Electric Field at a Distance R/2 from the Centre of a Metallic Sphere
To determine the electric field at a distance of R/2 from the centre of the metallic sphere, we can use the principle of Gauss's law.

Gauss's Law:
Gauss's law states that the electric flux through a closed surface is proportional to the total charge enclosed by that surface. Mathematically, it can be written as Φ = q/ε₀, where Φ is the electric flux, q is the charge enclosed, and ε₀ is the permittivity of free space.

Electric Field at 2R from the Centre:
Given that the electric field at 2R from the centre of the sphere is E, we can use Gauss's law to determine the total charge enclosed by a sphere of radius 2R. The electric field at a distance of 2R from the centre is given by E = kQ/2R², where k is the Coulomb constant.

Determination of Electric Field at R/2:
To find the electric field at a distance of R/2 from the centre, we need to consider a Gaussian surface of radius R/2. By Gauss's law, the electric flux through this surface will be proportional to the charge enclosed within it. As the sphere is non-conducting, the charge is uniformly distributed. Hence, the charge enclosed within the Gaussian surface is proportional to the volume of the sphere of radius R/2.

Calculation of Electric Field:
Since the charge enclosed within the Gaussian surface is proportional to the volume of the sphere of radius R/2, the electric field at a distance of R/2 from the centre can be determined using the formula E = kQ/(R/2)². Substituting the value of Q in terms of the charge enclosed within the sphere of radius R/2, we can calculate the electric field at the desired distance.
Therefore, by applying Gauss's law and considering the distribution of charge within the metallic sphere, we can determine the electric field at a distance of R/2 from the centre.