Question

Since \[p\cos\theta = \vec{p} \cdot \hat{r}\], the dipole potential at a point (r >> a) written in compact vector form is:

Options

• \[V = \frac{1}{4\pi\varepsilon_0}\frac{|\vec{p}|}{r^2}\]

• \[V = \frac{1}{4\pi\varepsilon_0}\frac{\vec{p} \times \hat{r}}{r^2}\]

• \[V = \frac{1}{4\pi\varepsilon_0}\frac{\vec{p} \cdot \hat{r}}{r}\]

• \[V = \frac{1}{4\pi\varepsilon_0}\frac{\vec{p} \cdot \hat{r}}{r^2}\]

Answer 1

p cos θ is the component of the dipole moment along the unit vector r̂, which is exactly the dot product p⃗ · r̂. The cross product would produce a vector, which cannot represent the scalar potential.