Question

If temperature of sun is decreased by \[1\% \] then the value of solar constant will change by
(A) \[2\% \]
(B) \[ - 4\% \]
(C) \[ - 2\% \]
(D) \[4\% \]

Answer 1

Hint Solar constant is proportional to fourth power of temperature so find the final solar constant in terms of initial solar constant. Substitute in the percentage increase formula to calculate the change in solar constant.

Complete step-by-step solution:
The solar constant is given by
$S = {T^4}\sigma {\left( {\dfrac{R}{r}} \right)^2}$
From, this we know that
$S \propto {T^4}$
So, let \[{S_1}\] and \[{S_2}\] be the initial and final state of the solar constant at temperatures \[{T_1}\] and \[{T_2}\] respectively.
\[{T_1} = {\text{ }}tK\]
\[{T_2} = {\text{ }}t - {\text{ }}1\% = 0.99{\text{ }}K\]
Using the temperature and solar constant relation,
\[
  \dfrac{{{S_1}}}{{{S_2}}} = \dfrac{{{T_1}^4}}{{{T_2}^4}} \\
  {S_2} = \dfrac{{{{(0.99t)}^4} \times {S_1}}}{{{t^4}}} \\
  {S_2} = 0.96{S_1} \\
 \]
Now, the percentage increase in the solar constant is given by,
$
  S = \dfrac{{{S_2} - {S_1}}}{{{S_1}}} \times 100 \\
  S = \dfrac{{0.96{S_1} - {S_1}}}{{{S_1}}} \times 100 \\
  S = - 0.0394 \times 100 \\
  S = - 3.94\% \simeq - 4\% \\
 $

Hence, the change in solar constant is by \[ - 4\% \] and the correct option is B.

Note

The solar constant depends on temperature and the surface area. The value of the constant is approximately equally \[1.366{\text{ }}kW{m^{ - 2}}\]. This constant increases by around \[0.2\% \] for each 11 year solar cycle.