QuestionJuly 19, 2025

The radius of our Sun is 6.96times 10^8m and its total power output is 3.85times 10^26W (a) Assuming the Sun's surface emits as a black body, calculate its surface temperature. __ K (b) Using the result of part (a), find lambda _(max) for the Sun. square nm

The radius of our Sun is 6.96times 10^8m and its total power output is 3.85times 10^26W (a) Assuming the Sun's surface emits as a black body, calculate its surface temperature. __ K (b) Using the result of part (a), find lambda _(max) for the Sun. square nm
The radius of our Sun is 6.96times 10^8m and its total power output is 3.85times 10^26W (a) Assuming the Sun's surface emits as a black body, calculate its surface temperature. __ K (b) Using the result of part (a), find lambda _(max) for the Sun. square nm

Solution
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Answer

(a) T \approx 5778 K ### (b) \lambda_{max} \approx 502 nm Explanation 1. Calculate the surface area of the Sun The surface area A of a sphere is given by **A = 4\pi r^2**. For the Sun, r = 6.96 \times 10^8 m. So, A = 4\pi (6.96 \times 10^8)^2. 2. Use Stefan-Boltzmann Law to find temperature The power output P of a black body is given by **P = \sigma A T^4**, where \sigma = 5.67 \times 10^{-8} W/m²K⁴. Rearrange to find T: T = \left(\frac{P}{\sigma A}\right)^{1/4}. Substitute P = 3.85 \times 10^{26} W and calculated A. 3. Calculate \lambda_{max} using Wien's Displacement Law Wien's Displacement Law is **\lambda_{max} = \frac{b}{T}**, where b = 2.898 \times 10^{-3} m K. Use the temperature from Step 2 to find \lambda_{max}.

Explanation

1. Calculate the surface area of the Sun<br /> The surface area $A$ of a sphere is given by **$A = 4\pi r^2$**. For the Sun, $r = 6.96 \times 10^8$ m. So, $A = 4\pi (6.96 \times 10^8)^2$.<br /><br />2. Use Stefan-Boltzmann Law to find temperature<br /> The power output $P$ of a black body is given by **$P = \sigma A T^4$**, where $\sigma = 5.67 \times 10^{-8}$ W/m²K⁴. Rearrange to find $T$: $T = \left(\frac{P}{\sigma A}\right)^{1/4}$. Substitute $P = 3.85 \times 10^{26}$ W and calculated $A$.<br /><br />3. Calculate $\lambda_{max}$ using Wien's Displacement Law<br /> Wien's Displacement Law is **$\lambda_{max} = \frac{b}{T}$**, where $b = 2.898 \times 10^{-3}$ m K. Use the temperature from Step 2 to find $\lambda_{max}$.
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