Electric Potential Due to an Electric Dipole
This section explains the calculation of electrostatic potential resulting from an electric dipole configuration, consisting of two charges (+q and -q) separated by a distance 2a. The total charge of the dipole is zero, which distinguishes it from a single charge. The potential (V) at a point in space due to a dipole is derived using the principle of superposition:
$$ V = \frac{1}{4 \pi \epsilon_0} \left(\frac{q}{r_1} - \frac{q}{r_2}\right) $$
where r1 and r2 are the distances from the point of interest to the respective charges. The excerpt highlights that for distances far greater than the size of the dipole (r >> a), the higher-order terms become negligible, leading to the dipole potential being expressed as:
$$ V = \frac{1}{4 \pi \epsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2} $$
where p is the dipole moment vector, and \hat{r} is the unit vector along the position vector from the dipole. It is also noted that the potential depends on the angle between the position vector and the dipole moment, a feature that distinctly differentiates dipole potential from point charge potential, which only depends on the distance from the charge. Understanding these concepts is paramount in analyzing systems with multiple charges and their resulting electric fields.