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[itex][/itex]

The intensity of the Sun's radiation just outside the Earth's atmosphere is approximately [itex] 8 \cdot 10^4 \frac {joules}{m^2 \cdot min} [/itex]. Echo II is a spherical shell of radius [itex] r_0 = [/itex] 20.4m. Its skin consists of a layer of Mylar plastic ... between two layers of aluminum. ... Aluminum's reflection coefficient is .81.

Approximately what force does [the Sun's] radiation exert on the Echo II reflecting satellite balloon?

(From experiment explained earlier in the chapter)

[tex]c = \frac {W(1 + ρ)}{F}[/tex]

(Solved for force)

[tex]F = \frac {W(1 + ρ)}{c}[/tex]

Where W is the rate of arrival of energy, F is the rate of change of momentum (force), ρ is the reflection coefficient, and c is the speed of light in a vacuum.

This seemed like a flux problem, so

[tex]\iint_D{F(r(\varphi,\vartheta)) \cdot (r_\varphi \times r_\vartheta) dA}[/tex]

Where

[tex] r(\varphi,\vartheta) = r_o sin \varphi cos \vartheta \textbf{i} + r_o sin \varphi sin \vartheta \textbf{j} + r_o cos \varphi \textbf{k} [/tex]

[tex]F = -\frac{W(1 + ρ)}{c}\textbf{k}[/tex]

[tex]r_\varphi \times r_\vartheta = r_o^2 sin^2 \varphi cos \vartheta \textbf{i} + r_o^2 sin^2 \varphi sin \vartheta \textbf{j} + r_o^2 sin \varphi cos \varphi \textbf{k}[/tex]

so

[tex]F(r(\varphi,\vartheta)) \cdot (r_\varphi \times r_\vartheta) = -\frac{W (1 + ρ)}{c} r_o^2 sin \varphi cos \varphi \textbf{k}[/tex]

[tex]-\frac{W (1 + ρ)}{c} r_0^2 \int_0^{2\pi}{\int_0^\frac{\pi}{2}{sin \varphi cos \varphi d\varphi d\vartheta}}[/tex]

Did I set something up wrong?

## Homework Statement

The intensity of the Sun's radiation just outside the Earth's atmosphere is approximately [itex] 8 \cdot 10^4 \frac {joules}{m^2 \cdot min} [/itex]. Echo II is a spherical shell of radius [itex] r_0 = [/itex] 20.4m. Its skin consists of a layer of Mylar plastic ... between two layers of aluminum. ... Aluminum's reflection coefficient is .81.

Approximately what force does [the Sun's] radiation exert on the Echo II reflecting satellite balloon?

## Homework Equations

(From experiment explained earlier in the chapter)

[tex]c = \frac {W(1 + ρ)}{F}[/tex]

(Solved for force)

[tex]F = \frac {W(1 + ρ)}{c}[/tex]

Where W is the rate of arrival of energy, F is the rate of change of momentum (force), ρ is the reflection coefficient, and c is the speed of light in a vacuum.

## The Attempt at a Solution

This seemed like a flux problem, so

[tex]\iint_D{F(r(\varphi,\vartheta)) \cdot (r_\varphi \times r_\vartheta) dA}[/tex]

Where

[tex] r(\varphi,\vartheta) = r_o sin \varphi cos \vartheta \textbf{i} + r_o sin \varphi sin \vartheta \textbf{j} + r_o cos \varphi \textbf{k} [/tex]

[tex]F = -\frac{W(1 + ρ)}{c}\textbf{k}[/tex]

[tex]r_\varphi \times r_\vartheta = r_o^2 sin^2 \varphi cos \vartheta \textbf{i} + r_o^2 sin^2 \varphi sin \vartheta \textbf{j} + r_o^2 sin \varphi cos \varphi \textbf{k}[/tex]

so

[tex]F(r(\varphi,\vartheta)) \cdot (r_\varphi \times r_\vartheta) = -\frac{W (1 + ρ)}{c} r_o^2 sin \varphi cos \varphi \textbf{k}[/tex]

[tex]-\frac{W (1 + ρ)}{c} r_0^2 \int_0^{2\pi}{\int_0^\frac{\pi}{2}{sin \varphi cos \varphi d\varphi d\vartheta}}[/tex]

**my problem :**I get .0105 N when the back of the book solution is .00576 NDid I set something up wrong?

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