Question

The electric potential due to an electric dipole
(A) depends on r, where r is the magnitude of position vector \(\vec{r}\)
(B) depends on the angle between the position vector \(\vec{r}\) and the dipole moment vector \(\vec{p}\)
(C) falls off at long distances, as \(1/r^2\)
(D) does not depend upon the distance separating the charges
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (A), (B) and (C) only
• C. (A), (B), (C) and (D)
• D. (B), (C) and (D) only

Hint:

Remember the distance dependence for different charge distributions: Point charge potential \(V \propto 1/r\), dipole potential \(V \propto 1/r^2\), and quadrupole potential \(V \propto 1/r^3\). This pattern is useful for quickly identifying the correct relationship in multiple-choice questions.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
An electric dipole consists of two equal and opposite charges separated by a small distance. The electric potential at a point due to this dipole is the sum of the potentials due to each individual charge. We need to analyze the properties of this potential, especially at large distances from the dipole.

Step 2: Key Formula or Approach:
The electric potential \(V\) at a point with position vector \(\vec{r}\) due to an electric dipole with dipole moment \(\vec{p}\) is given by: \[ V = \frac{1}{4\pi\varepsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2} = \frac{1}{4\pi\varepsilon_0} \frac{p \cos\theta}{r^2} \] where \(r = |\vec{r}|\) is the distance from the center of the dipole to the point, and \(\theta\) is the angle between the dipole moment vector \(\vec{p}\) and the position vector \(\vec{r}\). The dipole moment is defined as \(\vec{p} = q\vec{d}\), where \(\vec{d}\) is the vector from the negative to the positive charge.

Step 3: Detailed Explanation:
Let's evaluate each statement based on the formula:
(A) depends on r, where r is the magnitude of position vector \(\vec{r}\):
The formula \(V \propto \frac{1}{r^2}\) clearly shows that the potential depends on the distance \(r\). So, statement (A) is correct.
(B) depends on the angle between the position vector \(\vec{r}\) and the dipole moment vector \(\vec{p}\):
The term \(\cos\theta\) in the formula indicates that the potential depends on the angle \(\theta\) between \(\vec{p}\) and \(\vec{r}\). So, statement (B) is correct.
(C) falls off at long distances, as \(1/r^2\):
The formula shows that \(V\) is inversely proportional to the square of the distance \(r\) (\(V \propto 1/r^2\)). This confirms that the potential falls off as \(1/r^2\) at long distances. So, statement (C) is correct.
(D) does not depend upon the distance separating the charges:
The dipole moment is defined as \(p = qd\), where \(d\) is the distance separating the two charges. Since the potential \(V\) depends on \(p\), it indirectly depends on the separation distance \(d\). Therefore, statement (D) is incorrect.

Step 4: Final Answer:
Statements (A), (B), and (C) are correct, while (D) is incorrect. The correct choice is the option that includes only (A), (B), and (C).