At a first glance, electric dipoles seem like one of the electrostatic phenomena we don't come across too often in our daily lives; however, electric dipoles actually appear more commonly than you might have first thought. Analysing the disturbance of the balance of positive and negative charges in an electric dipole and finding their polarity provides us with a better understanding of concepts such as contact forces, dielectrics and the behaviour of molecules. Electric dipoles are especially relevant when it comes to microscopic systems such as the dipoles found in molecules, which are far too close together to notice the effects of on a macroscopic scale, yet have very distinguishable charges. Before we dive into understanding what an electric dipole is and the closely related electric dipole moment, we will want to first establish some useful concepts like Coulomb's forces and dielectrics.
What is a conductor?
A conductor is a substance that facilitates the free flow of charged particles. Substances containing a relatively large number of free charge carriers are called conductors. Metals, for instance, are considered good electrical conductors. In metals, electrons can move freely as they’re not bound to the metal's crystal lattice and hence are free charge carriers. Let’s imagine a metal rod placed in an electric field equal to zero \((E=0)\). In such a case, electrons are uniformly distributed over the metal rod's surface. Now, if we place the same rod into an electric field not equal to zero, all of the free charge carriers will move to one end of the object caused by the electric force experienced by the charge carriers due to the Coulomb force.
The Coulomb force is the force which acts between two point charges causing them to attract or repel each other in a vacuum, and can be expressed as:
$$F=k\frac{\left|q_1\right|\left|q_2\right|}{r^2}.$$
The constant of proportionality \(k\) is known as Coulomb's constant:
$$k=\frac1{4\pi\varepsilon_0}=8.99\times10^9\;\mathrm{Nm}^2\mathrm C^{-2}$$
Coulomb's force is proportional to the product of the magnitude of two point charges (\(q_1\) and \(q_"\)) and inversely proportional to the distance \(r\) between the charges squared. In other words, as the distance between two point charges increases, the force between them decreases with the square of their separation. This phenomenon is called electrostatic induction and results in the separation of negative and positive charges in a conductor.
What is a dielectric?
More generally, a material that does not easily facilitate the flow of free charge carriers is called an electrical insulator.
Substances that have hardly any free charge carriers, but can be or become dielectrically polarised, are known as dielectrics. Familiar examples from everyday life include air, plastic, and alcohol. There are two categories of dielectrics: polar and non-polar. If we place a non-polar dielectric in an electric field equal to zero, the centers of the positive and negative charges coincide. If the same dielectric is placed in a non-zero electric field, the centers no longer coincide and therefore a dipole is formed.
In a polar dielectric, each molecule forms its own randomly oriented dipole due to thermal motion, without the need for an external electric field. If an external electric field is applied, the dipoles line up in the direction of the field. This process is known as the polarization of a dielectric.
Definition of an electric dipole
An electric dipole is a system that consists of two equally sized particles of opposite charges, separated by a small distance.
A diagram showcasing an electric dipole can be seen below.
Two particles of equal magnitude and opposite charge separated by small distance creating an electric dipole, Vaia Originals
Because the charges are of equal magnitude, the opposite signs cancel out, making the overall charge of the system zero. The midpoint between the two charges is known as the center of the dipole.
Electric dipole field
Let's see what would happen to an electric dipole placed in an electric field. As visible in the figure below, the electric field lines extend from the positive charge to the negative charge. They are directed radially outwards from the positive charge and radially inwards to the negative charge. As a result, they are also perpendicular to the surface of the point charges and therefore the equipotential lines.
Equipotential lines represent sections of constant electric potential and are always perpendicular to the electric field lines. If a charge was to be moved along either one of these lines, no work would be done.
Electric dipole placed in an electric field. The curved electric field lines expand from the positive charge to the negative charge. The equipartition lines are perpendicular to the point charges, Wikimedia Commons
If a dipole is created by a dielectric, it will have its own electric field, acting in the opposite direction of the external electric field. This interaction will result in an overall reduction of the electric field. Depending on the dielectric used, the extent of this reduction will differ. A parameter used to distinguish the level of change is known as dielectric permeability \(\epsilon\). Essentially, it describes the factor by which the dielectric will weaken the external electric field.
Dielectric molecules either already have or develop electric dipoles in the presence of an external electric field. For instance, in an electrically neutral water molecule (H2O), the arrangement of three atoms is such that the center of the positive charges of the atomic nuclei is not the same as the center of the negative charge of all the electrons of the water molecule. The total electric charge of the molecule is equal to zero, but it forms, as it were, a dumbbell with charges of opposite signs at the ends. Such a system of two charges, equal in magnitude but opposite in sign, is called an electric dipole.
