Comparison of electric field of short dipole at axial and equatorial points Why axial is two times??E equatorial=1/4π€.×p/r cubeE axial= 2× E equatorial

To understand the comparison of the electric field of a short dipole at axial and equatorial points, we first need to clarify what a dipole is and how its electric field behaves in different regions. A dipole consists of two equal and opposite charges separated by a small distance. The electric field generated by this configuration varies depending on where you measure it—either along the axis of the dipole (axial point) or perpendicular to it (equatorial point).

Electric Field of a Dipole

The electric field (\(E\)) due to a dipole can be expressed mathematically. For a short dipole, the electric field at different points can be derived from the dipole moment (\(p\)), which is defined as the product of the charge (\(q\)) and the distance (\(d\)) between the charges:

• Dipole Moment: \(p = q \cdot d\)

Field at Axial Points

At an axial point, which is along the line extending from the positive charge to the negative charge, the electric field can be calculated using the formula:

• Electric Field (Axial): \(E_{\text{axial}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{r^3}\)

Here, \(r\) is the distance from the center of the dipole to the point where the field is being measured, and \(\epsilon_0\) is the permittivity of free space. This formula shows that the electric field at axial points is directly proportional to the dipole moment and inversely proportional to the cube of the distance.

Field at Equatorial Points

In contrast, at an equatorial point, which is located on the perpendicular bisector of the dipole, the electric field is given by:

• Electric Field (Equatorial): \(E_{\text{equatorial}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{2r^3}\)

Notice that the electric field at the equatorial point is also dependent on the dipole moment and the distance, but it is only half the strength of the axial field at the same distance.

Why is the Axial Field Twice as Strong?

The reason the electric field at the axial point is twice that at the equatorial point can be understood through the geometry of the dipole and the superposition principle. At the axial point, the contributions to the electric field from both charges add constructively, while at the equatorial point, they partially cancel each other out.

Visualizing the Concept

Imagine standing at the axial point. The electric field vectors from both charges point in the same direction, enhancing the overall field strength. Conversely, at the equatorial point, the electric field vectors from the two charges point in opposite directions, leading to a reduction in the net electric field strength.

Summary of the Comparison

To summarize:

• The electric field at the axial point is given by \(E_{\text{axial}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{r^3}\).
• The electric field at the equatorial point is given by \(E_{\text{equatorial}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{2r^3}\).
• Thus, \(E_{\text{axial}} = 2 \times E_{\text{equatorial}}\).

This relationship highlights the unique behavior of electric fields generated by dipoles and is fundamental in understanding more complex electrostatic interactions. If you have any further questions or need clarification on any points, feel free to ask!