Continuous Charge Distribution

Earlier, discrete charges such as q1, q2, …, qn were considered, where the mathematical treatment is simple and does not require calculus. However, in many real-world situations, charges are not discrete but are distributed continuously over an object.

In a continuous charge distribution, charges are closely packed with very little space between them and are distributed over a length, surface, or volume, so calculus is required to study their effects.

Charges can be distributed in three ways, including

Linear charge distribution.
Surface charge distribution.
Volume charge distribution.

Linear Charge Distribution

When charges are uniformly distributed along a length, such as along a straight wire or around the circumference of a circle, it is called a linear charge distribution. It is represented by the symbol λ (lambda). The linear charge density λ is defined as

\lambda=\frac{\Delta{Q}}{\Delta{l}}

where Δl is a small line element of the wire that still contains many microscopic charged particles, and ΔQ is the amount of charge present in that line element. The SI unit of linear charge density is C/m (coulomb per meter).

Surface Charge Distribution

When charge is distributed over the surface of a conductor, it is called surface charge distribution. Instead of considering individual charged particles, a small surface area element ΔS\Delta SΔS is taken on the conductor, which contains many microscopic charges. The amount of charge on this element is ΔQ\Delta QΔQ. The surface charge density is represented by σ (sigma) and is defined as

\sigma=\frac{\Delta{Q}}{\Delta{S}}

Surface charge density is a continuous function that represents the average distribution of charge over a small surface area. The SI unit of surface charge density is C/m² (coulomb per square meter).

Volume Charge Distribution

When a charge is uniformly distributed throughout a volume, it is called a volume charge distribution. Examples include a charge distributed inside a sphere or a cylinder. The volume charge density is represented by ρ (rho) and is defined as

\rho=\frac{\Delta{Q}}{\Delta{V}}

where ΔQ is the charge present in a small volume element ΔV, which is macroscopically small but contains many microscopic charged particles. The SI unit of volume charge density is C/m³ (coulomb per cubic meter).

Electric Field Calculation

The electric field produced by a distribution of charge can be determined in a way similar to that used for a system of discrete charges. Consider a charge distribution in space with charge density ρ, which may vary from point to point. Choose a suitable origin O, and let r be the position vector of a point within the charge distribution. The entire distribution can be divided into very small volume elements of size ΔV, each containing a charge ΔQ = ρΔV.

Now consider a general point P, either inside or outside the charge distribution, with position vector R. The electric field produced at point P by a small charge element ρΔV is given by Coulomb’s law:

\Delta{E}=\frac{1}{4\pi\epsilon_0}\frac{\rho\Delta{V}}{r'^2}\hat{r'}

where r′ is the distance between the charge element and the point P, and \hat{r'} is the unit vector directed from the charge element to P.

The total electric field due to the entire charge distribution is obtained by applying the principle of superposition, which means adding the contributions of electric fields from all the small volume elements:

\Delta{E}\cong\frac{1}{4\pi\epsilon_0}\sum_{all\text{ }\Delta{V} }\frac{\rho\Delta{V}}{r'^2}\hat{r'}

Sample Problems

Question 1: A wire of length 10 m carries a total charge of 50 C uniformly distributed along its length. Find the linear charge density (λ) and the charge present in a small segment of length 0.5 m.

Solution: \lambda = \frac{Q}{L}
\lambda = \frac{50}{10} = 5\,C/m
Charge in 0.5 m segment
\Delta Q = \lambda \Delta l
\Delta Q = 5 \times 0.5 = 2.5\,C

Question 2: A circular disc of radius 2 m has a uniform surface charge density of 4 C/m². Find the total charge on the disc.

Solution: Area of a disc
A = \pi r^2
A = \pi (2)^2 = 4\pi
Total charge
Q = \sigma A
Q = 4 \times 4\pi = 16\pi\,C

Question 3: A solid sphere of radius 3 m contains a total charge of 36π C uniformly distributed throughout its volume. Find the volume charge density (ρ).

Solution: Volume of sphere
V = \frac{4}{3}\pi r^3
V = \frac{4}{3}\pi (3)^3 = 36\pi
Charge density
\rho = \frac{Q}{V}
\rho = \frac{36\pi}{36\pi} = 1\,C/m^3

Question 4: A circular annulus has an inner radius of 1 m and an outer radius of 3 m. If the surface charge density is 2 C/m², find the total charge on the annulus.

Solution: Area of annulus
A = \pi (R^2 - r^2)
A = \pi (3^2 - 1^2)
A = \pi (9 - 1) = 8\pi
Total charge
Q = \sigma A
Q = 2 \times 8\pi = 16\pi\,C

Unsolved Problems

Question 1: A straight wire of length 15 m carries a total charge of 60 C uniformly distributed along its length. Find the linear charge density (λ) of the wire.

Question 2: A circular disc of radius 2 m has a uniform surface charge density of 5 C/m². Determine the total charge present on the disc.

Question 3: A solid sphere of radius 3 m carries a uniform volume charge density of 2 C/m³. Calculate the total charge inside the sphere.

Question 4: A rectangular metal sheet of dimensions 5 m × 4 m has a surface charge density of 3 C/m². Find the total charge on the sheet.

Question 5: A cylindrical rod of radius 2 m and height 4 m has a uniform volume charge density of 1.5 C/m³. Determine the total charge contained in the cylinder.