#dipole

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 2 mm is in a field \( 8 \times 10

**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. Dipole moment: p = q × 2a = 6 × 10⁻⁶ × 2 × 10⁻³ = 1.2 × 10⁻⁸ C m . Torque: tau = p E sin θ = 1.2 × 10⁻⁸ × 8 × 10⁴ × sin 60° = 9.6 × 10⁻⁴ × (√(3)/2) = 8.31 × 10⁻⁴ N

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Electric Field and Electric Field Lines

What explains why the electric field due to a dipole can be zero at certain points along its equatorial plane?

**Electric field concept** visualizes influence of source charge. Uniform field exerts constant force F = qE, and flux Φ = E·A = E A cosθ links field to area orientation, maximum when field normal to surface. In the equatorial plane, the fields from the dipole’s positive and negative charges are equal in magnitude and opposite in direction at the midpoint. Their vector sum cancels out, resulting in a zero field at that specific location. Substituting values gives Field cancellation, 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 \( p = 7 \times 10^{-9} \, \text{C m} \) is at 45° to a field \( E = 4 \times 10^4 \, \text{N/C} \). 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 = 7 × 10⁻⁹ × 4 × 10⁴ × sin 45° = 28 × 10⁻⁵ × (√(2)/2) = 1.98 × 10⁻⁴ N m . Substituting values gives 1.98 × 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 moment \( p = 5 \times 10^{-9} \, \text{C m} \) is at 60° to a uniform field \( E = 4 \times 10^4 \, \text

**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. Torque: tau = p E sin θ . tau = 5 × 10⁻⁹ × 4 × 10⁴ × sin 60° = 20 × 10⁻⁵ × (√(3)/2) = 1.732 × 10⁻⁴ N m . Substituting values gives 1.73 × 10⁻⁴ N m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's

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Electric Dipole - Moment, Field and Torque

A dipole with charges \( +4 \, \mu\text{C} \) and \( -4 \, \mu\text{C} \) separated by 2 mm is in a field \( 7 \times 10

**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. Dipole moment: p = q × 2a = 4 × 10⁻⁶ × 2 × 10⁻³ = 8 × 10⁻⁹ C m . Torque: tau = p E sin θ = 8 × 10⁻⁹ × 7 × 10⁴ × sin 30° = 5.6 × 10⁻⁴ × 0.5 = 2.8 × 10⁻⁴ N m . Substituting values gives 2.8 × 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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