Charge Distribution

Before explaining the topic of charge distribution, let us first understand the two main types of distributions we come across. For example, we have 20 candles and arrange them in the room during a festival. We can count the number of candles in our collection. This type of distribution of candles in the room is known as discrete distribution.

It seems like the room is still missing something. Let's bring some colored powder into the picture to add some flair. Now we start spreading the primary colored powder and making patterns. Can we count each particle in the powder? Not practically, right? This is because of the very large (effectively infinite) number of color particles in the powder. Because there are so many powder particles and because they can vary in color. We can mix the colored particles to produce a near-infinite selection of different color combinations depending on the relative concentration of each primary colored powder. Every color combination can be made this way, including colors that seem to lie somewhere between the primary colors. Because there are no gaps in the spectrum of colors that we can create this way, we call this distribution a continuous distribution.

Fig. 1 - A continuous distribution of rangoli colors can be formed when the sands are smeared and mixed together.

Previously we have studied how the individual charge behaves in an electric field. What if we have a large amount of charge uniformly distributed on the surface of a conductor? Is there an analogy we can draw between the colored powder and the infinite number of ways of mixing the colors? Like the colored powder, the charge comes in tiny discrete packets. There are several different fundamental particles that are charged, including electrons, protons, etc., but all these particles are relatively small compared to the macroscopic scale on which we operate from day to day. These charged particles are so tiny that if we take a whole ensemble of such particles, we can create a near-infinite variety of charge distributions by arranging the charges in different ways. Like colored powder, the surface of a conductor consists of discrete, countable electrons, but there are so many electrons. They are so small that they end up being practically uncountable and appear on our macroscopic scale to behave like a free-flowing fluid that can take on any shape and more or less occupy any volume (ignoring electromagnetic forces of attraction and repulsion), which means we can treat them like a continuous distribution.

Charge Distribution Meaning

We know that the mobility of charge carriers inside conductors is higher than in insulators. When some excess charge is provided to the conductor, the electrostatic force of repulsion between them causes them to start moving away from each other as much as possible. Also, they occupy as much space on the conductor's surface as possible to minimize this force of repulsion.

From figure 2, we can see that there is a uniform distribution of charge on the surface of the conductor. This distribution is known as a surface charge distribution.

Charge Distributions Examples and Types

Three main types of charge distribution are studied, which differ in geometry:

1. Linear charge distribution

2. Surface charge distribution

3. Volume charge distribution

Each type of charge distribution has a different charge density depending on the number of spatial dimensions in the situation. A brief description and explanation of each type are given below.

Linear Charge Distribution

When a charge is uniformly distributed along a one-dimensional length of material, then this type of arrangement of charge is known as a linear charge distribution. The charge density in such cases is known as linear charge density \(\left(\lambda\right)\). Electrically conductive wires are often modeled as linear charge distributions due to their approximately one-dimensional nature.

The SI unit of charge is the Coulomb \(\left(\text{C}\right)\), and the SI unit of length is the meter \(\left(\text{m}\right)\). Using these units, the SI unit of linear charge density is the Coulomb per meter \(\mathrm{C\,m^{-1}}\).

Surface Charge Distribution

When the charge is uniformly distributed on the surface of the conductor, then this type of distribution is known as a surface charge distribution. The charge density, in this case, is known as surface charge density \(\left(\sigma\right)\).

Let the surface area of a conductor be \(S\) and the net charge on the conductor, distributed across the surface of the conductor, be \(Q\).

Fig. 4 - The figure shows the surface charge distribution of the positive charges on the spherical conductor.

The surface charge density for this case is \(\sigma=\frac{Q}{S}\).

The SI unit of the surface area is the meter squared \(\left(\mathrm{m^{2}}\right)\), and the SI unit of charge is the Coulomb \(\left(\text{C}\right)\). By using these units, the SI unit of surface charge density is \(\mathrm{C\,m^{-2}}\).

