Continuous Charge Distribution

At the atomic level, charge exists in discrete units (multiples of e = 1.6 × 10⁻¹⁹ C) — this is called quantisation of charge. However, at the macroscopic scale (a charged rod, metal plate, or sphere), the number of charge carriers is so enormously large that treating each charge individually is impractical.​

Just as we describe water as a continuous fluid rather than counting individual molecules, we treat large assemblies of charges as a continuous distribution.​

Condition for continuous treatment: When charges are so closely and uniformly packed that there is no measurable gap between them on the scale of observation, the distribution is called a continuous charge distribution

The charge per unit area on a surface, is called surface charge density.

The charge per unit length along a line (such as a wire), is called linear charge density.

OR

When charge is distributed along a line, the charge distribution is called linear charge distribution.

The charge per unit volume in a region of space, is called volume charge density.

OR

When charge is distributed over the volume of an object, it is called volume charge distribution.

A charge distribution in which charge is treated as continuously spread over a line, surface, or volume (ignoring microscopic discreteness), is called continuous charge distribution.

When charge is distributed along a line, the charge distribution is called a linear charge distribution.

OR

The linear charge density λ is the charge per unit length at any point on the line.

When charge is distributed over a surface, the charge distribution is called surface charge distribution.

OR

The surface charge density σσ is the charge per unit area at any point on the surface.

When charge is distributed over the volume of an object, it is called volume charge distribution.

OR

The volume charge density ρρ is the charge per unit volume at any point inside the body.

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\sum\frac{\rho\Delta V}{r^{\prime2}}\hat{r}^{\prime}\]

λ = \[\frac {ΔQ}{Δl}\] ⇒ dq = λdl

where ΔQ is the charge distributed over a small length Δl of the wire.

• SI Unit: C m⁻¹ (coulomb per metre)
• Nature: Scalar quantity

σ = \[\frac {ΔQ}{ΔS}\] ⇒ dq = σ dS

where ΔQ is the charge distributed over a small surface area ΔS.

• SI Unit: C m⁻² (coulomb per square metre)
• Nature: Scalar quantity

ρ = \[\frac {ΔQ}{ΔV}\] ⇒ dq = ρ dV

where ΔQ is the charge distributed over a small volume ΔV of the material.

• SI Unit: C m⁻³ (coulomb per cubic metre)
• Nature: Scalar quantity

Principle: Extension of Superposition

The electric field due to a continuous distribution is derived by extending the Principle of Superposition. Since the distribution consists of infinitely many small charge elements dq, the net field is the vector sum (integral) of fields due to all elements.

For a small charge element dqdq located at position \[\vec{r}^{\prime}\], the field at observation point P (position \[\vec r\]) is:​

\[d\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{dq}{r^{\prime2}}\hat{r}^{\prime}\]

where \[\hat r\]′ is the unit vector from dq to point P, and r′ is the distance between them.

Total Electric Field (Integral Form)

Integrating over the entire distribution:

\[\vec E\] = \[\frac{1}{4\pi\varepsilon_{0}}\int\frac{dq}{r^{\prime2}}\hat{r}^{\prime}\]

Substituting the appropriate charge element: