[Physics Class Notes] on Continuous Charge Distribution Pdf for Exam

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{klambda 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 dEint_{0}^{s}frac{ksigma 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 dEint_{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}{4pi epsilon _0}frac{pDelta 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}{4pi epsilon _0}sum_{allDelta v}frac{pDelta 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}{4pi epsilon _0}frac{pDelta v}{r^2}widehat{r’}Delta Eapprox frac{1}{4pi epsilon _0}sum frac{pDelta v}{r^2}widehat{r’}]