EBK AN INTRODUCTION TO MODERN ASTROPHYS
EBK AN INTRODUCTION TO MODERN ASTROPHYS
2nd Edition
ISBN: 9781108390248
Author: Carroll
Publisher: YUZU
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Chapter 6, Problem 6.1P
To determine

The proof to show that the solid angle subtended about P gives Ωtot=dΩ which is equal to 4π.

Expert Solution & Answer
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Answer to Problem 6.1P

The total solid angle is 4π.

Explanation of Solution

Write the expression for solid angle subtended at the centre due to a small area.

    dΩ=sinθdθ        (1)

Conclusion:

Integrate Equation (1) to get Ωtot

    Ωtot=dΩ=02π0πsinθdθdϕ=02πdϕ0πsinθdθ=[ϕ]02π[cosθ]0π

Solving further,

    Ωtot=(2π0)[(cosπcos0)]=(2π)(2)=4π

Thus, the total solid angle is 4π.

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