Section 2.6: Electric Dipoles
An electric dipole consists of two equal and opposite charges separated by a small distance. Dipoles are fundamental in molecular physics and form the basis of dielectric behavior.
The dipole moment is defined as \[ \vec{p} = q \cdot \vec{d} \] where \( q \) is the magnitude of each charge and \( \vec{d} \) is the displacement vector from negative to positive charge.
On the axis: \[ E_{\text{axis}} \approx \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p}{r^3} \] On the perpendicular bisector: \[ E_{\text{equatorial}} \approx \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{r^3} \]
Example 1
Find the torque on an electric dipole of moment \( p \) placed in a uniform electric field \( E \) making an angle \( \theta \) with the field.
The forces on the two charges are equal and opposite, creating a torque but no net force.
\[ \tau = p E \sin \theta \]
The dipole tends to align with the electric field.
Practice Problems
- Derive the expression for the electric field along the axis of a dipole.
- Derive the expression for the electric field along the perpendicular bisector of a dipole.
- Explain why the dipole field decreases as \( 1/r^3 \), unlike a single charge field (\( 1/r^2 \)).
- Compute the torque on a dipole of moment \( 3 \times 10^{-29} \, \text{C·m} \) in a \( 2 \times 10^5 \, \text{N/C} \) field at \( 30^\circ \).
- Conceptual: Why do polar molecules tend to align with external electric fields?