Question

Problem 4. How many solid angle are subtended at
the center of a sphere of radius 5 cm by its surface of area
25 cm^2?
​

Answer 1

Explanation:

here's the answer............

Answer 2

Explanation:

Method 1: Solid angle subtended by any surface completely surrounding the center of a sphere is equal to the solid angle subtended by a sphere at its center

Ω=surface area of sphere(radius)2=4πr2r2=4π srΩ=surface area of sphere(radius)2=4πr2r2=4π sr

Method 2: Solid angle subtended by any object (2D or 3D) at any point in the space is given by following formula in spherical coordinates

Ω=∫∫sinθ dθ dϕΩ=∫∫sin⁡θ dθ dϕ

where, θθ & ϕϕ are polar & azimuth angles respectively

Applying proper limits for surface enclosing the center of sphere, we get

Ω=∫2π0∫π0sinθ dθ dϕΩ=∫02π∫0πsin⁡θ dθ dϕ

=∫2π0[−cosθ]π0 dϕ=∫02π[−cos⁡θ]0π dϕ

=∫2π0[1+1]π0 dϕ=∫02π[1+1]0π dϕ

=2∫2π0 dϕ=2∫02π dϕ

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