To determine the ratio of the wavelength of the radiation to the present value, we can use Wien's displacement law, which relates the temperature of a black body (like the Sun) to the peak wavelength of its emitted radiation.
Wien's displacement law states that the peak wavelength (λ) is inversely proportional to the temperature (T). Mathematically, it can be expressed as:
λ ∝ 1 / T
Let's denote the present temperature of the Sun as T1 and the doubled radius of the Sun as R2. The new temperature, T2, is given as half of T1, and we need to find the ratio of the wavelength (λ2) to the present value (λ1).
Using Wien's displacement law, we can write:
λ2 / λ1 = T1 / T2
Substituting the values, we have:
λ2 / λ1 = T1 / (T1 / 2) λ2 / λ1 = 2
Therefore, the ratio of the wavelength of the radiation to the present value would be 2:1.