMoon≈0. Which statement best explains why gravity dominates their interaction even though the electric force constant k is much larger than G?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. Additionally, gravity is always attractive (masses always pull together) while electric force can be attractive (opposite charges) or repulsive (same charges), and gravity depends on mass while electric force depends on charge. For scale dependence: The ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of distance r (since both forces have r² in denominator which cancels in the ratio), meaning the relative strength depends only on the charge-to-mass ratios of the objects—this is why electric forces dominate at atomic scales (where charge-to-mass ratio is high for particles like electrons) while gravity dominates at cosmic scales (where objects are electrically neutral with charge-to-mass ratio near zero, but masses are enormous). Choice B is correct because it correctly explains that gravity dominates at cosmic scale because massive objects are electrically neutral. Choice A suggests that the distance dependence is different for the two forces (claiming one decreases faster with distance than the other), when actually both follow inverse square laws (F ∝ 1/r²) and decrease at the same rate with distance—the ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of r. When comparing gravitational and electric forces: (1) both follow inverse square laws F ∝ 1/r², so distance affects them equally, (2) the ratio of strengths F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charges and masses but not distance, (3) electric force is intrinsically stronger by a factor of k/G ≈ 10²⁰ when comparing equal numerical values, (4) at atomic scales electric dominates because particles have charge but tiny mass (gravity ≈ 10⁻⁴⁷ N is negligible), and (5) at cosmic scales gravity dominates because objects are massive but electrically neutral (equal + and - charges cancel, so net F_elec ≈ 0). The key insight is that electric force is far stronger intrinsically (k >> G by 10²⁰), yet gravity controls the cosmos—this apparent paradox resolves when you realize that charge comes in two types (+ and -) that cancel when combined, while mass comes in only one type (positive) that always adds up, so large objects like planets inevitably have huge total mass but near-zero net charge, making F_grav = G(M_planet)(M_star)/r² enormous while F_elec ≈ k(~0)(~0)/r² ≈ 0.
Two objects are separated by the same distance r. Object 1 has m1=2.0 kg and q1=+3.0 μC; object 2 has m2=4.0 kg and q2=−3.0 μC. Which expression correctly gives the ratio of magnitudes ∣Felec∣/∣Fgrav∣ for this pair?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. The ratio F_elec/F_grav = (kq₁q₂/r²)/(Gm₁m₂/r²) = (kq₁q₂)/(Gm₁m₂) is independent of distance r (since both forces have r² in denominator which cancels in the ratio), meaning the relative strength depends only on the charge-to-mass ratios of the objects. Choice A is correct because it properly shows the ratio as k|q₁q₂|/(Gm₁m₂), with the r² terms canceling out and absolute value bars ensuring we compare magnitudes regardless of whether charges attract or repel. Choice C incorrectly includes r² in the numerator, suggesting the ratio increases with distance squared, when actually both forces decrease equally with distance so their ratio is constant—this error comes from forgetting that r² appears in both denominators and cancels when forming the ratio. When comparing gravitational and electric forces: (1) the ratio F_elec/F_grav is independent of separation distance, (2) it depends only on the intrinsic properties (masses and charges) of the objects, (3) this explains why the relative importance of these forces is determined by the charge-to-mass ratio of objects, not their separation, and (4) for any given pair of objects, electric force will always be stronger than gravity by the same factor regardless of how far apart they are.
Two identical objects have both mass and charge and are separated by the same distance r in two trials.
Trial 1: each object has mass m and charge q. Trial 2: each object has mass 2m and charge 2q.
How does the ratio ∣Felec∣/∣Fgrav∣ change from Trial 1 to Trial 2?
Explanation: This question tests understanding of how gravitational and electric forces compare in magnitude and significance at different scales. Both gravitational force (F = Gm₁m₂/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, decreasing with the square of the distance between objects, but they differ dramatically in strength: the Coulomb constant k = 9.0 × 10⁹ N·m²/C² is about 10²⁰ times larger than the gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg², making electric forces intrinsically much stronger than gravitational forces for comparable numerical values. The ratio F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) is independent of distance r. In Trial 1: ratio = (kq²)/(Gm²), and in Trial 2: ratio = (k(2q)²)/(G(2m)²) = (k×4q²)/(G×4m²) = (kq²)/(Gm²), which is the same as Trial 1. Choice B is correct because the ratio is unchanged—when both mass and charge double, the electric force increases by factor of 4 (since F_elec ∝ q²) and gravitational force also increases by factor of 4 (since F_grav ∝ m²), so their ratio remains constant. Choice C incorrectly suggests the ratio becomes 4 times larger, failing to recognize that both forces scale quadratically with their respective properties (charge for electric, mass for gravitational), so doubling both charge and mass increases both forces by the same factor of 4, leaving their ratio unchanged. When analyzing how force ratios change: (1) F_elec/F_grav = (kq₁q₂)/(Gm₁m₂) depends on charge-to-mass ratios, (2) if charges and masses scale by the same factor, the ratio is unchanged, (3) this explains why the relative importance of electric vs gravitational forces for an object depends on its charge-to-mass ratio, not its absolute size, and (4) doubling all properties (mass and charge) doesn't change which force dominates.