Find the flux of F⃗ = (x³, y³, z³) across the surface of the sphere x² + y² + z² = a².
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Step-by-Step Solution
Step 1: Identify the vector field and surface
We are given the vector field F=(x3,y3,z3) and the surface of a sphere defined by the equation x2+y2+z2=a2. We need to find the flux of F across this closed surface.
Step 2: Apply the Divergence Theorem
Since the surface is a closed sphere, we can use the Divergence Theorem (also known as Gauss's Theorem) to simplify the calculation. The theorem states that the flux of a vector field across a closed surface is equal to the triple integral of the divergence of the field over the volume enclosed by the surface.
Step 3: Calculate the divergence of the vector field
First, we need to compute the divergence of the vector field F. The divergence is the sum of the partial derivatives of each component with respect to its corresponding coordinate. This gives us 3x2+3y2+3z2, which can be factored as 3(x2+y2+z2).
Step 4: Substitute divergence into the integral
Now we substitute the calculated divergence into the volume integral from the Divergence Theorem. The volume V is the interior of the sphere x2+y2+z2=a2.
Step 5: Convert to spherical coordinates
To evaluate this triple integral over a sphere, it is most convenient to convert to spherical coordinates. In spherical coordinates, x2+y2+z2=r2 and the volume element dV=r2sinϕdrdϕdθ. The limits of integration for a sphere of radius a are r from 0 to a, ϕ from 0 to π, and θ from 0 to 2π.
Step 6: Evaluate the integrals
Now we evaluate each of the separated integrals. The integral with respect to θ gives 2π. The integral with respect to ϕ gives 2. The integral with respect to r gives 5a5. Multiplying these results together with the constant 3 gives the total flux.