A sphere of radius \(R\) has a volume density of charge \(\rho=k r,\) where \(r\) is the distance from the centre of the sphere and \(k\) is constant. The magnitude of the electric field which exists at the surface of the sphere is given by \(\left(\varepsilon_{0}=\right.\) permittivity of free space)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.