(a) Continuous charge distribution.
Point charges do not exist in real practice and are only imaginary charges. In practice we always deal with charged bodies having definite length, area or volume which possesses a continuous distribution of charges.
Types. There are three main types of charge distribution:
- Linear charge distribution
- Surface charge distribution and
- Volume charge distribution.
1. Linear charge distribution. When the charge is distributed uniformly along a line, and the distribution is one dimensional, it is called linear charge distribution.
If λ (r') be the line density of charge i.e. charge per unit length at position r' and dl be a small length element about that position, then charge on the element becomes λ (r') dl [Fig.(a)].
The total charge is given by ∫L λ (r') dl. The expression is called line integral of linear charge distribution.
Expression for force
Let point charge q lie at a point on position vector \(\vec r\).
∴ Force on charge q due to linear charge distribution is
\(\vec F=\frac{1}{4\piɛ_o}q∫ λ (r') dl.\ \frac{(\vec r-\vec r')}{|\vec r- \vec r'|^3}\)
2. Surface charge distribution. When the charge is distributed over a surface and the distribution is two-dimensional, it is called surface charge distribution.
If σ(r') be the surface density of charge i.e., charge per unit area at position r' and ds be a small area element about that position, then charge on the element becomes σ(r') ds [Fig.(b)].
The total charge is given by ∫s δ (r')ds. The expression is called surface integral of superficial charge distribution.
Expression for Force. Let point charge q lie at a point on position r.
∴ Force on charge q due to surface charge distribution is
\(\vec F=\frac{1}{4\piɛ_o}q'∫_s\sigma(r') ds.\ \frac{(\vec r-\vec r')}{|\vec r- \vec r'|^3}\)
3. Volume charge distribution. When charge is continuously distributed over a volume and distribution is three-dimensional, it is called volume charge distribution.
If ρ(r') be the volume density of charge i.e. charge per unit volume at position r' and dV be small volume element about that position, then charge on the element becomes ρ(r') dV [Fig.(c)]
The total charge is given by ∫v P (r') d V. The expression is called volume integral of volume charge distrinution.
Expression for force. Let point charge q lie on position r.
∴ Force on charge q due to volume distribution is
\(\vec F=\frac{1}{4\piɛ_o}q∫_vP(r') dv.\ \frac{(\vec r-\vec r')}{|\vec r- \vec r'|^3}\)
(b) Total force on a point charge. Let point charge q' lie at a point in position r'. Let λ (r), σ(r') and ρ(r') be the line charge density, the surface charge density and the volume charge density respectively at same position r'.
Let point charge q, lie at a point on position r.
Then Force on charge, q due to point charge q'
Total force on charge q, at position \(\vec r\)becomes
