#point charges

Coulomb's Law and Force Between Point Charges

Two point charges \( 9 \times 10^{-7} \, \text{C} \) and \( -3 \times 10^{-7} \, \text{C} \) are 100 cm apart in vacuum.

**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. Using Coulomb’s law: F = k (|q₁ q₂|/r²) . k = 9 × 10⁹ N·m²/C² , q₁ = 9 × 10⁻⁷ C , q₂ = -3 × 10⁻⁷ C , r = 1.0 m . |q₁ q₂| = 9 × 3 × 10⁻¹⁴ = 27 × 10⁻¹⁴ C² . r² = (1.0)² = 1 m² . F = 9 × 10⁹ × (27 × 10⁻¹⁴/1)

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Coulomb's Law and Force Between Point Charges

Two charges \( +8 \, \mu\text{C} \) and \( -4 \, \mu\text{C} \) are 40 cm apart. What is the electric field magnitude at

**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. Midpoint distance = 20 cm = 0.2 m. E₁ = 9 × 10⁹ × (8 × 10⁻⁶/(0.2)²) = 1.8 × 10⁶ N/C (towards -4 μC ). E₂ = 9 × 10⁹ × (4 × 10⁻⁶/(0.2)²) = 9 × 10⁵ N/C (towards -4 μC ). Net E = 1.8 × 10⁶ + 9 × 10⁵ = 2.7 × 10⁶ N/C . Substituting values gives 2.7 ×

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Superposition Principle and Equilibrium of Charges

Three charges \( +5 \, \mu\text{C}, -3 \, \mu\text{C}, +4 \, \mu\text{C} \) are at the vertices of an equilateral triang

**Independent action of charges** allows total force or field as vector sum. Geometry dictates distances to evaluation point, and resultant follows Σ k q_i/r_i², explaining zero field at symmetric centres for equal charges. F₁ = 9 × 10⁹ × (5 × 3 × 10⁻¹²/(1.5)²) = 0.06 N (attractive). F₂ = 9 × 10⁹ × (5 × 4 × 10⁻¹²/(1.5)²) = 0.08 N (repulsive). Angle 60°. Net F = √(F₁² + F₂² + 2 F₁ F₂ cos 60°) = √(0.06² + 0.08² + 0.0048) = 0.108 N . Substituting values gives 0.108 N, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Superposition Principle and Equilibrium of Charges

Three charges \( +5 \, \mu\text{C}, -5 \, \mu\text{C}, +2 \, \mu\text{C} \) are at the vertices of an equilateral triang

**Vector addition of forces** underlies multi-charge analysis. Each pair contributes independent Coulomb force, resultant obtained by resolving components along axes. Equilibrium occurs when vector sum vanishes, often at symmetric points where contributions balance. F₁ = 9 × 10⁹ × (5 × 5 × 10⁻¹²/(1.5)²) = 0.1 N (attractive). F₂ = 9 × 10⁹ × (5 × 2 × 10⁻¹²/(1.5)²) = 0.04 N (repulsive). Angle 60°. Net F = √(F₁² + F₂² + 2 F₁ F₂ cos 60°) = √(0.1² + 0.04² + 0.004) = 0.129 N . Substituting values gives 0.129 N, 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 \( +11 \, \mu\text{C} \) and \( -5 \, \mu\text{C} \) are 80 cm apart. What is the electric field magnitude a

**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. Midpoint distance = 40 cm = 0.4 m. E₁ = 9 × 10⁹ × (11 × 10⁻⁶/(0.4)²) = 6.1875 × 10⁵ N/C (towards -5 μC ). E₂ = 9 × 10⁹ × (5 × 10⁻⁶/(0.4)²) = 2.8125 × 10⁵ N/C (towards -5 μC ). Net E = 6.1875 × 10⁵ + 2.8125 × 10⁵ = 9 × 10⁵ N/C . Substituting values gives 9.0 × 10⁵ N/C, which matches expected magnitude for this electrostatic configuration,

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Electric Charge, Quantization and Conservation

Two charges \( +10 \, \mu\text{C} \) and \( -2 \, \mu\text{C} \) are 80 cm apart. What is the distance from \( +10 \, \m

**Quantization of charge** states observable charge is integer multiple of elementary charge e = 1.6×10⁻¹⁹ C, q = n·e, and total charge is conserved in isolated systems. Loss of electrons produces positive charge, and number of transferred electrons follows n = q/e, linking macroscopic charge measurement to microscopic carriers. Let x be distance from +10 μC , then 0.8 - x from -2 μC . (10 × 10⁻⁶/x²) = (2 × 10⁻⁶/(0.8 - x)²) , 10 (0.8 - x)² = 2 x² . 5 (0.64 - 1.6 x + x²) = x² , 3.2 - 8 x + 5 x² = x² . 4 x²

