The Question is from neet-physics electrostatics , and it was created by the author VENKATESWARARAO

Question

An electric dipole with dipole moment $\mathbf{p} = p_0\mathbf{i}$ is placed at the origin. The electric potential at a point $(r, 	heta)$ is $V$.  If $r >> a$, where $2a$ is the dipole separation,  which of the following is correct regarding the equipotential surfaces?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION ✓ **Answer**: C ## CONCEPT FRAMEWORK **Primary Concept**: Electric potential due to a dipole and equipotential surfaces. **Related Topics**: Electrostatics, dipole moment, potential, spherical coordinates. **Prerequisites**: Understanding of electric potential, dipole moment, and spherical coordinate system. ## SOLUTION PATHWAY **Given Information**: * Dipole moment: $\mathbf{p} = p_0\mathbf{i}$ (along the x-axis) * Point in space: $(r, heta)$ in spherical coordinates * Condition: $r >> a$ (far-field approximation) * Electric potential: V (constant on equipotential surfaces) **Method & Steps**: 1. The electric potential due to a dipole at a point $(r, heta)$ in the far-field approximation ($r >> a$) is given by: $V(r, heta) = \frac{1}{4\pi\epsilon_0} \frac{p_0 \cos heta}{r^2}$ Reasoning: This formula is derived from the general expression for the electric potential of a dipole and simplifies significantly in the far-field limit. Formula/Rule: $V = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \mathbf{r}}{r^3}$ 2. For an equipotential surface, the potential V is constant. Therefore, we have: $V = \frac{1}{4\pi\epsilon_0} \frac{p_0 \cos heta}{r^2} = constant$ This implies that $r^2 \cos heta$ is constant. This equation represents a conical surface. Working: Rearranging the equation, we get $r^2 = k/\cos heta$, where k is a constant. This is the equation of a cone with its apex at the origin and axis along the dipole moment (x-axis). Key Point: The far-field approximation simplifies the potential expression, making the analysis tractable. 3. The equipotential surfaces are approximately conical surfaces with their axes along the dipole moment. Result: The correct option is C. Verification: The equation obtained satisfies the condition of constant potential for different values of r and θ, forming conical surfaces. ## OPTION ANALYSIS A :x: Spheres centered at the origin would have a potential dependent solely on r, not θ. B :x: Cylinders would have potential independent of θ, contradicting the potential formula. C ✓: Conical surfaces correctly reflect the potential's dependence on both r and θ as shown in the derived equation. D :x: Planes perpendicular to the dipole moment would have a potential dependent on the distance from the plane, not following the derived potential formula. ## LEARNING AIDS **Common Mistakes**: 1.Using the near-field approximation ($r << a$) instead of the far-field approximation ($r >> a$). This would lead to a completely different potential expression. How to avoid: Carefully examine the given conditions and use the appropriate approximation for the potential. 2.Incorrectly interpreting the equation of the equipotential surface. Misunderstanding the relationship between r and θ can lead to incorrect conclusions. How to avoid: Practice analyzing equations in spherical coordinates and visualize the surfaces they represent. **Quick Tips**: 1.Remember the far-field potential formula for a dipole: $V(r, heta) = \frac{1}{4\pi\epsilon_0} \frac{p\cos heta}{r^2}$ for quick calculations. 2.Visualize the equipotential surfaces to understand their shape and orientation relative to the dipole moment.

VENKATESWARARAO · Answer

They are spheres centered at the origin.

VENKATESWARARAO · Answer

They are cylinders with their axes along the dipole moment.

VENKATESWARARAO · Answer

The correct Option is "They are approximately conical surfaces with their axes along the dipole moment."

VENKATESWARARAO · Answer

They are planes perpendicular to the dipole moment.

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