Electric dipole moment
Every electric dipole comes with a dipole moment. The length of a dipole is the distance between its positive and negative charges. The electric dipole moment is represented as a vector directed from the negative charge of the dipole to the positive charge.
In chemistry, the vector representing a dipole moment has an opposite direction: directed from the positive charge to the negative charge.
Electric dipole moment allows us to describe the effect of a dipole at a certain distance, by telling us the strength of the electric field that will be produced.
Electric dipole moment formula
The equation used to calculate the electric dipole moment is:
$$\overset\rightharpoonup p=q\overset\rightharpoonup d$$
where \(p\) is the electric dipole moment in Coulomb-meters \((\mathrm{C m})\), \(q\) is either of the charges (positive or negative) in Coulombs \(\mathrm{C}\), and \(d\) is the displacement vector pointing from the negative charge to the positive charge in meters \((\mathrm{m})\). We can use either of the charges because the electric dipole moment represents the overall polarity of the system. In other words, it depends only on the magnitude of the point charge rather than the sign. Let's apply the equation mentioned above to an example problem to test our knowledge.
Two point charges \(+6\;\mathrm{C}\) and \(-6\;\mathrm{C}\) are separated by a distance of \(10\;\mathrm{cm}\). What is the electric dipole moment of this system?
As discussed earlier, the charges in an electric dipole are of equal magnitude and opposite sign, so in this case for the value of \(q\) we use \(6\;\mathrm{C}\). Now we simply plug this value and the distance in SI units \((0.1\;\\mathrm{m})\) into the equation for the electric dipole moment as follows.
$$\overset\rightharpoonup p=q\overset\rightharpoonup d=6\;\mathrm{C}\times0.1\;\mathrm{m} = 0.6\;\mathrm{Cm}$$
The magnitude of this dipole moment is \(0.6\;\mathrm{Cm}\), and the direction is from the negative charge \((-6\;\mathrm{C})\) to the positive charge \((+6\;\mathrm{C})\).
Careful, linear momentum and electric dipole moment have the same symbol \(p\), yet represent entirely different concepts!
Electric dipole potential
Before understanding what the electric dipole potential is, we might want to recall what the electric potential is in general. The equation used to calculate the electric potential of a system consisting of a point charge \(q\) at a distance \(r\) is $$V=k\frac qr,$$ where \(k\) is the same Coulomb's constant as discussed earlier.
We can use this relation to find the electric dipole potential of two charged particles at a point P. Visually, it can be represented as the diagram visible below.
Schematic representation of finding the electric dipole potential at a point P between two point charges, Vaia Originals
To find the electric dipole potential, we must consider each individual charge and sum them together. $$V=V_++V_-=k\left[\frac q{r_+}+\frac{(-q)}{r_-}\right]$$ $$V=\frac q{4\pi\varepsilon_0}\left[\frac1{r_+}-\frac1{r_-}\right]=\frac q{4\pi\varepsilon_0}\left[\frac{r_--r_+}{r_+r_-}\right].$$ Assuming that the arbitrary point P is located really far from the dipole, the values of \(r_+\) and \(r_-\) are set equal. By connecting the center of the dipole and the point P, we obtain the position vector \(r\). The cosine law can be used to derive the expression for \(r_{-\;}-\;r_+\), yielding the following equation: $$V=\frac q{4\pi\varepsilon_0}\left[\frac{d\cos\theta}{r^2}\right].$$ Now we can rewrite the product of \(q\) and \(d\) as \(p\) and as (electric dipole moment), based on the definition mentioned earlier, and obtain the final expression for the electric dipole potential at point P: $$V=\frac1{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}.$$
Electric Dipole - Key takeaways
- Substances that have hardly any free charge carriers are known as dielectrics.
- When a non-polar dielectric is placed in a non-zero electric field, an electric dipole is formed.
- An electric dipole is a system that consists of two equally sized particles of opposite charges, separated by a small distance.
- If an electric dipole is placed in an electric field, the curved electric field lines expand from the positive charge to the negative charge.
- The equation used to calculate the electric dipole moment is \(p=qd\).
- The electric dipole moment is a vector quantity directed from the negative charge of the dipole to the positive.
- To find the electric dipole potential at a certain point, each individual charge must be considered and summed together using the general equation for electric potential \(V=k\frac{q}{r}.\)
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