Continuous Charge Distribution Class 12 | Formula, Derivation & Numericals

    Continuous charge distribution is a system where electric charge is spread over a line, surface, or volume instead of being concentrated at a single point, so it cannot be treated as discrete particles.

    Master Continuous Charge Distribution Class 12 Physics with simple explanations, formulas, derivations, and numericals. Learn linear, surface, and volume charge density easily and boost your exam score with this complete guide. 

Types of Continuous Charge Distribution

There are three types of continuous charge distribution -

Linear Charge Distriubution `(\lamda)`

    In this distribution, charges are distributed uniformly along a line, such as a wire or a ring.

Formula

        `\lamda = \frac {dq}{dl}`

Unit 

Coulomb per meter `(C/m)`.

Surface Charge Distribution `(\sigma)`

    In this distribution, charges are spread continuously over a two-dimensional surface area, such as a charged sheet or plate.

Formula

        `\sigma = \frac{dq}{ds}`

Unit

Coulomb per square meter `(\frac{C}{m^2})`

Volume Charge Distrubution `(\rho)`

    In this distribution, charges are uniformly distributed throughout a three-dimensional volume, such as within a charged sphere.

Formula

`\rho = \frac{dq}{dv}`

Unit

Coulomb per cubic meter `(\frac{C}{m^3})`

Electric Field due to a Continuous Charge of Distribution

    When electric charge is not concentrated at a single point but is continuously spread over a conductor or object, it is called a continuous charge distribution. In such cases, we cannot directly apply Coulomb's law for a point charge. Instead, we divide the charge into very small elements and calculate the total electric field using integration.

According to Coulomb's law:

The electric field due to a small charge element dq at a distance r is:

`d\vce{E} = \frac{1}{4 \pi \epsilon_0}\frac{dq}{r^2}\hat{r}`

    This field has a direction along the line joining the charge element and the observation point.

Total Electric Field

    The total electric due to the entire charge distribution is obtained by adding (integrating) all small contributions:

        `\vec{E} = \int d\vec{E}`

 `\vec{E} = \frac {1}{4 \pi \epsilon_0}\frac{dq}{r^2}\hat{r}`

Here, the value of dq depends on the type of charge distribution:

• For a linear charge distribution: `dq = \lamda dl`

• For surface charge distribution: `dq = \sigma dA`

• For volume charge distribution: `dq = \rho dV`