Charge Density explained in simple words | Linear, Surface and Volume

# Types of Charge Densities: Line, Surface & Volume

Charge density is the charge distributed per unit length, surface or volume. If charge is distributed over the body then there may be three possible distributions i.e. over the line, over the surface and within the volume. This arises the term charge density in respective distribution.

## What is a Point Charge?

Theoretically, we represent any charge by the point assuming that the complete charge is concentrated at that point. This is called as the point charge. Off course, the point charge is way to present the charge on the paper theoretically. Practically, any charge has to be deposited on certain body giving rise to charge distribution.

Do you know? Coulomb’s Law is defined for the point charges!

## Charge Density in different Charge Distributions

No charge can be point charge except the electron. Given charge is deposited on certain body and hence the charge is distributed over that body. The nature of this distribution of the charge depends on various factors like the dimensions, nature, type etc.

If the given charge is distributed along the length of the body then such kind of distribution is called as the Line Charge. In this case we define the term line or linear charge density which is the total charge per unit length.

If the given charge is distributed over the surface of the body then such kind of distribution is called as the Surface Charge. In this case we define the term surface charge density which is the total charge per unit area.

Finally, if the given charge is distributed within the volume of the body then such kind of distribution is called as the Volume Charge. In this case we define the term volume charge density which is total charge per unit volume.

#### Linear Charge Density

Let Q is the charge deposited uniformly over the line of length L. Then linear charge density is defined as follows:

$\rho_L=\frac QL$

Obviously, its unit is C/m.

For example, if total 10 C of charge is deposited over the length of 2 m then its charge density will be (10/2) = 5 C/m.

Conversely, if we are given with ρL then we can fine the total charge present on the given length. For example, if the line charge density is 5 C/m and we are supposed to find the total charge contained by the 10 m of line; then it is quite simple as below:

$\rho_L=\frac QL\;\;\;\;\rightarrow\;\;\;\;\;\;Q=\rho_L\times L$

$\therefore Q=5\times10=50C$

Now, the above cases are very simple. You can easily observe that ρis constant so leading to very simple calculations. But this ρmay be a function and in such cases you cannot find the total charge merely by multiplying the ρL with the length. Here we have to use the concept of differential and the total value of the function is calculated by the integration.

For example, let’s say we have a line charge parallel to X axis having charge density $\rho_L=0.1x\;C/m$. And we are supposed to find the total charge contained by this line from x = 1 to x = 5.

In this case we have to find the total charge using the concepts of integration. So first of all we need to find the infinitesimal charge considering infinitesimal length of the line charge.

$\rho_L=\frac{charge}{length}\Rightarrow Charge=\rho_L\times Length$

So, for the infinitesimal length, say ‘dl’, the infinitesimal charge ‘dq’ is given as,

$dq=\rho_Ldl$

For our case, $dq=0.1xdl$

Now, this infinitesimal charge, differential element are collected over the required length using the proper limits of the integration.

$Q=\int\limits_Ldq=\int\limits_L\rho_Ldl$

Now as I have mentioned in the Line integration, the differential ‘dl’ should be properly written according to the nature of the line along with the limits.

In our case the given line is parallel to the X axis, so ‘dl’ will be ‘dx’ with limits of x from 1 to 5.

$\therefore Q=\int_{x=1}^50.1xdx=0.1\left[\frac{x^2}2\right]_1^5=1.2C$

So concluding all,

1. The line charge distribution posses the line charge density ρL
2. ρcan be constant or function.
3. The total charge can be calculated using, $Q=\int_L\rho_Ldl$
4. The differential and the limits for the integration are written accordingly to the concepts of the line integration.

Suggested Read: What is Line Integration in Electromagnetics?

#### Surface charge density

Let Q is the charge deposited uniformly over the surface of area A. Then surface charge density is defined as follows:

$\rho_S=\frac QA$

Obviously, its unit is C/m2.

As discussed above, the surface charge density can be a constant or a function. If it is a function then the total charge over the required surface must be calculated using the concepts of the surface integration.

$Q=\int\limits_S\rho_sds$

The surface integration is the double integration. The differential ‘ds’ and the limits for the double integration must be properly written according to the concepts of the surface integration.

Suggested Read: What is Surface Integration in Electromagnetics?

#### Volume Charge Density

Let Q is the charge deposited uniformly within the volume V. Say the sphere filled with charge or medium having free charge etc. Then volume charge density is defined as follows:

$\rho_V=\frac QV$

Obviously, its unit is C/m3.

As discussed above, the volume charge density can be a constant or a function. If it is a function then the total charge within the required volume must be calculated using the concepts of the volume integration.

$Q=\int\limits_V\rho_vdv$

The volume integration is the triple integration. The differential ‘dv’ and the limits for the triple integration must be properly written according to the concepts of the volume integration.

Suggested Read: What is Volume Integration in Electromagnetics?

Suggested Community: Electromagnetics for GATE & ESE

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