A small, rigid object carries positive and negative 2.00 nC charges. It is orient [Free] B116
A small, rigid object carries positive and negative 2.00 nC charges. It is oriented so that the positive charge has coordinates (-1.20mm,1.60mm)1.50mm,-1.30mm Cmhat(i)+,Cmhat(j) (b) The object is placed in an electric field vec(E)=(7.80×103(hat(i))-4.90×103(hat(j))) NC.Find the torque acting on the object. N*m—Select— v (c) Find the potential energy of the object-field system when the object is in this orientation. J (d) Assuming the orientation of the object can change, find the difference between the maximum and the minimum potential energies of the system. J
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Answer
Electric Dipole Moment, Torque & Potential Energy
Step 1: Electric Dipole Moment (⃗p)
Given:
- Charge: q = 2.00 × 10⁻⁹ C
- Positive charge location: (-1.20 × 10⁻³, 1.60 × 10⁻³) m
- Negative charge location: (1.50 × 10⁻³, -1.30 × 10⁻³) m
Formula:
Where ⃗d is the vector from negative to positive charge.
Displacement vector:
Dipole moment:
⃗p = (-5.40 × 10⁻¹², 5.80 × 10⁻¹²) C·m
Final Result: ⃗p = -5.40 × 10⁻¹² î + 5.80 × 10⁻¹² ĵ C·m
Step 2: Torque on the Dipole (τ⃗)
Given Electric Field: ⃗E = (7.80 × 10³, -4.90 × 10³) N/C
Formula:
Cross product in 2D:
= (-5.40 × 10⁻¹²)(-4.90 × 10³) – (5.80 × 10⁻¹²)(7.80 × 10³)
= 2.646 × 10⁻⁸ – 4.524 × 10⁻⁸
= -1.878 × 10⁻⁸ N·m
Final Result: τ = -1.878 × 10⁻⁸ N·m
Step 3: Potential Energy of Dipole in Electric Field
Formula:
Dot product:
= -[-4.212 × 10⁻⁸ – 2.842 × 10⁻⁸]
= 7.054 × 10⁻⁸ J
Final Result: U = 7.054 × 10⁻⁸ J
Step 4: Maximum and Minimum Potential Energy
Formulas:
- Umax = +|⃗p||⃗E|
- Umin = -|⃗p||⃗E|
- ΔU = Umax – Umin
Magnitude of Dipole Moment:
= √62.80 × 10⁻¹² ≈ 7.93 × 10⁻¹² C·m
Magnitude of Electric Field:
= √84.85 × 10³ ≈ 9.21 × 10³ N/C
Maximum Potential Energy:
Energy Range:
Final Result:
- Umax = 7.423 × 10⁻⁸ J
- ΔU = 1.485 × 10⁻⁷ J
Final Summary
- Dipole Moment: ⃗p = -5.4010 × 10⁻¹² î + 5.8010 × 10⁻¹² ĵ C·m
- Torque: τ = -1.878 × 10⁻⁸ N·m
- Potential Energy: U = 7.05410 × 10⁻⁸ J
- Energy Range: ΔU = 1.485 × 10⁻⁷ J