The electric field in certain region is acting radially outward and is given by E=Ar.Acharge contained inasphere of radius `a` centered at the origin of the field,will be given by:-

The electric field you've described, which acts radially outward and is given by \( E = Ar \), suggests a scenario where the electric field strength increases linearly with distance from the origin. To determine the charge contained within a sphere of radius \( a \) centered at the origin, we can apply Gauss's Law, which relates the electric field to the charge enclosed by a surface. Let's break this down step by step.

Understanding Gauss's Law

Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, it can be expressed as:

Φ = ∮ E · dA = Q_enc / ε₀

Where:

• Φ is the electric flux through the surface.
• E is the electric field.
• dA is the differential area vector on the surface.
• Q_enc is the total charge enclosed within the surface.
• ε₀ is the permittivity of free space.

Applying Gauss's Law to the Given Electric Field

In your case, the electric field \( E \) is given as \( E = Ar \), where \( A \) is a constant and \( r \) is the radial distance from the origin. To find the charge enclosed within a sphere of radius \( a \), we will calculate the electric flux through the surface of that sphere.

Calculating the Electric Flux

The surface area \( A \) of a sphere is given by:

A = 4πa²

Since the electric field is radial and uniform over the surface of the sphere, we can simplify the flux calculation:

Φ = E · A = (Ar) · (4πa²)

Substituting \( r \) with \( a \) (the radius of the sphere), we have:

Φ = A(a) · (4πa²) = 4πAa³

Finding the Enclosed Charge

Now, we can use Gauss's Law to find the enclosed charge \( Q_enc \):

Q_enc = ε₀ · Φ

Substituting the expression for flux we found earlier:

Q_enc = ε₀ · (4πAa³)

Final Expression for Charge

Thus, the total charge contained within the sphere of radius \( a \) centered at the origin is:

Q_enc = 4πε₀Aa³

This result shows how the charge enclosed within the sphere depends on the constant \( A \) and the cube of the radius \( a \). The linear relationship of the electric field with distance leads to a cubic relationship for the charge, illustrating how electric fields can influence charge distributions in space.