Divergence Theorem — Question 8

Question 8

For a constant $a$, let $$\mathbf F_a(x,y,z)=\langle ax,2y,3z\rangle.$$ The measured outward flux through the unit sphere is $24\pi$. Determine $a$.

Tasks

1. Express the divergence in terms of $a$.

2. Use the flux measurement and the unit-ball volume.

3. Verify the recovered value directly on the sphere.

Question 8 – Solution

Strategy. The divergence is constant, so the measured flux gives a linear equation for the unknown coefficient.

Step 1: Apply the theorem $$\nabla\cdot\mathbf F_a=a+2+3=a+5.$$ The unit ball has volume $4\pi/3$, so $$24\pi=(a+5)\frac{4\pi}{3}.$$ Therefore $a+5=18$, and $$\boxed{a=13}.$$

Step 2: Direct verification On the unit sphere, $\mathbf n=\langle x,y,z\rangle$, so with $a=13$, $$\mathbf F_{13}\cdot\mathbf n=13x^2+2y^2+3z^2.$$ By spherical symmetry, $$\iint_Sx^2\,dS=\iint_Sy^2\,dS=\iint_Sz^2\,dS=\frac{4\pi}{3}.$$ Hence the direct flux is $$(13+2+3)\frac{4\pi}{3}=24\pi.$$

Verification The recovered parameter reproduces the stated measurement by both methods.