Solid angle of a complete sphere: The total solid angle subtended at the center equals

Introduction / Context:Solid angle generalizes planar angle to three dimensions and is essential in radiometry, photometry, and sensor field-of-view calculations. Knowing the solid angle of a complete sphere provides a useful normalization for directional quantities like radiance and irradiance integration.

Given Data / Assumptions:

• Solid angle unit is the steradian (sr).
• A full sphere surrounds a point in all directions.
• Geometric relationships between surface area on a unit sphere and steradians apply.

Concept / Approach:By definition, the solid angle Ω corresponding to an area A on a unit-radius sphere is Ω = A (since A = r^2 * Ω and r = 1). The surface area of a unit sphere is 4π, hence the solid angle for the entire sphere is 4π sr. Hemispheres subtend 2π sr, quarter-spheres π sr, and so on.

Step-by-Step Solution:Recall: surface area of unit sphere = 4π.Use Ω = A (for r = 1) ⇒ full sphere Ω = 4π sr.Cross-check: hemisphere area = 2π ⇒ hemisphere Ω = 2π sr.Select 4π steradians as the correct total solid angle.

Verification / Alternative check:Integration of differential solid angle dΩ = sinθ dθ dφ over θ = 0..π and φ = 0..2π yields ∫∫ sinθ dθ dφ = 4π, confirming the result.

Why Other Options Are Wrong:2π sr: Hemisphere only.3π sr and 6π sr: No standard spherical fraction corresponds to these totals.8π sr: Would imply double-counting both sides of each direction.

Common Pitfalls:Confusing planar radians with steradians; forgetting that steradian corresponds to area on the unit sphere.

Final Answer:4π steradians.