To solve the problem of finding the radius of the circular ring such that the maximum illuminance on the screen is at a point (2h) away from the center, we follow these key steps:

Key Observations & Formula

Illuminance (E) from a point source is proportional to (\frac{\cos\theta}{r^2}), where (\theta) is the angle between the light ray and the normal to the screen, and (r) is the distance from the source to the point on the screen. For a ring of radius (R), the illuminance at a point ((x,0,h)) on the screen is an integral over the ring.

The maximum illuminance occurs when (\frac{dE}{dx}=0) at (x=2h). This leads to an integral equation that simplifies to finding (R) such that the balance between the constant and cosine terms in the integral is zero.

Simplification & Result

Normalizing variables by (h) (let (R = t h)), the equation reduces to finding (t) where the integral condition holds. Through standard integral identities and simplification, the value of (t) is (\sqrt{5}).

Answer: (\boxed{\sqrt{5}h})

(\boxed{\sqrt{5}h})

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