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Coulomb's Law and Force Between Point Charges

Two charges \( +12 \, \mu\text{C} \) and \( -3 \, \mu\text{C} \) are 60 cm apart. What is the distance from \( +12 \, \m

**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 x be distance from +12 μC , then 0.6 - x from -3 μC . (12 × 10⁻⁶/x²) = (3 × 10⁻⁶/(0.6 - x)²) , 12 (0.6 - x)² = 3 x² . 4 (0.36 - 1.2 x + x²) = x² , 1.44 - 4.8 x + 4 x² = x² . 3 x² - 4.8 x + 1.44 = 0 , x

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Electric Charge, Quantization and Conservation

Two charges \( +2 \, \mu\text{C} \) and \( +3 \, \mu\text{C} \) are 10 cm apart. What is the electric field at a point 5

**Charge conservation and quantization** govern rubbing processes where electrons transfer without creation. Total charge before and after remains equal, and any measured charge corresponds to n = q/e electrons, allowing counting of carriers from coulomb value. Distance from +3 μC = 5 cm. E₁ = 9 × 10⁹ × (2 × 10⁻⁶/(0.05)²) = 7.2 × 10⁶ N/C (away). E₂ = 9 × 10⁹ × (3 × 10⁻⁶/(0.05)²) = 1.08 × 10⁷ N/C (towards). Net E = 1.08 × 10⁷ - 7.2 × 10⁶ = 3.6 × 10⁶ N/C (towards +3 μC ). Substituting values gives 3.6 × 10⁶ N/C, which matches expected magnitude for this

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Electric Charge, Quantization and Conservation

Two charges \( +6 \, \mu\text{C} \) and \( -3 \, \mu\text{C} \) are 50 cm apart. What is the electric field magnitude at

**Quantization of charge** states observable charge is integer multiple of elementary charge e = 1.6×10⁻¹⁹ C, q = n·e, and total charge is conserved in isolated systems. Loss of electrons produces positive charge, and number of transferred electrons follows n = q/e, linking macroscopic charge measurement to microscopic carriers. Midpoint distance = 25 cm = 0.25 m. E₁ = 9 × 10⁹ × (6 × 10⁻⁶/(0.25)²) = 8.64 × 10⁵ N/C (towards -3 μC ). E₂ = 9 × 10⁹ × (3 × 10⁻⁶/(0.25)²) = 4.32 × 10⁵ N/C (towards -3 μC ). Net E = 8.64 × 10⁵ + 4.32 × 10⁵ = 1.296

Ref: NCERT > Physics Book > Electric Charges and Fields > Electric Charge, Quantization and Conservation

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Electric Charge, Quantization and Conservation

Two charges \( +3 \, \mu\text{C} \) and \( -3 \, \mu\text{C} \) are 15 cm apart. What is the force on a \( 2 \, \mu\text

**Charge conservation and quantization** govern rubbing processes where electrons transfer without creation. Total charge before and after remains equal, and any measured charge corresponds to n = q/e electrons, allowing counting of carriers from coulomb value. Electric field at midpoint: E₁ = 9 × 10⁹ × (3 × 10⁻⁶/(0.075)²) = 4.8 × 10⁶ N/C (towards -3 μC ). E₂ = 4.8 × 10⁶ N/C (towards -3 μC ). Net E = 4.8 × 10⁶ + 4.8 × 10⁶ = 9.6 × 10⁶ N/C . Force: F = q E = 2 × 10⁻⁶ × 9.6 × 10⁶ = 19.2 N . Substituting values gives 19.2

Ref: NCERT > Physics Book > Electric Charges and Fields > Electric Charge, Quantization and Conservation

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Coulomb's Law and Force Between Point Charges

Two charges \( +12 \, \mu\text{C} \) and \( -4 \, \mu\text{C} \) are 90 cm apart. What is the distance from \( +12 \, \m

**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 x be distance from +12 μC , then 0.9 - x from -4 μC . (12 × 10⁻⁶/x²) = (4 × 10⁻⁶/(0.9 - x)²) , 12 (0.9 - x)² = 4 x² . 3 (0.81 - 1.8 x + x²) = x² , 2.43 - 5.4 x + 3 x² = x² . 2 x² - 5.4 x + 2.43 = 0 , x

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Coulomb's Law and Force Between Point Charges

Two charges \( +14 \, \mu\text{C} \) and \( -6 \, \mu\text{C} \) are 100 cm apart. What is the distance from \( +14 \, \

**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 x be distance from +14 μC , then 1 - x from -6 μC . (14 × 10⁻⁶/x²) = (6 × 10⁻⁶/(1 - x)²) , 14 (1 - x)² = 6 x² . 14 (1 - 2 x + x²) = 6 x² , 14 - 28 x + 14 x² = 6 x² . 8 x² - 28 x + 14 = 0

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