Volume Charge Distribution

When the charge is uniformly distributed throughout the material in question, this type of distribution is known as volume charge distribution. The charge density, in this case, is known as volume charge density \(\left(\mathrm{\rho}\right)\), or charge density.

The SI unit of volume is a meter cube \(\left(\mathrm{m^{3}}\right)\) and the SI unit of charge is Coulomb \(\left(\text{C}\right)\). By using these units, the SI unit of the volume charge density is \(\mathrm{C\,m^{-3}}\).

Charge Distribution in Conductors and Insulators

In insulators, electrons are firmly bound to the nucleus. Therefore, a lot of energy is required to free electrons to increase their mobility inside insulators. On the other hand, electrons inside conductors are free to move as they are only loosely bound to the nucleus. Thus, the charge carriers' mobility is more significant in conductors than in insulators.

Now think about what will happen if we supply some charge to a conductor and to an insulator.

In the case of Conductors

The charge carriers in a conductor can move quickly due to high mobility. Unbounded by a nucleus and under the electrostatic forces of repulsion between each electron, the charges spread as far apart as possible, distributing themselves across the surface of the conductor.

In the case of Insulators

The charge carriers in insulators cannot move quickly due to the strong electrostatic force of attraction between the nuclei of the atoms and their electrons. In other words, the mobility of the charge carriers (electrons) is low in insulators. Therefore, even under the action of the electrostatic repulsive force, they continue to remain bound to the nuclei of the atoms, and there is no re-distribution of charge that takes place in the case of insulators.

Charge Distribution during Different Charging Methods

There are three main charging methods that you should be aware of:

1. Charging by friction

2. Charging by induction

3. Charging by Conduction

Charging by Friction

Insulators can become charged by the method of friction. For example, when two insulators rub against each other, electrons are transferred from one insulator to another, resulting in a charge redistribution between the insulators. As a result, one insulator becomes positively charged, and the other becomes negatively charged.

Fig. 6 - Charging of an insulator using a charging-by-friction method.

The charge on each insulator does not become re-distributed across the insulator material like in a conductor. So even though there is an electrostatic repulsion between the charges, the mobility of charge carriers is still low, meaning the electrons aren't free to move.

Charging by Induction

When a positively charged metal rod is brought near an uncharged metal sphere, an equal and opposite charge is induced on the side of the initially uncharged metal sphere facing the rod due to the electrostatic force of attraction between the net positive charge in the rod and the electrons in the metal sphere. Likewise, due to electrostatic repulsion, a net positive charge forms on the opposite side of the initially uncharged sphere. Thus, the presence of a charge can turn an uncharged conductor into a polarized conductor.

By grounding the other face of the metal sphere, the positive charge is transferred to the ground.

Fig. 7 - The charging of conduction with a charged rod using a charging-by-induction method.

By ungrounding the sphere, there is a negative charge left on it, which redistributes itself such that the charge spreads around the surface of the conductor.

Fig. 8 - The figure shows the charge distribution in the conductor after charging by induction method.

The redistribution of the charges takes place in the conductor due to the electrostatic force of repulsion. However, this redistribution is only possible when the charged rod is removed because the electrostatic force of attraction between the positive charge in the rod and the negative charge of the electrons in the sphere keeps the electrons attracted to one side of the sphere. Hence, the sphere is still polarised until the rod is removed.

Hence, from the above explanation, we can say that the charge distribution is affected by the electrostatic force (or the electric field).

Charging by Conduction

When a charged metal sphere is brought into contact with an uncharged but identical metal sphere, the charge is transferred from one sphere to another. After the transfer of charge, the net charge on each conductor is the same.

To let the charge re-distribute itself around the surface, we have to move the spheres, as an electrostatic force of repulsion acts between the charges on either sphere due to the net-like charge on each sphere.

Fig. 9 - The figure shows the charge re-distribution in the conductor after charging by the conduction method.

The above diagram shows that the presence of electric charge affects the distribution of charges. We can also say that the electric field/electrostatic force affects the distribution of charges.