Electric Dipole and Its Electric Field

Electric Dipole

Definition and Structure

An electric dipole consists of two equal and opposite charges (+q and -q) separated by a fixed distance d. The dipole moment p is defined as:

• Dipole Moment: $\mathbf{p} = q \mathbf{d}$, where q is the magnitude of each charge and d is the vector pointing from the negative to the positive charge.

• Physical Example: Many molecules, such as water, possess a permanent dipole moment.

• Applications: Dipoles are fundamental in understanding molecular interactions, electric fields in materials, and biological systems.

Electric Field of a Dipole: Perpendicular Bisector

The electric field produced by a dipole at a point on the perpendicular bisector (a distance L from the center) can be calculated using superposition.

• Key Steps:

  1. Calculate the field from the positive charge (+q).

  2. Calculate the field from the negative charge (-q).

  3. Add the two fields vectorially.

• Symmetry: The y-components of the fields cancel due to symmetry, leaving only the x-component.

• Resulting Field: $E = \frac{p}{4\pi\epsilon_0 r^3} \hat{i}$, where r is the distance from the dipole center to point P, and p = qd.

• Direction: The field points opposite to the dipole direction on the perpendicular bisector.

Electric Field of a Dipole: Along the Axis

For points located on the axis of the dipole (y-axis), at distances much greater than the dipole size (r ≫ d), the field is derived using the binomial expansion.

• Superposition Principle: Calculate the field from each charge and sum.

• Binomial Expansion: Used to approximate the field for r ≫ d.

• Resulting Field: $E = \frac{p}{2\pi\epsilon_0 r^3} \hat{j}$

• Comparison: The field along the axis is twice as strong as the field at the same distance on the perpendicular bisector.

Summary Table: Electric Field of a Dipole

Location	Electric Field Expression	Direction
Perpendicular Bisector	$E = \frac{p}{4\pi\epsilon_0 r^3}$	Opposite to dipole moment
Along Axis (r ≫ d)	$E = \frac{p}{2\pi\epsilon_0 r^3}$	Same as dipole moment

Physical and Biological Applications

• Biological Example: The African knifefish generates its own electric field using a dipole-like mechanism, allowing it to sense its environment in murky waters.

• Mechanism: The fish charges its tail and head like a dipole, creating a sensory map.

• Relevance: This demonstrates the practical importance of dipole fields in nature.

Key Equations and Constants

• Dipole Moment: $p = qd$

• Electric Field (Perpendicular Bisector): $E = \frac{p}{4\pi\epsilon_0 r^3}$

• Electric Field (Axis): $E = \frac{p}{2\pi\epsilon_0 r^3}$

• Coulomb's Constant: $k = \frac{1}{4\pi\epsilon_0}$

Additional info:

• The binomial expansion is used to approximate the electric field for points far from the dipole, where $\frac{d}{2r} \ll 1$.

• These equations are valid only for points much farther from the dipole than the separation of the charges.