Coulomb's Law and Force Between Point Charges

Coulomb's Law and Force Between Point Charges

What ensures that electric field lines never cross each other in a static field configuration?

**Electrostatic force** described by F = (1/4π ε₀)·q₁q₂/r² obeys Newton's third law. Magnitude depends on q₁q₂ and 1/r², enabling quantitative estimation at given separation, with sign indicating attraction or repulsion. The electric field has a unique direction at each point, defined by the force on a test charge. Crossing lines would imply multiple directions at one point, which is impossible in a static field where the field vector is well-defined. Substituting values gives Unique direction, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

What ensures that the electric field inside a hollow conductor remains zero even if an external charge is placed nearby?

**Inverse-square law** for charges states F ∝ 1/r² while increasing with charge product. Using k = 9×10⁹ N·m²/C², force at distance r follows F = k q₁q₂/r², forming basis for pairwise force calculation. Electrostatic shielding occurs as free charges in the conductor redistribute to cancel any external field inside. This adjustment creates an induced field that neutralizes the external influence, maintaining zero field within the hollow region. Substituting values gives Electrostatic shielding, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

What explains why the electric field due to a charged object can penetrate an insulator but not a conductor?

**Inverse-square law** for charges states F ∝ 1/r² while increasing with charge product. Using k = 9×10⁹ N·m²/C², force at distance r follows F = k q₁q₂/r², forming basis for pairwise force calculation. Insulators lack free charges to redistribute and cancel an external field, allowing it to penetrate. Conductors, with mobile charges, redistribute them to create an opposing field, shielding the interior from penetration. Substituting values gives Charge mobility, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

A plastic rod gains a charge of \( -1.28 \times 10^{-7} \, \text{C} \) when rubbed. How many electrons were transferred

**Electrostatic force** described by F = (1/4π ε₀)·q₁q₂/r² obeys Newton's third law. Magnitude depends on q₁q₂ and 1/r², enabling quantitative estimation at given separation, with sign indicating attraction or repulsion. Negative charge means electrons gained. q = n e , e = -1.6 × 10⁻¹⁹ C . n = (q/|e|) = (1.28 × 10⁻⁷/1.6 × 10⁻¹⁹) = 8 × 10¹¹ . Substituting values gives 8 × 10¹¹, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

A conducting sphere of radius 20 cm has an electric field of \( 3 \times 10^3 \, \text{N/C} \) at 40 cm from its center.

**Coulomb's law** gives force between point charges as F = k·|q₁q₂|/r², k = 1/(4π ε₀) = 9×10⁹ N·m²/C², directed along line joining charges. Like charges repel, opposite attract, magnitude scales with product of charges and inverse square of separation r². For a conductor, E = (k q/r²) outside. 3 × 10³ = 9 × 10⁹ × (q/(0.4)²) . q = (3 × 10³ × 0.16/9 × 10⁹) = 5.33 × 10⁻⁸ C . Substituting values gives 5.33 × 10⁻⁸ C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

A point charge \( q = 5 \, \mu\text{C} \) is placed at the origin. What is the electric field magnitude at a point 2 m a

**Inverse-square law** for charges states F ∝ 1/r² while increasing with charge product. Using k = 9×10⁹ N·m²/C², force at distance r follows F = k q₁q₂/r², forming basis for pairwise force calculation. Electric field: E = (k |q|/r²) . k = 9 × 10⁹ N·m²/C² , q = 5 × 10⁻⁶ C , r = 2 m . E = 9 × 10⁹ × (5 × 10⁻⁶/(2)²) = 9 × 10⁹ × (5 × 10⁻⁶/4) = 1.125 × 10⁴ N/C . Substituting values gives 1.125 × 10⁴ N/C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

Two charges \( +6 \, \mu\text{C} \) and \( -4 \, \mu\text{C} \) are 25 cm apart. What is the distance from \( +6 \, \mu\

**Coulomb's law** gives force between point charges as F = k·|q₁q₂|/r², k = 1/(4π ε₀) = 9×10⁹ N·m²/C², directed along line joining charges. Like charges repel, opposite attract, magnitude scales with product of charges and inverse square of separation r². Let distance from +6 μC be x , then from -4 μC is 0.25 - x . (k × 6 × 10⁻⁶/x²) = (k × 4 × 10⁻⁶/(0.25 - x)²) . (6/x²) = (4/(0.25 - x)²) , 6 (0.25 - x)² = 4 x² . 6 (0.0625 - 0.5 x + x²) = 4 x² , 0.375 - 3 x + 6 x² = 4

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Coulomb's Law and Force Between Point Charges

Which property of electric field lines ensures they never intersect, even in complex charge distributions?

