An electric dipole of dipole moment 5 × 10⁻⁶ Cm is placed in a uniform electric field of 10⁻² N/C making an angle of 30° with the direction of the field. The torque exerted by the electric field on the dipole is:
2.5 × 10⁻⁸ Nm
Calculating Torque on an Electric Dipole in a Uniform Field
This problem requires us to calculate the torque experienced by an electric dipole when placed in a uniform electric field. The torque depends on the magnitude of the dipole moment, the strength of the electric field, and the angle between the dipole moment vector and the electric field vector.
Understanding Electric Dipole Torque
An electric dipole consists of two equal and opposite charges separated by a small distance. The electric dipole moment (\( \vec{p} \)) is a vector quantity defined as the product of the charge magnitude and the separation distance, pointing from the negative charge to the positive charge.
When an electric dipole is placed in a uniform electric field (\( \vec{E} \)), forces act on the two charges. These forces are equal in magnitude and opposite in direction, forming a couple. This couple exerts a torque on the dipole, tending to align it with the direction of the electric field.
Formula for Torque
The magnitude of the torque (\( \tau \)) experienced by an electric dipole in a uniform electric field is given by the formula:
\( \tau = pE \sin(\theta) \)
Where:
- \( p \) is the magnitude of the electric dipole moment.
- \( E \) is the magnitude of the electric field.
- \( \theta \) is the angle between the direction of the electric dipole moment vector (\( \vec{p} \)) and the direction of the electric field vector (\( \vec{E} \)).
The direction of the torque is perpendicular to the plane containing \( \vec{p} \) and \( \vec{E} \), and it is given by the vector cross product \( \vec{\tau} = \vec{p} \times \vec{E} \).
Applying the Formula to the Given Problem
We are given the following values:
- Electric dipole moment, \( p = 5 \times 10^{-6} \) Cm
- Uniform electric field, \( E = 10^{-2} \) N/C
- Angle between the dipole moment and the field, \( \theta = 30^\circ \)
We need to find the torque \( \tau \).
Using the formula \( \tau = pE \sin(\theta) \), we substitute the given values:
\( \tau = (5 \times 10^{-6} \text{ Cm}) \times (10^{-2} \text{ N/C}) \times \sin(30^\circ) \)
We know that \( \sin(30^\circ) = 0.5 \).
So, the calculation becomes:
\( \tau = (5 \times 10^{-6}) \times (10^{-2}) \times 0.5 \text{ Nm} \)
\( \tau = (5 \times 10^{-8}) \times 0.5 \text{ Nm} \)
\( \tau = 2.5 \times 10^{-8} \text{ Nm} \)
Result of the Torque Calculation
The calculated torque exerted by the electric field on the electric dipole is \( 2.5 \times 10^{-8} \) Nm.
Summary of Calculation
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Dipole Moment | \(p\) | \(5 \times 10^{-6}\) | Cm |
| Electric Field | \(E\) | \(10^{-2}\) | N/C |
| Angle | \( \theta \) | 30 | Degrees |
| \( \sin(\theta) \) | \( \sin(30^\circ) \) | 0.5 | dimensionless |
| Torque (\( \tau = pE \sin(\theta) \)) | \( \tau \) | \(2.5 \times 10^{-8}\) | Nm |
The result \( 2.5 \times 10^{-8} \) Nm matches one of the given options.
Revision Table: Key Concepts
| Concept | Description | Relevant Formula |
|---|---|---|
| Electric Dipole Moment | Vector quantity representing separation of charges, points from -q to +q. \( \vec{p} = q \vec{d} \). | \(p = qd\) |
| Torque on Dipole | Rotational effect on dipole in electric field, tends to align dipole with field. | \( \vec{\tau} = \vec{p} \times \vec{E} \) or \( \tau = pE \sin(\theta) \) (magnitude) |
| Potential Energy of Dipole | Energy stored due to dipole's orientation in field. Minimum when aligned with field. | \( U = -\vec{p} \cdot \vec{E} \) or \( U = -pE \cos(\theta) \) |
Additional Information: Electric Dipoles and Fields
Electric dipoles are fundamental in understanding the behavior of molecules in electric fields (like polarization in dielectrics). The torque calculated here is responsible for rotating the dipole. When the dipole aligns parallel to the electric field (\( \theta = 0^\circ \)), the torque is zero, and the dipole is in a state of stable equilibrium (minimum potential energy). When it is anti-parallel (\( \theta = 180^\circ \)), the torque is also zero, but this is a state of unstable equilibrium (maximum potential energy). The maximum torque occurs when the dipole is perpendicular to the field (\( \theta = 90^\circ \)), where \( \sin(90^\circ) = 1 \).
Understanding the torque on an electric dipole is crucial for studying the response of materials to electric fields and the dynamics of polar molecules.
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