Section 3.6: Dipole Potential
For an electric dipole consisting of charges +q and -q separated by a distance 2a, the electric potential at a point in space is given by:
Dipole Potential:
\[ V(\vec{r}) = \frac{1}{4 \pi \epsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2}, \quad \vec{p} = q \cdot 2\vec{a} \]
where \( \vec{p} \) is the dipole moment vector, \( r \) is distance from the center of the dipole, and \( \hat{r} \) is the unit vector pointing from the dipole to the point.
Example 1
Compute the potential along the axial line of a dipole at distance \( z \) from its center.
For axial points, \( \hat{r} \) is aligned with \( \vec{p} \), so \( \vec{p} \cdot \hat{r} = p \).
\[ V(z) = \frac{1}{4 \pi \epsilon_0} \frac{p}{z^2} \]
Practice Problems
- Compute the potential along the equatorial line of a dipole at distance \( r \) from center.
- Show that \( \vec{E} = -\nabla V \) for a dipole potential.
- Determine the potential at a point midway between the charges of a dipole.
- Explain the directional dependence of dipole potential.
- Compare the axial and equatorial electric fields of a dipole.