Electric Dipole - Moment, Field and Torque

Electric Dipole - Moment, Field and Torque

A conducting sphere of radius 21 cm has an electric field of \( 8 \times 10^3 \, \text{N/C} \) at 42 cm from its center.

**Dipole moment** governs torque and energy in external field. Axial field stronger than equatorial, torque maximum at θ = 90°, zero when aligned. Work done rotating dipole relates to ΔU = pE(1 - cosθ), explaining stable equilibrium at θ = 0°. E = (k q/r²) . 8 × 10³ = 9 × 10⁹ × (q/(0.42)²) . q = (8 × 10³ × 0.1764/9 × 10⁹) = 1.57 × 10⁻⁷ C . Substituting values gives 1.57 × 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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Electric Dipole - Moment, Field and Torque

Which property of electric charge explains why the total charge of an isolated system remains constant even when objects

**Electric dipole** consists of charges +q and -q separated by 2a, dipole moment p = q·2a, vector from negative to positive, unit C·m. In uniform field E, torque τ = p × E, magnitude τ = p E sinθ, tending to align p with E, potential energy U = -p·E = -p E cosθ. The conservation of electric charge states that the total charge in an isolated system remains constant over time. When objects are rubbed together, charge is transferred from one to another (e.g., electrons move), but no new charge is created or destroyed. This ensures the net charge of the system stays the

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Electric Dipole - Moment, Field and Torque

What allows an electric dipole to experience a net force in a non-uniform electric field but not in a uniform one?

**Dipole moment** governs torque and energy in external field. Axial field stronger than equatorial, torque maximum at θ = 90°, zero when aligned. Work done rotating dipole relates to ΔU = pE(1 - cosθ), explaining stable equilibrium at θ = 0°. In a non-uniform field, the field strength varies across the dipole, causing unequal forces on the positive and negative charges. This results in a net force, unlike in a uniform field where equal and opposite forces cancel out. Substituting values gives Field gradient, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Electric Dipole - Moment, Field and Torque

Two charges \( +9 \, \mu\text{C} \) and \( -6 \, \mu\text{C} \) are 45 cm apart. What is the distance from \( +9 \, \mu\

**Dipole moment** governs torque and energy in external field. Axial field stronger than equatorial, torque maximum at θ = 90°, zero when aligned. Work done rotating dipole relates to ΔU = pE(1 - cosθ), explaining stable equilibrium at θ = 0°. Let x be distance from +9 μC , then 0.45 - x from -6 μC . (9 × 10⁻⁶/x²) = (6 × 10⁻⁶/(0.45 - x)²) , 9 (0.45 - x)² = 6 x² . 1.5 (0.2025 - 0.9 x + x²) = x² , 0.30375 - 1.35 x + 1.5 x² = x² . 0.5 x² - 1.35 x + 0.30375 = 0 ,

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Electric Dipole - Moment, Field and Torque

A metallic sphere of radius 15 cm has a charge of \( 6 \, \mu\text{C} \). What is the electric field just outside its su

**Electric dipole** consists of charges +q and -q separated by 2a, dipole moment p = q·2a, vector from negative to positive, unit C·m. In uniform field E, torque τ = p × E, magnitude τ = p E sinθ, tending to align p with E, potential energy U = -p·E = -p E cosθ. For a conductor, E = (k q/r²) at surface ( r = 0.15 m ). E = 9 × 10⁹ × (6 × 10⁻⁶/(0.15)²) = 2.4 × 10⁶ N/C . Substituting values gives 2.4 × 10⁶ 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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Electric Dipole - Moment, Field and Torque

What property of electric charges explains why the smallest observable charge is that of an electron or proton?

