Question

Determine the components of the electric field, Ex and Ey, as a function of x and y in the xy plane due to a dipole shown in (Figure 1) starting with the equation v = 1/(4πε₀) * p * cosθ / r². Assume r = (x² + y²)¹/² ≫.

A) Ex = -∂v/∂x, Ey = -∂v/∂y
B) Ex = ∂v/∂x, Ey = ∂v/∂y
C) Ex = ∂v/∂y, Ey = -∂v/∂x
D) Ex = -∂v/∂y, Ey = ∂v/∂x

Answer 1

The components of the electric field due to a dipole in the xy plane are Ex = -∂v/∂x and Ey = -∂v/∂y, which are calculated by taking the negative gradient of the potential.

Step-by-step explanation:

To determine the components of the electric field, Ex and Ey, as a function of x and y in the xy plane due to a dipole, we can use the potential v given by the equation:

v = \( \frac{1}{4\pi\varepsilon_0} \cdot \frac{p \cdot \cos\theta}{r^2} \)

Assuming that r equals \( (x^2 + y^2)^{\frac{1}{2}} \) and is much larger than the separation of the charges in the dipole, we can use this to calculate the electric field. The correct components of the electric field are given by the negative gradient of the potential:

• Ex = -\( \frac{\partial v}{\partial x} \)
• Ey = -\( \frac{\partial v}{\partial y} \)

Therefore, the correct answer to the student's question is:

A) Ex = -\( \frac{\partial v}{\partial x} \), Ey = -\( \frac{\partial v}{\partial y} \)