Continuous Charge Distribution: Definition, Types, and Electric Field Calculations

A continuous charge distribution explains how charges are distributed across a surface. Charge can be continuously distributed over a line, volume, or surface. According to this concept, these charges are too densely packed at the atomic level. At the human scale, they appear as continuous and smooth. 

In reality, when dealing with wires and plates, physicists do not really use the quantisation property, which is in multiples of the elementary charge value of 1.6 x 10^19C. This quantisation at a large scale is not feasible for calculations of electric fields when there are way too many charges to even count. 

The solution, or the need for continuous distribution of charges Class 12, comes here. It works like an accurate assumption, where we can simply apply calculus and integration to find electric fields (and electric potentials) with a large number of point charges.   

Example to ponder: The surface of a spherical conductor or a charged metal rod displays a continuous charge distribution as charges are close to its surface. 

You learnt before about forces between multiple charges, where vector summing rules like the triangle (or parallelogram) method apply. Till here, the conceptual basis was still about a discrete charge distribution. The charges are individual points, separated at certain distances. And we’re calculating the electrostatic force on a single charge from other charges, with Coulomb’s Law along with the superposition principle.
The case is different and not as direct as for continuous charge distribution calculation. See the difference table below. 

Basis	Discrete Charge Distribution	Continuous Charge Distribution
Definition	Charges exist as separate points.	Charges are spread smoothly.
Representation	Finite number of point charges (q_1, q_2, q_3...)	Charge element (dq) used for integration.
Calculation Method	Simple vector addition.	Requires calculus and integration.
Example	Two charged spheres.	A charged wire or sheet.

Just remember, a continuous charge is a limit of an infinite number of small point charges. 

Continuous charge distribution exists in three main forms, measured in terms of charge density - a related but important concept. What you need to remember at the Class 12 Physics level is that charge density tells us how much electric charge there is per length, area, or volume, when there is a continuous distribution. 

Look into the dimensionality of the distribution to explain types. 

Linear Charge Distribution 

Charges going along a line, which could be a wire, tells us about this type of continuous charge distribution. 

Remember, it’s a one-dimensional surface where the charges are spread out. We use lambda (λ) to describe the linear charge distribution.

The general formula for linear charge density is λ = Q/L, where Q is the total charge and L is the length of the line in one dimension. But since we are talking about a very small part of the line, the formula for linear charge distribution would be: 

λ = dq/dl

The unit of Linear charge distribution would be Coulomb per metre or C/m. You might want to refer to the SI unit of electric charge to see how Coulomb is derived from Ampere, for additional clarity.  

Surface Charge Distribution

We go for surface charge distribution to learn and calculate how charges are spread across a surface, or better put, an area (A or s), which is usually a two-dimensional one. The sigma 𝜎 symbol is used here. A sheet is the most common example of it. (Can you think of something different?)

Same as the surface charge density formula, which is 𝜎 = Q/s, for surface charge distribution, we’ll use 

 𝜎 = dq/ds

As area is 2D, the unit of the surface charge distribution would be Coulomb per metre-squared, or C/m^2. 

Volume Charge Distribution

Charges can spread over the volume, much like on a three-dimensional object.

rho or ρ is how we denote, so equation for this would be

ρ = dq/dv

Coulomb per cubic metre or C/m^3 is the unit. 

The types mentioned above will be useful when we need to find the total electric field for a large number of point charges.

We know how to use the vector of E to define the electric field that a point charge creates.

E = (1/4πε_0) x (q/r^2) x r

Differentiating

dE = (1/4πε_0) x (dq/r^2) x r

Using integral

E = (1/4πε_0)  ∫(dq/r^2) x r

Now, if you have to find across the different types,

• A line, we’d go for dq=λdl
• On 2d, it’s dq=σdA
• For area, dq=ρdV

Then we simply need these to get these equations. 

If it’s linear, we have E = (1/4πε_0)  ∫(λdl/r^2) x r

For a surface it is E = (1/4πε_0)  ∫(σdA/r^2) x r

In case of volume, we have E = (1/4πε_0)  ∫(ρdV/r^2) x r

Also, we can use the Coulomb constant (k) for (1/4πε_0) in vacuum. 

 

If preparing for Class 12 CBSE exams, check these below pages that are fully aligned with the recent NCERT Textbooks for Physics, Chemistry, and Maths. 

Complete CBSE Class 12 Study Material
NCERT Class 12 Maths Solutions	NCERT Physics Class 12 Solutions	Class 12 Chemistry NCERT Solutions
Class 12 Maths NCERT Notes	Physics Class 12 NCERT Notes	Class 12 Chemistry NCERT Notes
CBSE Sample papers for class 12 Maths	CBSE Class 12 Physics Sample Papers	CBSE Class 12 Sample Paper for Chemistry
CBSE Class 12 Maths Previous Year Question Papers	CBSE Class 12 Physics Previous Year Question Papers	CBSE Chemistry Class 12 Question Papers