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Chapter 14: Problem 7

Suppose div \(\mathbf{F}=0\) in a region enclosed by two concentric spheres. What is the relationship between the outward fluxes across the two spheres?

Short Answer

Expert verified
Answer: The outward fluxes across the two concentric spheres are equal when the divergence of the vector field F is 0 in the region enclosed by the spheres.

Step by step solution

01

Apply the Divergence theorem

The Divergence theorem says that the outward flux of a vector field F through a closed surface S is equal to the triple integral of the divergence of F, div F, over the volume V enclosed by S. Mathematically, the Divergence theorem is given by: \(\iint_S \mathbf{F} \cdot d\mathbf{A} = \iiint_V (\nabla \cdot \mathbf{F})dV\) We need to apply this theorem to both the spheres and find the relationship between the outward fluxes across them.
02

Calculate the outward flux for the inner sphere

Let's label the inner sphere as sphere 1 with surface S1 and the outer sphere as sphere 2 with surface S2. Applying the Divergence theorem to sphere 1, we have: \(\iint_{S1} \mathbf{F} \cdot d\mathbf{A} = \iiint_{V1} (\nabla \cdot \mathbf{F})dV\) Since div F = 0 in the volume enclosed by the two concentric spheres, the integral of div F over the volume enclosed by sphere 1 (V1) will be 0. Thus, the outward flux across sphere 1 is 0.
03

Calculate the outward flux for the outer sphere

Now, let's apply the Divergence theorem to sphere 2: \(\iint_{S2} \mathbf{F} \cdot d\mathbf{A} = \iiint_{V2} (\nabla \cdot \mathbf{F})dV\) Here V2 is the volume enclosed by sphere 2 and outside sphere 1. Since div F = 0 in the volume enclosed by the two concentric spheres, the integral of div F over the volume V2 will also be 0. Thus, the outward flux across sphere 2 is 0.
04

State the relationship between the outward fluxes across the two spheres

Since the outward flux across both sphere 1 and sphere 2 is 0, their relationship can be expressed as: Flux across sphere 1 = Flux across sphere 2 This means that the outward fluxes across the two concentric spheres are equal when the divergence of the vector field F is 0 in the region enclosed by the spheres.

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Most popular questions from this chapter

Consider the radial vector fields \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle\) Let \(C\) be any circle in the \(x y\) -plane centered at the origin. a. Evaluate a line integral to show that the field has zero circulation on \(C\) b. For what values of \(p\) does Stokes' Theorem apply? For those values of \(p,\) use the surface integral in Stokes' Theorem to show that the field has zero circulation on \(C\).

Prove the following identities. Assume that \(\varphi\) is \(a\) differentiable scalar-valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \times(\varphi \mathbf{F})=(\nabla \varphi \times \mathbf{F})+(\varphi \nabla \times \mathbf{F}) \quad \text { (Product Rule) }$$

Let \(S\) be the paraboloid \(z=a\left(1-x^{2}-y^{2}\right),\) for \(z \geq 0,\) where \(a>0\) is a real number. Let \(\mathbf{F}=\langle x-y, y+z, z-x\rangle .\) For what value(s) of \(a\) (if any) does \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) have its maximum value?

Prove that for a real number \(p,\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla\left(\frac{1}{|\mathbf{r}|^{p}}\right)=\frac{-p \mathbf{r}}{|\mathbf{r}|^{p+2}}.\)

Suppose that a surface \(S\) is defined as \(z=g(x, y)\) on a region \(R\). Show that \(\mathbf{t}_{x} \times \mathbf{t}_{y}=\left\langle-z_{x},-z_{y}, 1\right\rangle\) and that \(\iint_{S} f(x, y, z) d S=\iint_{R} f(x, y, z) \sqrt{z_{x}^{2}+z_{y}^{2}+1} d A\).
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