Electric Field and Charge Density

The electric field in a certain region of space is given by E(r) = Cr/ε₀a³, where a has a dimension of length and C is a constant. To determine the charge density, we need to understand the relationship between the electric field and charge.

Electric Field and Charge

The electric field (E) is a vector quantity that describes the force experienced by a charged particle in the presence of other charges. It is defined as the force per unit charge. Mathematically, it can be expressed as E = F/q, where F is the force experienced by the charge q.

Electric Field Due to a Point Charge

The electric field due to a point charge q at a distance r from it is given by Coulomb's Law: E = kq/r², where k is the electrostatic constant.

Electric Field from Continuous Charge Distribution

In the given scenario, the electric field E(r) is given as Cr/ε₀a³. This implies that the field is not generated by a single point charge, but rather by a continuous charge distribution.

Charge Density

The charge density (ρ) represents the amount of charge per unit volume. To determine the charge density in this scenario, we need to relate the electric field to the charge distribution.

Relation between Electric Field and Charge Density

The electric field generated by a continuous charge distribution can be expressed as an integral over the charge distribution. Mathematically, it can be written as:

E = ∫ (kρ)/r² dV,

where ρ is the charge density, r is the distance from the charge element, and dV represents a small volume element.

Applying the Given Electric Field

Comparing the given electric field E(r) = Cr/ε₀a³ with the general expression, we can equate the terms:

(kρ)/r² = Cr/ε₀a³.

Simplifying this expression, we find:

ρ = (Cε₀a)/k.

Therefore, the charge density in the given region of space is ρ = (Cε₀a)/k.

Conclusion

In conclusion, the charge density in the given region of space is ρ = (Cε₀a)/k. The electric field E(r) = Cr/ε₀a³ is generated by a continuous charge distribution, and the charge density represents the amount of charge per unit volume. By relating the electric field to the charge density, we can determine the expression for the charge density in terms of the given electric field.