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Chapter 14: Problem 5

What is the net outward flux of the radial field \(\mathbf{F}=\langle x, y, z\rangle\) across the sphere of radius 2 centered at the origin?

Short Answer

Expert verified
Answer: The net outward flux of the radial field across the sphere is \(16\pi\).

Step by step solution

01

Find the divergence of the vector field

To compute the divergence of the vector field \(\mathbf{F}=\langle x, y, z\rangle\), we need to find the partial derivatives of each component with respect to the corresponding variable, then add them together: $$ \nabla\cdot\mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1+1+1 = 3. $$The divergence is a scalar and its value is 3.
02

Understanding the Divergence Theorem

The Divergence Theorem states that the net outward flux of a vector field across a closed surface \(S\) is equal to the triple integral of the divergence of the field over the volume enclosed by the surface \(S\): $$ \oiint_S \mathbf{F}\cdot\hat{n} \, dS = \iiint_V (\nabla\cdot\mathbf{F}) \, dV, $$where \(\hat{n}\) is the unit normal vector to the surface \(S\). Since we know the divergence of \(\mathbf{F}\) and the surface is a sphere, we can apply the Divergence Theorem to find the net outward flux across the sphere.
03

Calculate the volume integral

Now convert the surface integral into a volume integral using the divergence of the field. The volume \(V\) is enclosed by the sphere of radius 2 centered at the origin: $$ \iiint_V (\nabla\cdot\mathbf{F}) \, dV = \int_{-\text{radius}}^{\text{radius}} \int_{-\text{radius}}^{\text{radius}} \int_{-\text{radius}}^{\text{radius}} 3\, dx\, dy\, dz. $$Using the spherical coordinate system with \(r\), \(\theta\), and \(\phi\), the volume integral becomes: $$ \iiint_V (\nabla\cdot\mathbf{F}) \, dV = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{2} 3 \cdot r^2 \cdot \sin{\phi}\, dr\, d\phi\, d\theta. $$
04

Solve the triple integral

Calculate the net outward flux by solving the triple integral: $$ \begin{aligned} \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{2} 3r^2\sin{\phi} \, dr\, d\phi\, d\theta &= 3\int_{0}^{2\pi} \, d\theta \int_{0}^{\pi} \sin{\phi}\, d\phi \int_{0}^{2} r^2 \, dr \\ &= 3\cdot (2\pi) \cdot [-\cos{\phi}\Big|_{0}^{\pi}] \cdot [\dfrac{1}{3}r^3\Big|_{0}^{2}] \\ &= 3\cdot(2\pi)\cdot(2)\cdot(\dfrac{8}{3}) \\ &= 16\pi. \end{aligned} $$Thus, the net outward flux of the radial field \(\mathbf{F}=\langle x, y, z\rangle\) across the sphere of radius 2 centered at the origin is \(16\pi\).

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Most popular questions from this chapter

Consider the rotational velocity field \(\mathbf{v}=\mathbf{a} \times \mathbf{r},\) where \(\mathbf{a}\) is a nonzero constant vector and \(\mathbf{r}=\langle x, y, z\rangle .\) Use the fact that an object moving in a circular path of radius \(R\) with speed \(|\mathbf{v}|\) has an angular speed of \(\omega=|\mathbf{v}| / R\). a. Sketch a position vector a, which is the axis of rotation for the vector field, and a position vector \(\mathbf{r}\) of a point \(P\) in \(\mathbb{R}^{3}\). Let \(\theta\) be the angle between the two vectors. Show that the perpendicular distance from \(P\) to the axis of rotation is \(R=|\mathbf{r}| \sin \theta\). b. Show that the speed of a particle in the velocity field is \(|\mathbf{a} \times \mathbf{r}|\) and that the angular speed of the object is \(|\mathbf{a}|\). c. Conclude that \(\omega=\frac{1}{2}|\nabla \times \mathbf{v}|\).

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Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. $$\mathbf{F}=\langle x, 2\rangle \text { from } A(0,0) \text { to } B(2,4)$$

The area of a region \(R\) in the plane, whose boundary is the closed curve \(C,\) may be computed using line integrals with the formula $$\text { area of } R=\int_{C} x d y=-\int_{C} y d x$$ These ideas reappear later in the chapter. Let \(R=\\{(r, \theta): 0 \leq r \leq a, 0 \leq \theta \leq 2 \pi\\}\) be the disk of radius \(a\) centered at the origin and let \(C\) be the boundary of \(R\) oriented counterclockwise. Use the formula \(A=-\int_{C} y d x\) to verify that the area of the disk is \(\pi r^{2}.\)

Prove the following properties of the divergence and curl. Assume \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields and \(c\) is a real number. a. \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\) b. \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\) c. \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\) d. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)
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