Superposition Principle and Equilibrium of Charges

Superposition Principle and Equilibrium of Charges

Two point charges \( 5 \times 10^{-7} \, \text{C} \) and \( -7 \times 10^{-7} \, \text{C} \) are 150 cm apart in vacuum.

**Superposition principle** asserts net Coulomb force on charge equals vector sum of forces from each other charge independently, F_net = Σ F_i, where F_i = k q q_i/r_i² r̂_i. In equilateral triangle or square symmetry, components may cancel at centroid, producing equilibrium. Using Coulomb’s law: F = k (|q₁ q₂|/r²) . k = 9 × 10⁹ N·m²/C² , q₁ = 5 × 10⁻⁷ C , q₂ = -7 × 10⁻⁷ C , r = 1.5 m . |q₁ q₂| = 5 × 7 × 10⁻¹⁴ = 35 × 10⁻¹⁴ C² . r² = (1.5)² = 2.25 m² . F = 9 × 10⁹ × (35

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Superposition Principle and Equilibrium of Charges

Which characteristic of insulators prevents them from shielding their interior from an external electric field?

**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. Insulators lack free charges that can move to cancel an external field. Unlike conductors, where mobile electrons redistribute, insulators’ fixed charges allow the field to penetrate, as no shielding mechanism exists. Substituting values gives Absence of free charges, 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

What causes electric field lines to emerge perpendicularly from the surface of a charged conductor?

**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. In equilibrium, the field inside a conductor is zero. Any tangential component outside would drive surface charge motion, violating equilibrium. Thus, the field must be perpendicular to avoid such motion, maintaining static conditions. Substituting values gives Equilibrium condition, 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

Two charges \( +15 \, \mu\text{C} \) and \( -5 \, \mu\text{C} \) are 75 cm apart. What is the distance from \( +15 \, \m

**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. Let x be distance from +15 μC , then 0.75 - x from -5 μC . (15 × 10⁻⁶/x²) = (5 × 10⁻⁶/(0.75 - x)²) , 15 (0.75 - x)² = 5 x² . 3 (0.5625 - 1.5 x + x²) = x² , 1.6875 - 4.5 x + 3 x² = x² . 2 x² - 4.5 x + 1.6875 = 0 , x = (4.5 ± √(20.25 - 13.5)/4) =

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Superposition Principle and Equilibrium of Charges

Three charges \( +2 \, \mu\text{C} \) each are at the vertices of an equilateral triangle of side 2 m. What is the force

**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. Force between two charges: F = 9 × 10⁹ × ((2 × 10⁻⁶)²/(2)²) = 9 × 10⁻³ N . Two forces at 60°. Net force: Fₙₑt = √(F² + F² + 2 F² cos 60°) = √(3) × 9 × 10⁻³ = 1.56 × 10⁻² N . Substituting values gives 1.56 × 10⁻² 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

What characteristic of an electric field ensures that work done in moving a charge along a closed path is zero in electr

**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. The conservative nature of the electrostatic field means the work done depends only on the potential difference between points, not the path. For a closed path, the start and end points are the same, so the net work is zero. Substituting values gives Conservative nature, 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

A conducting sphere of radius 25 cm has an electric field of \( 5 \times 10^3 \, \text{N/C} \) at 50 cm from its center.

**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. E = (k q/r²) . 5 × 10³ = 9 × 10⁹ × (q/(0.5)²) . q = (5 × 10³ × 0.25/9 × 10⁹) = 1.389 × 10⁻⁷ C . Substituting values gives 1.39 × 10⁻⁷ C, 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

What property of electric charges explains why two objects with identical charges repel each other?

**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. The polarity of charge (positive or negative) determines interaction: like charges repel due to the repulsive force described by Coulomb’s law, where the force direction depends on the sign of the charges, causing repulsion for identical signs. Substituting values gives Polarity, 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

A conducting sphere of radius 12 cm has an electric field of \( 6 \times 10^3 \, \text{N/C} \) at 24 cm from its center.

**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. E = (k q/r²) . 6 × 10³ = 9 × 10⁹ × (q/(0.24)²) . q = (6 × 10³ × 0.0576/9 × 10⁹) = 3.84 × 10⁻⁸ C . Substituting values gives 3.84 × 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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Superposition Principle and Equilibrium of Charges

Two point charges \( q_1 = 4 \, \mu\text{C} \) and \( q_2 = -6 \, \mu\text{C} \) are placed 20 cm apart in vacuum. What

**Superposition principle** asserts net Coulomb force on charge equals vector sum of forces from each other charge independently, F_net = Σ F_i, where F_i = k q q_i/r_i² r̂_i. In equilateral triangle or square symmetry, components may cancel at centroid, producing equilibrium. Using Coulomb's law: F = k (|q₁ q₂|/r²) . Given: k = 9 × 10⁹ N·m²/C² , q₁ = 4 × 10⁻⁶ C , q₂ = -6 × 10⁻⁶ C , r = 0.2 m . Magnitude: |q₁ q₂| = (4 × 10⁻⁶) × (6 × 10⁻⁶) = 24 × 10⁻¹² C² . r² = (0.2)² = 0.04 m² . F = 9

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Superposition Principle and Equilibrium of Charges

Two point charges \( 6 \times 10^{-7} \, \text{C} \) and \( 9 \times 10^{-7} \, \text{C} \) are 45 cm apart in air. What

**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. Using Coulomb’s law: F = k (|q₁ q₂|/r²) . k = 9 × 10⁹ N·m²/C² , q₁ = 6 × 10⁻⁷ C , q₂ = 9 × 10⁻⁷ C , r = 0.45 m . |q₁ q₂| = 6 × 9 × 10⁻¹⁴ = 54 × 10⁻¹⁴ C² . r² = (0.45)² = 0.2025 m² . F = 9 × 10⁹ × (54 × 10⁻¹⁴/0.2025) = 9 × 10⁹ × 2.667 ×

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Superposition Principle and Equilibrium of Charges

What characteristic of conductors allows charges to distribute uniformly over their surface when placed in an external e

**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. In conductors, charges (free electrons) can move freely. In an external field, they redistribute until the internal field cancels the external field, achieving equilibrium. This results in charges residing only on the surface, distributed uniformly for a spherical conductor due to symmetry. Substituting values gives Mobility of charges, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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