Electric Dipole
Claimed By: Yashwin Thammiraju, Spring 2026
Summary
An electric dipole is made up of two point charges that have equal but opposite electric charges (q) and are separated by a short distance (d).
The electric field of a dipole is inversely proportional to the cube of the distance from the dipole ($1/r^3$), unlike a single point charge which falls off by $1/r^2$. The field's magnitude and direction are highly dependent on whether you are observing it along the parallel axis (the line separating the two charges) or the perpendicular axis (the bisector).
A temporary dipole can be created when you place a neutral atom in an external electric field. Due to the movement of the electron cloud relative to the nucleus, the atom polarizes (shifting negative charge to one side and positive charge to the other), yielding a separation of charge.
Electric dipoles are characterized by their dipole moment ($\vec{p}$), a vector quantity measuring the strength and separation of the positive and negative electrical charges within a system. For two point charges, $+q$ and $-q$, separated by a distance $d$, the magnitude of the dipole moment is:
[math]\displaystyle{ p = qd }[/math]
A prime example of a permanent dipole in nature is the water molecule ($H_2O$), which forms a 105-degree angle between the two hydrogen atoms connected to the oxygen. Because oxygen has a greater electronegativity, it pulls more strongly on the shared electrons. Consequently, the oxygen end of the molecule becomes more negatively charged compared to the hydrogen end, and the net electric dipole moment points towards the oxygen atom.
Computational Model
To better visualize the electric field generated by an electric dipole, we can use a computational model. The GlowScript simulation below calculates the exact superposition of the point charges and displays the resulting electric field vectors at various observation locations.
<html><iframe src="https://trinket.io/embed/glowscript/31d0f9ad9e" width="100%" height="400" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe></html>
Mathematical Models
An Exact Model
An electric dipole is constructed from two point charges: one at position [math]\displaystyle{ [\frac{d}{2}, 0] }[/math] and one at position [math]\displaystyle{ [\frac{-d}{2}, 0] }[/math]. These point charges have equal and opposite charge magnitudes. We wish to calculate the exact electric field due to the dipole at some observation point [math]\displaystyle{ P }[/math] in the plane (see the figure). Point [math]\displaystyle{ P }[/math] can be defined by its coordinates [math]\displaystyle{ [p_x, p_y] }[/math] from the midpoint of the dipole, or by a distance [math]\displaystyle{ r }[/math] and an angle [math]\displaystyle{ \theta }[/math].
Using the superposition principle, the net electric field at [math]\displaystyle{ P }[/math] is [math]\displaystyle{ E_{net} = E_{q_+} + E_{q_-} }[/math]. We can break this down into x and y components:
- [math]\displaystyle{ E_{net_x} = E_{q_{+x}} + E_{q_{-x}} }[/math]
- [math]\displaystyle{ E_{net_y} = E_{q_{+y}} + E_{q_{-y}} }[/math]
Let [math]\displaystyle{ \theta_+ }[/math] be the angle from [math]\displaystyle{ q_{+} }[/math] to [math]\displaystyle{ P }[/math]. The y-component of the positive charge's field is [math]\displaystyle{ E_{q_{+y}} = E_{q_+} \sin(\theta_+) }[/math].
To find [math]\displaystyle{ \theta_+ }[/math] and its counterpart [math]\displaystyle{ \theta_- }[/math], we look at the geometry. [math]\displaystyle{ \theta_+ }[/math] belongs to a right triangle with an opposite side length of [math]\displaystyle{ p_y }[/math] and an adjacent side length of [math]\displaystyle{ p_x - \frac{d}{2} }[/math]. Therefore: [math]\displaystyle{ \sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}} }[/math]
The denominator here represents the hypotenuse, which is simply the distance [math]\displaystyle{ |\vec r_+| }[/math] from the positive charge to the observation point. A similar geometric analysis for the negative charge gives: [math]\displaystyle{ \sin(\theta_-) = \frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}} }[/math] where the denominator is [math]\displaystyle{ |\vec r_-| }[/math].
The general formula for the magnitude of an electric field from a point charge is [math]\displaystyle{ |E| = \frac{1}{4\pi\epsilon_0} \frac{q}{|\vec r|^2} }[/math]. Applying this to both charges: [math]\displaystyle{ E_{net_y} = \frac{1}{4\pi\epsilon_0} \frac{q_+}{|\vec r_+|^2} \sin(\theta_+) + \frac{1}{4\pi\epsilon_0} \frac{q_-}{|\vec r_-|^2} \sin(\theta_-) }[/math]
Noting that [math]\displaystyle{ q_+ = -q_- }[/math], we can factor out the charge: [math]\displaystyle{ E_{net_y} = \frac{q_+}{4\pi\epsilon_0} \Bigg(\frac{1}{|\vec r_+|^2}\sin(\theta_+) - \frac{1}{|\vec r_-|^2}\sin(\theta_-)\Bigg) }[/math]
Substituting our expanded radii and sines into the equation yields: [math]\displaystyle{ E_{net_y} = \frac{q_+}{4\pi\epsilon_0} \Bigg(\frac{1}{(p_x - \frac{d}{2})^2 +p_y^2 } \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}} - \frac{1}{(p_x + \frac{d}{2})^2 +p_y^2 } \frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}} \Bigg) }[/math]
Combining the denominators simplifies this to the exact analytical form for the y-component: [math]\displaystyle{ E_{net_y} = \frac{q_+}{4\pi\epsilon_0} \Bigg(\frac{p_y}{\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2} } - \frac{p_y}{\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2} } \Bigg) }[/math]
The derivation for the x-direction follows the exact same logic, using cosine (adjacent over hypotenuse) instead of sine. The result is: [math]\displaystyle{ E_{net_x} = \frac{q_+}{4\pi\epsilon_0} \Bigg(\frac{p_x - \frac{d}{2}}{\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2} } - \frac{p_x + \frac{d}{2}}{\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2} } \Bigg) }[/math]
These formulae provide the exact electric field due to an electric dipole anywhere on the 2D plane.
Special Cases (Approximations)
When the observation distance is much greater than the separation distance ($r \gg d$), we can simplify the exact models into the standard dipole approximations. Let [math]\displaystyle{ a = \frac{d}{2} }[/math].
On the Parallel Axis
On the axis running through the two charges, [math]\displaystyle{ p_y = 0 }[/math], meaning [math]\displaystyle{ E_{net_y} = 0 }[/math]. Plugging [math]\displaystyle{ p_y = 0 }[/math] into our exact [math]\displaystyle{ E_{net_x} }[/math] formula:
<math>E_{net_x} = \frac{q_+}{4\pi\epsilon_0} \Bigg(\frac{1}{(p_x - a)^2 } - \frac{1}{(p_x + a)^2