Discussion
The flux of \( \mathbf{F} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) out of a sphere of radius \( a \) centred at the origin is
Answer: B
On the sphere the field is radial with magnitude \( a \), and the outward normal is radial too, so \( \mathbf{F} \cdot \mathbf{n} = a \) everywhere:
\[ \Phi \;=\; a \times 4\pi a^{2} \;=\; 4\pi a^{3}. \]
The divergence theorem confirms it, since \( \nabla \cdot \mathbf{F} = 3 \) and \( 3 \times \tfrac{4}{3}\pi a^{3} = 4\pi a^{3} \)
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