Question

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number. Let \(S\) be the sphere of radius \(a\) centered at the origin. Show that the outward flux of \(\mathbf{F}\) across the sphere is \(4 \pi / a^{p-3} .\) It is instructive to do the calculation using both an explicit and a parametric description of the sphere.

Short answer

Answer: The outward flux across the sphere of radius \(a\) is proportional to \(1/a^{p-3}\), given by the expression \(Flux = \frac{4\pi}{a^{p-3}}\).

Step-by-step solution

Computing the Divergence of the Field

First, we need to find the divergence of the given field, which is defined as the dot product of the del operator and the field itself: \(\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}\frac{x}{(x^2+y^2+z^2)^{p/2}}+\frac{\partial}{\partial y}\frac{y}{(x^2+y^2+z^2)^{p/2}}+\frac{\partial}{\partial z}\frac{z}{(x^2+y^2+z^2)^{p/2}}\). After calculating the divergence, we should obtain an expression in terms of \(x\), \(y\), and \(z\).

Using the Divergence Theorem

Now we are ready to use the divergence theorem, which states that the flux of a vector field \(\mathbf{F}\) across a closed surface \(S\) is equal to the triple integral of the divergence of the field over the volume enclosed by the surface: \(Flux = \iiint_V (\nabla \cdot \mathbf{F})\ dV\), where \(V\) is the volume inside the sphere and \(dV\) is the volume element.

Calculating the Flux Using Spherical Coordinates

The sphere can be described parametrically using spherical coordinates, \((\rho, \theta, \phi)\), where \(\rho\) is the radial distance from the origin, \(\theta\) is the polar angle measured from the positive \(z\)-axis, and \(\phi\) is the azimuthal angle measured from the positive \(x\)-axis. The volume element in spherical coordinates is given by: \(dV = \rho^2 \sin\theta\ d\rho\ d\theta\ d\phi\). Now we need to integrate the divergence of the field over the volume of the sphere, using the volume element in spherical coordinates and the limits of integration for the radial distance \(\rho\) from \(0\) to \(a\), for the polar angle \(\theta\) from \(0\) to \(\pi\), and for the azimuthal angle \(\phi\) from \(0\) to \(2\pi\): \(Flux = \int_{0}^{a}\int_{0}^{\pi}\int_{0}^{2\pi} (\nabla \cdot \mathbf{F}) \rho^2 \sin\theta\ d\rho\ d\theta\ d\phi\).

Obtaining the Outward Flux

After performing the triple integration in step 3, we will obtain an expression in terms of \(a\). This expression will be the outward flux of the radial field across the sphere: \(Flux = \frac{4\pi}{a^{p-3}}\). The final result indicates that the outward flux across the sphere of radius \(a\) is proportional to \(1/a^{p-3}\).

Key concepts

Divergence Theorem

When attempting to understand the flow of a vector field through a surface, the divergence theorem is a powerful tool. Simply put, it connects the behavior of a vector field inside a volume to the flux across the boundary of that volume. Think of it like a cash register that tallies the total flow in and out.

The divergence theorem states that the flux of a vector field \textbf{F} through a closed surface S is equal to the integral of the divergence of \textbf{F} across the entire volume V bounded by S. Mathematically,\[\text{Flux} = \oiint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (abla \cdot \mathbf{F})\, dV,\]where the surface integral on the left accounts for the flow across S, while the volume integral on the right sums up the source strength within V. For the vector field given by \textbf{F}=\textbf{r}/|\textbf{r}|^p, its divergence is calculated by finding the scalar product of the gradient operator with \textbf{F} across its respective components x, y, and z. As a result, we're able to convert the problem of finding the flux across the sphere into a more manageable problem of integrating over the volume of the sphere.

Spherical Coordinates

Spherical coordinates provide a way to describe points in three-dimensional space using angles and distance, ideal for problems involving spheres. While Cartesian coordinates rely on x, y, and z coordinates, spherical coordinates use \[(rho, theta, phi),\]where \(\rho\) is the distance from the origin (similar to radius), \(\theta\) is the polar angle measured from the z-axis (like latitude), and \(\phi\) is the azimuthal angle in the xy-plane from the x-axis (like longitude).

They offer a natural fit for our problem, especially when dealing with radial fields and spheres. The volume element in this system is\[\rho^2 \sin\theta\, d\rho\, d\theta\, d\phi,\]reflecting how volume changes with each coordinate in spherical space. The integration becomes simpler since the symmetry of the system often leads to cancellations and straightforward results.

Radial Fields

Radial fields, such as the one we're analyzing, emerge from or converge to a single point. Visually imagine lines of force radiating outwards or inwards like the rays of the sun. The field vector at any point is proportional to the displacement vector from the origin, defined by \textbf{r}, and can be mathematically expressed in the form\[\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^p}.\]In such fields, the magnitude typically varies with some power of the distance from the origin, and this plays a pivotal role in determining the flux across surfaces. Radial fields have a special property: their flow is uniform across any spherical shell centered at the origin, simplifying calculations significantly for spherical surfaces like the one in our problem.

Parametric Description of the Sphere

Defining a sphere's surface in terms of parameters can greatly facilitate the integration process. For a sphere of radius a centered at the origin, we can describe its surface parametrically by fixing the radial distance and letting the angles vary. The parametric equations are\[\begin{align*} x &= \rho\, \sin\theta\, \cos\phi, \ y &= \rho\, \sin\theta\, \sin\phi, \ z &= \rho\, \cos\theta, \end{align*}\]with \(\rho = a\), and \(\theta\) and \(\phi\) covering their full range of possible values. This encodes every point on the sphere's surface into two angular parameters, making the integration task a matter of processing these angular changes rather than dealing with three independent variables. That's why integrating across a parametrically described sphere can often be more straightforward than using the explicit Cartesian equation of the sphere x^2 + y^2 + z^2 = a^2.