How to Calculate Electric Field Due to Continuous Charge Distributions
The continuous load distribution system is a system in which the charge is uniformly distributed over the conductor. For a continuous charging device, the infinite number of charges is closely packed and there is no space between them. Unlike the discrete charging system, the continuous load distribution in the conductor is uninterrupted and continuous. There are 3 types of continuous charge distribution system -
Linear Charge Distribution
Surface Charge Distribution
Volume Charge Distribution
Linear Charge Density
If the charge is not evenly distributed over the length of the conductor, it is called linear charge distribution. It is often referred to as linear charge density and is denoted by the Lambda (λ ) symbol. Mathematically, there is a linear charge density -
λ = dq/ dl
The unit of the linear load density is C / m. If we find a conductor with a length of 'L' with a surface load density of λ and take an aspect of dl on it, then a small charge will be on it.
dq = λ dl
So, there will be an electrical field on the small charge element dq.
dE = kdq/r2
dE = kλdl / /r2
In order to calculate the net electrical field, we will integrate both sides with the correct limit, i.e.
\[\int dE=\int_{0}^{l}\frac{k\lambda dl}{r^2}\]
\[\int dE=\frac{k}{r^2}\int_{0}^{l}\lambda dl\]
Surface Charge Density
When the charge is uniformly distributed over the conductor surface, it is called Surface Charge Density or Surface Charge Distribution. It is denoted by the symbol σ (sigma) and the unit is C / m2. It is also defined as a charge/per area of the unit. Mathematically the density of the surface charge is
σ = dq / ds
Where dq is a small element of charge over a small ds surface. So, there will be a small charge on the driver -
dq = σ ds
The electric field due to a small charge at a distance of 'r' can be evaluated as
dE = kdq/r2
dE = kσdl / /r2
Integrating both sides with proper limits we get
\[\int dE\int_{0}^{s}\frac{k\sigma ds}{r^2}\]
\[\int dE=\frac{k}{r^2}\int_{0}^{s}\sigma ds\]
Volume Charge Density
When the charge is distributed over a driver's volume, it is called Volume Charge Distribution. It is denoted by the ρ (rho) symbol. In other words, the charge per unit volume is called Volume Charge Density and its unit is called C / m3. Mathematically, the density of the volume charge is
ρ = dq/ dv
where dq is a small charge element located in a small volume dv. To find the total charge, we must integrate dq with the correct limits. The electric field due to dq will be
dq = ρ dv
dE = kdq / r2
dE = kρdv / /r2
Integrating both sides with proper limits we get
\[\int dE\int_{0}^{v}\frac{vpdv}{r^2}\]
\[\int dE=\frac{k}{r^2}\int_{0}^{v}pdv\]
Calculation of Electric Field
Let us consider the case of the continuous distribution of charges in the body. Here, because of this charge, we'll measure the electrical field at point P. We can say that the charge density of different volumetric elements can be different so that we divide the body into different elements so that the charge density of a particular element can be considered to be a fixed quantity. Consider one of those elements of volume Δv, the charge density of which is given by ρ. Let the distance between the volume element and point P be given as r. The charge in the volume element could be given as ρΔv. As per the law of Coulomb, the electrical field can be given as ρΔv due to the charge.
\[\Delta E=\frac{1}{4\pi \epsilon _0}\frac{p\Delta v}{r^2}\overbrace{r}\]
Here, r is the distance between the charged part and the point P at which the field is to be measured, and ř is the unit vector in the direction of the electrical field from the charge to point P. By the principle of superposition, the electrical field can be given as a result of the total distribution of the charge in the body, which is divided into a number of such volume elements.
\[E=\frac{1}{4\pi \epsilon _0}\sum_{all\Delta v}\frac{p\Delta v}{r^2}\overbrace{r}\]
Types of Charge Distribution
Electrical field due to the above volume charge distribution (for 1 and n volume elements) will be as follows -
\[\Delta E=\frac{1}{4\pi \epsilon _0}\frac{p\Delta v}{r^2}\widehat{r'}\Delta E\approx \frac{1}{4\pi \epsilon _0}\sum \frac{p\Delta v}{r^2}\widehat{r'}\]
FAQs on Continuous Charge Distribution Explained: Concepts & Types
1. What is meant by a continuous charge distribution in the context of Class 12 Physics?
A continuous charge distribution refers to a system where charge is spread continuously over a line, surface, or volume, without any gaps. On a macroscopic level, we treat the charge as smoothly distributed, even though it is made up of discrete individual charges (like electrons) at the microscopic level. This model simplifies the calculation of electric fields for charged objects like rods, sheets, or spheres, which is a key concept in the CBSE Class 12 syllabus for 2025-26.
2. What are the three main types of continuous charge distribution?
The three types of continuous charge distribution are defined by the geometry over which the charge is spread:
- Linear Charge Distribution: The charge is distributed uniformly along a one-dimensional line or curve, like a charged wire.
- Surface Charge Distribution: The charge is spread over a two-dimensional surface, such as the surface of a thin conducting sheet or a hollow sphere.
- Volume Charge Distribution: The charge is distributed throughout a three-dimensional volume, for example, within a non-conducting charged sphere.
3. What are the formulas and SI units for the different types of charge densities?
Each type of charge distribution is characterised by its density:
- For linear charge density (λ), the formula is λ = dq/dl, representing charge per unit length. Its SI unit is Coulombs per meter (C/m).
- For surface charge density (σ), the formula is σ = dq/dS, representing charge per unit area. Its SI unit is Coulombs per square meter (C/m²).
- For volume charge density (ρ), the formula is ρ = dq/dV, representing charge per unit volume. Its SI unit is Coulombs per cubic meter (C/m³).
4. Why do we use the concept of a continuous distribution when electric charge is fundamentally quantised?
We use the continuous charge distribution model as a practical approximation. While charge is indeed quantised (exists in discrete multiples of the elementary charge, e), the number of charges on any macroscopic object is enormous. The separation between these individual charges is so small compared to the overall dimensions of the object that treating the distribution as continuous greatly simplifies calculations using calculus (integration) instead of summing up trillions of individual fields. It's a highly accurate model for problems at a macroscopic scale.
5. How is the electric field from a continuous charge distribution calculated?
The electric field from a continuous charge distribution is calculated by applying the principle of superposition along with integration. The process involves:
- Dividing the entire charge distribution into infinitesimal elements (dq).
- Calculating the small electric field (dE) produced by this element 'dq' at the desired point using Coulomb's Law.
- Integrating the electric field contributions (dE) from all such elements over the entire distribution (line, surface, or volume) to find the net electric field E.
This method effectively sums up the vector contributions from every part of the charged body.
6. What is the difference between a uniform and a non-uniform charge distribution?
The key difference lies in how the charge density varies:
- In a uniform charge distribution, the charge density (λ, σ, or ρ) is constant at all points throughout the distribution. For example, a perfectly conducting sphere isolated in space will have a uniform surface charge density.
- In a non-uniform charge distribution, the charge density varies from point to point. An example is a conducting object with an irregular shape, where charge tends to accumulate more at sharper points, leading to a higher charge density in those regions.
7. How does the shape of a conductor influence its surface charge distribution?
The geometry of a conductor significantly impacts its surface charge distribution. For a conductor in electrostatic equilibrium, charges rearrange themselves to ensure the electric field inside is zero. On an isolated, perfectly spherical conductor, the charge distributes uniformly. However, on an irregularly shaped conductor, the charge density becomes non-uniform. The charge density is highest at points with the smallest radius of curvature (i.e., the sharpest points or edges) and lowest on flatter portions. This phenomenon is why electric discharge often occurs from sharp points.