**Interaction of dipole with uniform field** produces pure couple without net force, equal opposite forces forming torque. Potential energy minimum -pE at alignment, maximum +pE at anti-alignment, governing orientation dynamics. Quantization of charge means that charge exists in discrete units, with the smallest unit being the charge of an electron or proton ( e ). All observable charges are integer multiples of this fundamental unit, reflecting this property. Substituting values gives Quantization, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Electric Dipole - Moment, Field and Torque

A dipole with \( p = 6 \times 10^{-9} \, \text{C m} \) is at 60° to a field \( E = 7 \times 10^4 \, \text{N/C} \). What

**Dipole moment** governs torque and energy in external field. Axial field stronger than equatorial, torque maximum at θ = 90°, zero when aligned. Work done rotating dipole relates to ΔU = pE(1 - cosθ), explaining stable equilibrium at θ = 0°. tau = p E sin θ . tau = 6 × 10⁻⁹ × 7 × 10⁴ × sin 60° = 42 × 10⁻⁵ × (√(3)/2) = 3.64 × 10⁻⁴ N m . Substituting values gives 3.64 × 10⁻⁴ N m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Electric Dipole - Moment, Field and Torque

Why can’t the electric field inside a charged insulator be zero, unlike in a conductor?

**Dipole moment** governs torque and energy in external field. Axial field stronger than equatorial, torque maximum at θ = 90°, zero when aligned. Work done rotating dipole relates to ΔU = pE(1 - cosθ), explaining stable equilibrium at θ = 0°. In insulators, charges are fixed and cannot move to cancel an internal field. If charges are present inside, they generate a field that persists, as there are no free charges to redistribute and neutralize it, unlike in conductors. Substituting values gives Lack 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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Electric Dipole - Moment, Field and Torque

A dipole with \( p = 9 \times 10^{-9} \, \text{C m} \) is at 90° to a field \( E = 3 \times 10^4 \, \text{N/C} \). What

**Electric dipole** consists of charges +q and -q separated by 2a, dipole moment p = q·2a, vector from negative to positive, unit C·m. In uniform field E, torque τ = p × E, magnitude τ = p E sinθ, tending to align p with E, potential energy U = -p·E = -p E cosθ. tau = p E sin θ . tau = 9 × 10⁻⁹ × 3 × 10⁴ × sin 90° = 27 × 10⁻⁵ × 1 = 2.7 × 10⁻⁴ N m . Substituting values gives 2.7 × 10⁻⁴ N m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and

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Electric Dipole - Moment, Field and Torque

Why does the electric field due to a dipole have both radial and tangential components at a general point?

**Electric dipole** consists of charges +q and -q separated by 2a, dipole moment p = q·2a, vector from negative to positive, unit C·m. In uniform field E, torque τ = p × E, magnitude τ = p E sinθ, tending to align p with E, potential energy U = -p·E = -p E cosθ. The dipole’s field results from two opposite charges, creating a complex pattern. At a general point, the field vectors from each charge have different directions, resolving into radial (along the line from the dipole) and tangential (perpendicular) components due to asymmetry. Substituting values gives Vector addition, which matches expected magnitude for

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Electric Dipole - Moment, Field and Torque

A dipole with \( p = 8 \times 10^{-9} \, \text{C m} \) is at 90° to a field \( E = 6 \times 10^4 \, \text{N/C} \). What

**Interaction of dipole with uniform field** produces pure couple without net force, equal opposite forces forming torque. Potential energy minimum -pE at alignment, maximum +pE at anti-alignment, governing orientation dynamics. tau = p E sin θ . tau = 8 × 10⁻⁹ × 6 × 10⁴ × sin 90° = 48 × 10⁻⁵ × 1 = 4.8 × 10⁻⁴ N m . Substituting values gives 4.8 × 10⁻⁴ N m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

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Electric Dipole - Moment, Field and Torque

A dipole with charges \( +6 \, \mu\text{C} \) and \( -6 \, \mu\text{C} \) separated by 4 mm is in a field \( 5 \times 10

**Dipole moment** governs torque and energy in external field. Axial field stronger than equatorial, torque maximum at θ = 90°, zero when aligned. Work done rotating dipole relates to ΔU = pE(1 - cosθ), explaining stable equilibrium at θ = 0°. Dipole moment: p = q × 2a = 6 × 10⁻⁶ × 4 × 10⁻³ = 2.4 × 10⁻⁸ C m . Torque: tau = p E sin θ = 2.4 × 10⁻⁸ × 5 × 10⁴ × sin 60° = 1.2 × 10⁻³ × (√(3)/2) = 1.04 × 10⁻³ N m . Substituting values gives 1.04 × 10⁻³ N m, which matches expected

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