To find the torque on an electric dipole in an electric field, we can use the formula for torque (\( \tau \)), which is given by:
τ = p × E
Here, \( p \) is the dipole moment, and \( E \) is the electric field strength. The dipole moment (\( p \)) can be calculated using the formula:
p = q × d
Where \( q \) is the magnitude of one of the charges and \( d \) is the separation distance between the charges. In this case, we have:
- Charge \( q = 2e \) (where \( e \) is the elementary charge, approximately \( 1.6 × 10^{-19} \) C)
- Separation \( d = 1.1 \) nm = \( 1.1 × 10^{-9} \) m
First, let's calculate the dipole moment:
p = (2 × 1.6 × 10^{-19} \, \text{C}) × (1.1 × 10^{-9} \, \text{m})
Calculating this gives:
p = 3.52 × 10^{-28} \, \text{C m}
Now that we have the dipole moment, we can calculate the torque for the three different orientations of the dipole moment relative to the electric field.
Torque Calculations
1. When the dipole moment is parallel to the electric field
In this case, the angle \( θ = 0° \). The torque is given by:
τ = pE sin(θ)
Since sin(0°) = 0, the torque is:
τ = 3.52 × 10^{-28} \, \text{C m} × 4.5 × 10^{6} \, \text{N/C} × 0 = 0
2. When the dipole moment is perpendicular to the electric field
Here, the angle \( θ = 90° \). The torque is maximized because sin(90°) = 1:
τ = pE sin(90°)
Calculating this gives:
τ = 3.52 × 10^{-28} \, \text{C m} × 4.5 × 10^{6} \, \text{N/C} × 1
τ = 1.584 × 10^{-21} \, \text{N m}
3. When the dipole moment is antiparallel to the electric field
In this scenario, the angle \( θ = 180° \). The torque is again calculated using:
τ = pE sin(180°)
Since sin(180°) = 0, the torque is:
τ = 3.52 × 10^{-28} \, \text{C m} × 4.5 × 10^{6} \, \text{N/C} × 0 = 0
Summary of Results
- Torque when parallel to the electric field: 0 N m
- Torque when perpendicular to the electric field: 1.584 × 10^{-21} N m
- Torque when antiparallel to the electric field: 0 N m
This analysis shows how the orientation of the dipole moment relative to the electric field significantly affects the torque experienced by the dipole. When aligned with the field, there is no torque, while maximum torque occurs when the dipole is perpendicular to the field.
