How much more light can the James Webb Space Telescope (with its 6-m diameter mirror) gather than the Hubble Space Telescope (with a diameter of 2-m) ? How much more light will a 15-inch telescope collect than a 3-inch telescope?

a)   I₁ / I₂ = 9,  b)    I₁ / I₂ = 25

Explanation:

a) The amount of light that can pass through a telescope is the intensity of the light

         I = P / A

where A is the area of ​​the mirror and P is the incident power that is the same in all telescopes

       P = I₁ A₁ = I₂ A₂

where subscript 1 is for the 2 m diameter telescope and subscript 2 for the 6 m diameter telescope

The mirror area is

        A = π r²

         r = d / 2

we substitute

        I₁ π d₁² /4 = I₂ π d₂² ​​/ 4

        I₁ / I₂ = (d₂ / d₁) ²

we calculate

        I₁ / I₂ = (6/2) ²

        I₁ / I₂ = 9

b) d₂ ​​= 15 inch and d₁ = 3 inc

let's calculate

          I₁ / I₂ = (15/3) ²

          I₁ / I₂ = 25