Step 1: Conceptual Foundation:
An electric dipole comprises two charges of equal magnitude but opposite sign, separated by a minimal distance. The electric potential at any given point due to this dipole is the aggregate potential from each constituent charge. Analysis will focus on the characteristics of this potential, particularly at significant distances from the dipole.
Step 2: Governing Equation:
The electric potential \(V\) at a point defined by the position vector \(\vec{r}\) caused by an electric dipole with dipole moment \(\vec{p}\) is articulated as:
\[ V = \frac{1}{4\pi\varepsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2} = \frac{1}{4\pi\varepsilon_0} \frac{p \cos\theta}{r^2} \]
Here, \(r = |\vec{r}|\) represents the distance from the dipole's center to the point of observation, and \(\theta\) is the angle between the dipole moment vector \(\vec{p}\) and the position vector \(\vec{r}\). The dipole moment itself is defined as \(\vec{p} = q\vec{d}\), where \(\vec{d}\) is the vector directed from the negative charge to the positive charge.
Step 3: Detailed Analysis:
Each statement is assessed against the established formula:
(A) Dependence on \(r\), the magnitude of the position vector \(\vec{r}\):
The proportionality \(V \propto \frac{1}{r^2}\) unequivocally demonstrates that the potential is contingent upon the distance \(r\). Therefore, statement (A) is valid.
(B) Dependence on the angle between \(\vec{r}\) and \(\vec{p}\):
The inclusion of the \(\cos\theta\) term within the formula indicates that the potential's value is influenced by the angle \(\theta\) formed between \(\vec{p}\) and \(\vec{r}\). Consequently, statement (B) is valid.
(C) Decay at long distances, proportional to \(1/r^2\):
The formula explicitly shows that \(V\) is inversely proportional to the square of the distance \(r\), denoted as \(V \propto 1/r^2\). This confirms that the potential diminishes with distance at a rate of \(1/r^2\) for large distances. Thus, statement (C) is valid.
(D) Independence from the separation distance between charges:
The dipole moment is defined as \(p = qd\), where \(d\) is the separation distance between the two charges. Since the potential \(V\) is dependent on \(p\), it is indirectly influenced by the separation distance \(d\). Therefore, statement (D) is invalid.
Step 4: Conclusion:
Statements (A), (B), and (C) are substantiated by the derived formula, while statement (D) is contradicted. The correct option will encompass only (A), (B), and (C).