**Inverse-square law** for charges states F ∝ 1/r² while increasing with charge product. Using k = 9×10⁹ N·m²/C², force at distance r follows F = k q₁q₂/r², forming basis for pairwise force calculation. Electric field lines represent the direction of the field at each point. If they intersected, it would imply two different field directions at the same point, which is impossible since the electric field is a unique vector quantity at any given position. Substituting values gives Uniqueness of direction, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

Two point charges \( 7 \times 10^{-7} \, \text{C} \) and \( -2 \times 10^{-7} \, \text{C} \) are 70 cm apart in vacuum.

**Electrostatic force** described by F = (1/4π ε₀)·q₁q₂/r² obeys Newton's third law. Magnitude depends on q₁q₂ and 1/r², enabling quantitative estimation at given separation, with sign indicating attraction or repulsion. Using Coulomb’s law: F = k (|q₁ q₂|/r²) . k = 9 × 10⁹ N·m²/C² , q₁ = 7 × 10⁻⁷ C , q₂ = -2 × 10⁻⁷ C , r = 0.7 m . |q₁ q₂| = 7 × 2 × 10⁻¹⁴ = 14 × 10⁻¹⁴ C² . r² = (0.7)² = 0.49 m² . F = 9 × 10⁹ × (14 × 10⁻¹⁴/0.49) = 9 × 10⁹ × 2.857 × 10⁻¹³ = 0.00257

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Coulomb's Law and Force Between Point Charges

Two point charges \( 6 \times 10^{-7} \, \text{C} \) and \( 10 \times 10^{-7} \, \text{C} \) are 120 cm apart in vacuum.

**Electrostatic force** described by F = (1/4π ε₀)·q₁q₂/r² obeys Newton's third law. Magnitude depends on q₁q₂ and 1/r², enabling quantitative estimation at given separation, with sign indicating attraction or repulsion. Using Coulomb’s law: F = k (|q₁ q₂|/r²) . k = 9 × 10⁹ N·m²/C² , q₁ = 6 × 10⁻⁷ C , q₂ = 10 × 10⁻⁷ C , r = 1.2 m . |q₁ q₂| = 6 × 10 × 10⁻¹⁴ = 60 × 10⁻¹⁴ C² . r² = (1.2)² = 1.44 m² . F = 9 × 10⁹ × (60 × 10⁻¹⁴/1.44) = 9 × 10⁹ × 4.167 × 10⁻¹³ = 0.00375

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Coulomb's Law and Force Between Point Charges

In an experiment, a charged object is brought near a neutral conductor, causing charge separation. What phenomenon is re

**Electrostatic force** described by F = (1/4π ε₀)·q₁q₂/r² obeys Newton's third law. Magnitude depends on q₁q₂ and 1/r², enabling quantitative estimation at given separation, with sign indicating attraction or repulsion. Electrostatic induction occurs when a charged object induces a separation of charges in a neutral conductor. The electric field of the charged object attracts opposite charges and repels like charges, redistributing them without direct contact. Substituting values gives Electrostatic induction, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Coulomb's Law and Force Between Point Charges

A conducting sphere of radius 15 cm has an electric field of \( 4 \times 10^3 \, \text{N/C} \) at 30 cm from its center.

**Coulomb's law** gives force between point charges as F = k·|q₁q₂|/r², k = 1/(4π ε₀) = 9×10⁹ N·m²/C², directed along line joining charges. Like charges repel, opposite attract, magnitude scales with product of charges and inverse square of separation r². E = (k q/r²) . 4 × 10³ = 9 × 10⁹ × (q/(0.3)²) . q = (4 × 10³ × 0.09/9 × 10⁹) = 4 × 10⁻⁸ C . Substituting values gives 4.0 × 10⁻⁸ C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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