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The differential surface element on a sphere of radius \( a \) is

  • A.\( a^2 \sin\theta \, da \, d\theta \)
  • B.\( a^2 \sin\theta \, d\theta \, d\phi \)
  • C.\( a \sin\theta \, d\theta \, d\phi \)
  • D.\( a^2 \, d\theta \, d\phi \)

Answer: B

The two arc lengths on the surface are \( a \, d\theta \) and \( a \sin\theta \, d\phi \), and their product is the element. Integrating it over the full angular ranges gives \( 4\pi a^2 \), the surface area of the sphere.

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